url
stringlengths 6
1.37k
| fetch_time
int64 1,368,856,924B
1,726,893,514B
| content_mime_type
stringclasses 3
values | warc_filename
stringlengths 108
138
| warc_record_offset
int32 46.4k
1.75B
| warc_record_length
int32 636
754k
| text
stringlengths 82
963k
| token_count
int32 42
397k
| char_count
int32 82
963k
| metadata
stringlengths 439
443
| score
float64 3.5
5.09
| int_score
int64 4
5
| crawl
stringclasses 93
values | snapshot_type
stringclasses 2
values | language
stringclasses 1
value | language_score
float64 0.05
1
|
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
http://www.pagalguy.com/discussions/official-quant-thread-for-cat-2011-part-8-25073031?page=52 | 1,469,660,333,000,000,000 | text/html | crawl-data/CC-MAIN-2016-30/segments/1469257827080.38/warc/CC-MAIN-20160723071027-00038-ip-10-185-27-174.ec2.internal.warc.gz | 632,170,326 | 18,906 | Last updated
Author
Pankaj Kamani @pkaman
Tags
Official Quant Thread for CAT 2011 [Part 8]
Order by:
Page 52 of 915
If we add the square of the digit in the tens place of the positive two digit number to the product of the digits of that number we get 52,and if we add the square of the digit in the unit's place to the same product of the digits we get 117.Find the two digit number??
i got till this step
x^2+xy=52
x(x+y)=52--(1)
y^2+xy=117
y(y+x)=117--(2)
after this hw to proceed the given answer is 30..pls explain in details with steps.
10x+y is the number
x*y+x^2 =52
x*y+y^2 =117
y^2-x^2 = 65
y+x = 13
y-x= 5
aading 2y = 18 y= 9
x= 4
so value is 49
Commenting on this post has been disabled.
Ani1308 Says
hi..a little help in remainder part...find the rem of (3^5^7^9)/41
5^7^9 = 5k
so it is 4k+1
3*3*3*3*3 /41 remainder 38
Commenting on this post has been disabled.
jain4444 Says
The sum of digits of a natural number (10^n 1) is 4707, where n is a natural number. The value of n is ?
The sum of digits of a natural number (10^n 1) is 4707, where n is a natural number. The value of n is ?
10^n-1 is obviously multiple of 9 and 11
so 4707/9 =523 (sum will be 9*n where n is no. by which 9 is repeated)
so value is 9999999999999999............523 times
P.S.:- jain sahab itna easy kaise de sakte hai
Commenting on this post has been disabled.
hi sry man...i dont have the answers bt i have one more question on the seme pattern...if u liketo practice dn..
5^25^125^3125 /11..
thanks all...
The remainder is 1.
E(11) = 10
25^anything has unit digit 5
5^(10K+5) % 11 = 5^5 % 11 = 25*25*5 = 45 = 1
Smoothest seas do not make tough sailors
Commenting on this post has been disabled.
hi sry man...i dont have the answers bt i have one more question on the seme pattern...if u liketo practice dn..
5^25^125^3125 /11..
thanks all...
Have fun in whatever you do, otherwise whats the point !!
Commenting on this post has been disabled.
E(41) = 40
We need to find 5^7^9 % 40
E(40) = 16
7^9%16 = 7
5^(16K+7) % 40 = 5 * 5^(16K+6)%8
Now, E(8 ) = 4
Therefore => (5^16K + 6) % 8 = 1
=> 5^(16K+7) % 40 = 5
3^(40K + 5) % 41 = (3*81) % 41 = -3 = 38
The remainder is 38
Ani1308 Says
but dude u cant apply euler in bold part...5 and 40 are nt co prime
I have not applied Euler in that part...I just calculated the remainder of 7^9 with 16....Then I represented 7^9 as (16K + 7), then I segregated common multiple from numerator and denominator. After that I applied Euler of 8....
What is the O.A.??
Smoothest seas do not make tough sailors
Commenting on this post has been disabled.
Ani1308 Says
hi..a little help in remainder part...find the rem of (3^5^7^9)/41
(3^5^7^9)/41
3 and 41 are coprimes.
=> E(41) = 40
5^7^9/40 => 40 = 5*8
=> 5^7^9/8 => E(8 ) = 4
=> 7^9/4 = 3^9/4 => -1 or 3 remainder
=> 5^3/8 = 125/8 => 5 remainder
5^7^9 => 5x = 8y+5 => 5 remainder
3^5/41 = 3*81/41 = 120/41 = 38 remainder
Sorry for x blunder.:-(
No problem.
• 1 Like
Commenting on this post has been disabled.
E(41) = 40
We need to find 5^7^9 % 40
E(40) = 16
7^9%16 = 7
5^(16K+7) % 40 = 5 * 5^(16K+6)%8
Now, E(8 ) = 4
Therefore => (5^16K + 6) % 8 = 1
=> 5^(16K+7) % 40 = 5
3^(40K + 5) % 41 = (3*81) % 41 = -3 = 38
The remainder is 38
but dude u cant apply euler in bold part...5 and 40 are nt co prime
Have fun in whatever you do, otherwise whats the point !!
Commenting on this post has been disabled.
Ani1308 Says
hi..a little help in remainder part...find the rem of (3^5^7^9)/41
E(41) = 40
We need to find 5^7^9 % 40
E(40) = 16
7^9%16 = 7
5^(16K+7) % 40 = 5 * 5^(16K+6)%8
Now, E(8 ) = 4
Therefore => (5^16K + 6) % 8 = 1
=> 5^(16K+7) % 40 = 5
3^(40K + 5) % 41 = (3*81) % 41 = -3 = 38
The remainder is 38
Smoothest seas do not make tough sailors
Commenting on this post has been disabled.
Ani1308 Says
hi..a little help in remainder part...find the rem of (3^5^7^9)/41
i am getting 38 ... oa?
C(41)=40
checking for 5^7^9 mod 40
C(40) = 4
7^9 mod 4= 3
so... 5^3 mod 40=5
and 3^5 mod 41 = 38
2012 :Joined NMIMS. Left after an year due to unfortunate circumstances. Target MBA 2016-18 :D
Commenting on this post has been disabled. | 1,595 | 4,158 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.84375 | 4 | CC-MAIN-2016-30 | latest | en | 0.83433 |
https://gateoverflow.in/85624/arrives-at-office-at-10am-regularly-arrives-at-11-am-every-day | 1,511,245,512,000,000,000 | text/html | crawl-data/CC-MAIN-2017-47/segments/1510934806317.75/warc/CC-MAIN-20171121055145-20171121075145-00078.warc.gz | 607,927,249 | 17,924 | +1 vote
134 views
A arrives at office at 8-10am regularly; B arrives at 9-11 am every day. Probability that one day B arrives before A? [Assume arrival time of both A and B are uniformly distributed]
A can come between 8 to 10 , which is 120 min duration . A can come on any instant from this 120 min window .
B can come between 9 to 11 , which is 120 min duration . B also can choose any instant for coming from this 120 min window
so tota possible cases of coming of A nd B =120*120
Now consider the case where B is coming early to A . here common window of 60 min which is from 9 to 10 .
we can choose two time instant from this window by 60C2 ways .
so probability is = 60c2/120*120 = 60*59
--------------= 59/480
2*120*120
selected
why are you choosing two time instants from 9-10??
and in this question,though the given answer is 1/8 which is very close to your answer but as it is given that data is uniformly distributed,does'nt we have to solve like that?
I choose two instant from 9 -10 because it is the common time window and two instant because one for A's arriving time and one for B's arriving time. and which value is less we take that for B's arriving time and other one for A's arriving . And it is satisfying the criteria .
And dont find any error in this method and in ans as well .
PS: I dont know the method of uniform distribution
thanks a lot..i got your point
+1 vote | 369 | 1,405 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.734375 | 4 | CC-MAIN-2017-47 | latest | en | 0.95542 |
https://nathanpeelphoto.com/how-many-days-in-a-year-2024/ | 1,685,245,236,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224643462.13/warc/CC-MAIN-20230528015553-20230528045553-00004.warc.gz | 488,585,420 | 10,907 | # How many days in a year 2024?
The year 2024 is a leap year, with 366 days in total.
## How many days are there in the year 2025?
The year 2025 is a common year, with 365 days in total.
## Will there be 366 days in 2024?
To be a leap year, the year number must be divisible by four – except for end-of-century years, which must be divisible by 400. This means that the year 2000 was a leap year, although 1900 was not. 2020, 2024 and 2028 are all leap years.
## Is there 29 days in February in 2024?
The next leap day is February 29, 2024. 2020 was also a leap year, so the last leap day was February 29, 2020.
## How many days does the year 2023 have?
There are a total of 260 working days in the 2023 calendar year. To compute partial month salary amounts, take monthly budgeted salary amount X # of days worked that month / total working days for that month.
## Is there ever 367 days in a year?
By adding a second leap day (Friday, February 30) Sweden reverted to the Julian calendar and the rest of the year (from Saturday, March 1) was in sync with the Julian calendar. Sweden finally made the switch from the Julian to the Gregorian calendar in 1753. This year has 367 days.
## How far is 2050 years from now?
2050 is only about 30 years from now, which means it’s close enough that we can imagine it happening, but far enough away that we can’t confidently say what it will look like.
## Will 2028 will be a leap year?
2020, 2024 and 2028 are all leap years.
## Why 1900 is not a leap year?
There is a leap year every year whose number is perfectly divisible by four – except for years which are both divisible by 100 and not divisible by 400. The second part of the rule effects century years. For example; the century years 1600 and 2000 are leap years, but the century years 1700, 1800, and 1900 are not.
## Is 2028 a leap year or not?
To be a leap year, the year number must be divisible by four – except for end-of-century years, which must be divisible by 400. This means that the year 2000 was a leap year, although 1900 was not. 2020, 2024 and 2028 are all leap years.
## Did 2000 have a leap year?
The year 2000, like the years 1996 and 2004, is a leap year – with 29 days in February; but the years 1900, 1999, 2001, 2002, 2003, 2005 and 2100 are not leap years – and have only 28 days in February.
## Is there a year Zero?
There is no year 0. Jesus was born before 4 B.C.E. The concept of a year “zero” is a modern myth (but a very popular one). In our calendar, C.E. 1 follows immediately after 1 B.C.E. with no intervening year zero.
## Do we have year zero?
Historians have never included a year zero. This means that between, for example, 1 January 500 BC and 1 January AD 500, there are 999 years: 500 years BC, and 499 years AD preceding 500. In common usage anno Domini 1 is preceded by the year 1 BC, without an intervening year zero.
## Does February 30th exist?
February 30. February 30 or 30 February is a date that does not occur on the Gregorian calendar, where the month of February contains only 28 days, or 29 days in a leap year. February 30 is usually used as a sarcastic date for referring to something that will never happen or will never be done.
## Why is there no February 30th?
The Julian Calendar added a little more than 10 days to each year, making each month either 30 or 31 days long, except for February. To account for the entire 365.25 day-long year, one day was added to February every four years, now known as a “leap year.” During most years, this left February with just 28 days.
## How hot will the earth be in 2050?
Since 1880, average global temperatures have increased by about 1 degrees Celsius (1.7° degrees Fahrenheit). Global temperature is projected to warm by about 1.5 degrees Celsius (2.7° degrees Fahrenheit) by 2050 and 2-4 degrees Celsius (3.6-7.2 degrees Fahrenheit) by 2100.
## What will Earth be in 100 years?
In 100 years, oceans will most likely rise, displacing many people, and it will continue to become warm and acidic. Natural disasters like wildfires and hurricanes will continue to be very common and water resources could be scarce. NASA is researching earth to make observations that will benefit everyone.
## What happens every 400 years for leap year?
So, 29th February will come 97 times in 400 years.
## Will the year 2300 be a leap year?
Also, if the year can be evenly divided by 100, it is NOT a leap year, unless the year is also evenly divisible by 400. This means that 2000 and 2400 are leap years, but the years 1800, 1900, 2100, 2200, 2300 and 2500 are not considered leap years.
## What would happen if you were born on February 29?
In non-leap years, that day is March 1. So for someone born on February 29, the first day they can legally drive, vote, join the Army, buy alcohol or start collecting Social Security is presumably March 1 in non-leap years.
## How many leap years in 400 years?
So, 29th February will come 97 times in 400 years.
## How do leap year babies age legally?
His legal thinking is that February 29 is the day after February 28, so a person born on February 29 is legally considered to have aged one year on the day after February 28. In non-leap years, that day is March 1.
## What happens every 400 years?
29) pops up on the calendar only on leap years, once almost every four years. It has taken millennia for our calendar, called the Gregorian calendar after the pope who modified it in 1582, to evolve to include this tweak — 97 leap years every 400 years.
## Was there a year 666?
Year 666 (DCLXVI) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. The denomination 666 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years.
## Who was born in the year 1?
Birth of Jesus, as assigned by Dionysius Exiguus in his anno Domini era according to at least one scholar. | 1,553 | 5,987 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.859375 | 4 | CC-MAIN-2023-23 | latest | en | 0.969753 |
https://www.physicsforums.com/threads/temperature-of-body-in-space.97579/ | 1,548,241,680,000,000,000 | text/html | crawl-data/CC-MAIN-2019-04/segments/1547584331733.89/warc/CC-MAIN-20190123105843-20190123131843-00072.warc.gz | 887,540,355 | 14,088 | # Temperature of body in space
1. Oct 30, 2005
### sid_galt
I want to determine the temperature of a body in space exposed to the Sun as it varies with time.
I tried this.
Let A be the area exposed to the Sun and 2A the total area of the body. Let 1380 W/m2 be the power of the sunlight falling on the body. Let $$\sigma$$ be the boltzmann constant, $$\epsilon$$ the emissivity of the body and T the temperature at a a particular instant of time, m the mass of the body and C the specific heat constant. Then
$$\displaystyle\frac{dT}{dt} = \displaystyle\frac{1380A - 2\sigma\epsilon A T^4}{mC}$$
$$\displaystyle\frac{mC}{A}\int\displaystyle\frac{1}{1380 - 2\sigma\epsilon T^4}dT = \int dt$$
I tried to integrate it on integrals.wolfram.com taking boltzmann constant as 5.6E-8 and emissivity as 0.7. The result was
$$\displaystyle\frac{0.156942mC}{A}(\arctan[0.00230862T]+arctanh[0.00230862T]) + C' = t$$
C' is here the integration constant
I dont know how to proceed further. Can anyone help please?
Thank you
Last edited: Oct 31, 2005
2. Oct 31, 2005
### sid_galt
3. Oct 31, 2005
### µ³
Are you sure it's T^4? My thermodynamics knowledge is limited but I thought the rate of heat transmition was proportional to deltaT, I might be wrong though. Otherwise, there is no analytic solution to the lower equation for T.
Edit - nevermind, it's blackbody radiation. Yeah, sorry, can't figure out what's wrong with either your physics or math. The last equation is not solvable for T.
Last edited: Oct 31, 2005
4. Oct 31, 2005
### sid_galt
But there must be someway to find temperature as a function of time for a body in space exposed to the Sun.
5. Oct 31, 2005
### Tide
Yes, there is. You can evaluate your integral numerically.
6. Oct 31, 2005
### sid_galt
But how do I integrate high temperatures into the equation, say if I want to evaluate it for a body with an initial temperature of 1000 K.
My Arctanh would give an unreal value for all temperatures higher than 434.78 K.
7. Oct 31, 2005
### µ³
if you notice that ArcTan[x]+ArcTanh[x] for x > 1 always give some value a - 1.5708i , well , make it so your integration constant takes out the imaginary part.
Edit- more specifically
for x>1
ArcTanh[x] = ArcTanh[1/x] -1/2pi*i
Edit: Another approximation:
for small x
ArcTanh[x] = ArcTan[x]
another nice identity:
(you might be able to solve for this actually)
$$tan^{-1}(x) +tan^{-1}(y) = tan^{-1}(\frac{x+y}{1-xy})$$
Last edit - I tried all the above, it doesn't work (not even with shoddy approximations).
Last edited: Oct 31, 2005
8. Oct 31, 2005
### sid_galt
I didn't notice that before. Thanks for the help. | 808 | 2,639 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.53125 | 4 | CC-MAIN-2019-04 | latest | en | 0.879689 |
https://easetolearn.com/smart-learning/web/physics/mechanics/gravitation/gravitational-potential-energy/gravitational-potential/5032 | 1,708,946,866,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947474659.73/warc/CC-MAIN-20240226094435-20240226124435-00359.warc.gz | 221,330,964 | 28,838 | # Gravitational Potential
Gravitational field:
In the case of non-contact force, the source mass and the test mass interact via a gravitational field. You can think of the gravitational force as a "command" and the gravitational field as a dialogue or speech used to give an order.
Gravitational field intensity
The strength of the gravitational field is called the gravitational field strength. This is the gravitational force acting on the unit mass.
where r cap represents the position vector of the test mass from the source mass. The intensity of the gravitational field depends only on the mass of the source and the distance of a unit of the test mass from the mass of the source.
The unit of gravitational field strength is N/kg.
The scale gives [M0L¹T–²]. The scale of gravitational field strength is identical to that of acceleration (from a gravitational perspective, we prefer to call it acceleration due to gravity).
The principle of superposition extends to the intensity of the gravitational field, for example
E = E1 + E2 + E3 + . . . . . . . + En
Thus, E1, E2, E3, . . . . . In are the gravitational field intensities at the point due to the nth particle of the system.
In the system, mass is always distributed in two different ways:
• Discrete mass distribution
• Continuous mass distribution
dE is the intensity of the gravitational field due to the main mass dm. The gravitational field formula is expressed as:
g = F/m
where, F = gravitational force and m = mass of the object.
Gravitational Field Intensity of a Point Mass:
Consider a point mass M, the gravitational force at a distance 'r' from it is given by the formula
Gravitational Intensity of the Gravitational Field created by the Ring:
Consider a ring of mass M with radius 'a'; the gravitational field at a distance x along its axis is found as:
Gravitational field intensity
Consider a small element along the circumference of the ring with mass 'dm'; the field strength resulting from that length element is given by the formula,
dE = Gdm/r²
The vertical components of the fields cancel each other due to ring symmetry, and only the horizontal components are preserved and added.
Gravitational field due to Uniform Spherical Shell:
Consider a thin uniform spherical shell in space of radius 'R' and mass 'M'. A 3D object divides space into three parts:
• Inside the spherical shell.
• On the surface of the spherical shell.
• Outside the spherical shell.
Our problem is to find out the value of gravitational field intensity in all these 3 regions
Outside the spherical shell
Consider a unit test mass at a point "P" at a distance "r" from the center of the spherical shell. Draw an imaginary spherical shell with point 'P' on its surface.
As we know, the gravitational field intensity at a point depends only on the mass of the source and the distance of the point from the mass of the source. We can say that the initial mass in the imaginary sphere is M and the separation distance is 'r'. That's what we get
E = -GM/r²
E -1/r²
On the surface of the spherical shell
Consider a unit test mass at a point "P" on the surface of the spherical shell at a distance "r" from the center of the spherical shell, where r = R. As mentioned above, the gravitational field intensity at the surface of the spherical shell is given by spherical shell
E = -GM/R²
Inside a spherical shell
If you consider a point inside a spherical shell, the entire mass of the shell is above the point. Draw an imaginary spherical shell around point 'P'; to leave this imaginary realm is zero.
We know that if the mass of the source is zero, the intensity of the gravitational field is also zero.
E = 0
Gravitational field due to a Uniform Solid Sphere:
Consider a uniform solid sphere of radius "R" and mass "M". Let's find the value of the gravitational field strength in all three regions:
• Within a solid sphere.
• On the surface of a solid sphere.
• Off a certain ball.
To find the gravitational field intensity at a point "P" at a distance "r" from the center outside the solid sphere, consider an imaginary sphere around P enclosing the entire mass "M".
E = – GM/r²
E -1/r²
On the surface of a solid sphere
To find the gravitational field intensity at point "P" on the surface of a solid sphere,
The distance from the surface point is r = R.
Then E = -GM/R² E = Constant.
Within a solid sphere
Find the gravitational effect at a point 'P' inside a uniform solid sphere at a distance 'r' from the center of the sphere. If we draw an imaginary sphere around this point, the mass of that imaginary sphere is given by 'm'.
For space (4/3) πR³, the existing mass is M; for volume (4/3) πr³ the existing mass is "m".
Since the density of a solid sphere remains the same at all times,
m = M × (r³/R³)
Then the gravitational field intensity at point "P" inside the solid sphere at a distance "r" from the center of the sphere is given by:
E = -Gm/r²
Where m is the initial mass on an imaginary sphere drawn around point "P". Substituting the value of m in the above equation, we get
E = -GMr/R³
E -r
Gravitational potential energy - formulas, derivatives, solved problems
Gravitational potential energy is the energy that an object possesses or gains from its change in position when it is in a gravitational field. Simply put, gravitational potential energy is the gravitational force or energy associated with gravity.
The most common example to help you understand the concept of gravitational potential energy is that you take two pencils, one of which is placed on a table and the other is held above the table. Now we can argue that a pencil in the air has more gravitational potential than a pencil on the table
Gravitational Potential Energy:
If a body of mass (m) is moved from infinity to a point under the gravitational influence of the initial mass (M) without accelerating it, the amount of work done in moving it to the initial field is stored in the form of potential energy. . . . This is called gravitational potential energy. It is denoted by the symbol Ug.
Explanation:
We know that the potential energy of a body at a particular location is defined as the energy stored in the body at that location. When the position of the body changes under the influence of external forces, the change in potential energy is equal to the amount of work done by the forces acting on the body. The work done by gravity does not depend on the path of change of position, so the force is a conservative force. All such forces also have a certain potential.
The effect of gravity on a body at infinity is zero; therefore, the potential energy is zero, called the reference point.
Gravitational Potential Energy Formula
The equation for gravitational potential energy is:
GPE = mgh
Where,
• m is the mass in kilograms
• g is the acceleration due to gravity (9.8 on earth)
• h is the height above the ground in metres
Derivation of Gravitational Potential Energy Equation:
Consider a source mass ‘M’ is placed at a point along the x-axis; initially, a test mass ‘m’ is at infinity. A small amount of work done in bringing it without acceleration through a very small distance (dx) is given by
dw = Fdx
Here, F is an attractive force, and the displacement is towards the negative x-axis direction, so F and dx are in the same direction. Then,
dw = (GMm/x2)dx
Integrating on both sides
If ri> rf then ΔU is negative.
Since the work done is stored as its potential energy U, the gravitational potential energy at a point which is at a distance ‘r’ from the source mass is given by;
U = -GMm/r
If a test mass moves from a point inside the gravitational field to another point inside the same gravitational field of source mass, then the change in potential energy of the test mass is given by;
ΔU = GMm (1/ri – 1/rf)
If ri> rf then ΔU is negative.
Expression for Gravitational Potential Energy at Height (h) –
Derive ΔU = mgh.
If a body is taken from the surface of the earth to a point at a height ‘h’ above the surface of the earth, then ri = R and rf= R + h, then,
ΔU = GMm [1/R – 1/(R+h)]
ΔU = GMmh/R(R + h)
When h<<R, then R + h = R and g = GM/R2.
On substituting this in the above equation, we get,
Gravitational Potential Energy ΔU = mgh
Note:
The weight of a body in the center of the earth is zero because the value of g in the center of the earth is zero.
At a point in the gravitational field where the gravitational potential energy is zero, the gravitational field is zero.
What Is Gravitational Potential?
The amount of work done in moving a unit test mass from infinity into the gravitational influence of source mass is known as gravitational potential.
Simply, it is the gravitational potential energy possessed by a unit test mass.
V = U/m
V = -GM/r
Important Points:
The gravitational potential at a point is always negative, and V is maximum at infinity.The SI unit of gravitational potential is J/s
Relation between Gravitational Field Intensity and Gravitational Potential
Integral Form:
V =
( If E is given and V has to be found using this formula)
Differential Form:
E = -dV/dr (If V is given and E has to be found using this formula)
(components along x, y, and z directions).
Gravitational Potential of a Point Mass
Consider a point mass M, the gravitational potential at a distance ‘r’ from it is given by;
V = – GM/r.
Gravitational Potential of a Spherical Shell
Consider a thin uniform spherical shell of the radius (R) and mass (M) situated in space. Now,
• Case 1: If point ‘P’ lies inside the spherical shell (r<R):
As E = 0, V is a constant.
The value of gravitational potential is given by, V = -GM/R.
Case 2: If point ‘P’ lies on the surface of the spherical shell (r=R):
On the surface of the earth, E = -GM/R2.
Using the relation
• Case 1: If point ‘P’ lies inside the spherical shell (r<R):
As E = 0, V is a constant.
The value of gravitational potential is given by, V = -GM/R.
• Case 2: If point ‘P’ lies on the surface of the spherical shell (r=R):
On the surface of the earth, E = -GM/R2.
Using the relation
over a limit of (0 to R), we get,
Gravitational Potential (V) = -GM/R.
• Case 3: If point ‘P’ lies outside the spherical shell (r>R):
Outside the spherical shell, E = -GM/r2.
Using the relation
V = -
over a limit of (0 to R), we get,
Gravitational Potential (V) = -GM/R.
Case 3: If point ‘P’ lies outside the spherical shell (r>R):
Outside the spherical shell, E = -GM/r2.
Using the relation
V = -
Gravitational Potential of a Uniform Solid Sphere:
Consider a thin, uniform solid sphere of radius (R) and mass (M) situated in space. Now,
• Case 1: If point ‘P’ lies inside the uniform solid sphere (r < R):
Inside the uniform solid sphere, E = -GMr/R3.
Using the relation
V = -
over a limit of (0 to r).
The value of gravitational potential is given by,
V = -GM [(3R2 – r2)/2R2]
• Case 2: If point ‘P’ lies on the surface of the uniform solid sphere ( r = R ):
On the surface of a uniform solid sphere, E = -GM/R2. Using the relation
V = -
over a limit of (0 to R) we get,
V = -GM/R.
• Case 3: If point ‘P’ lies outside the uniform solid sphere ( r> R):
Using the relation over a limit of (0 to r), we get, V = -GM/R.
• Case 4: Gravitational potential at the centre of the solid sphere is given by
V =(-3/2) × (GM/R).
Gravitational Self Energy
The gravitational self-energy of a body is defined as the work done by an external agent in assembling the body from the infinitesimal elements that are initially at an infinite distance apart.
Gravitational self-energy of a system of ‘n’ particles:
Let us consider n particle system in which particles interact with each other at an average distance ‘r’ due to their mutual gravitational attraction; there are n(n – 1)/2 such interactions, and the potential energy of the system is equal to the sum of the potential energy of all pairs of particles, i.e.,
Solved Problems
Example 1. Calculate the gravitational potential energy of a body of mass 10 kg and is 25 m above the ground.
Solution:
Given, Mass m = 10 Kg and Height h = 25 m
G.P.E is given as,
U = m × g × h = 10 Kg 9.8 m/s2 × 25 m = 2450 J. | 2,937 | 12,225 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4 | 4 | CC-MAIN-2024-10 | latest | en | 0.870954 |
https://www.physicsforums.com/threads/finding-inverse-matrices-using-guass-approach.358005/ | 1,723,274,273,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640790444.57/warc/CC-MAIN-20240810061945-20240810091945-00371.warc.gz | 739,593,492 | 19,269 | Finding inverse matrices using Guass approach
• thomas49th
In summary: The only thing that might have saved you some time would have been to recognize that the bottom row of I was not correct, because the bottom row of A was not all 0's.
thomas49th
Homework Statement
Use the determinate method and also the Guass elimination method to find the inverse of the following matrix. Check your results by direct multiplication
$$A =\left | \begin{array}{ccc} 2&1&0\\ 1&0&0\\ 4&1&2 \end{array}\right | =$$
Let's do Guass first
Homework Equations
Place A by I and attemp to get A into I. Everything I perform on A must be performed on I and when A is in I, the original I is the inverse?
$$A =\left | \begin{array}{ccc} 2&1&0\\ 1&0&0\\ 4&1&2 \end{array}\right |$$
$$I=\left | \begin{array}{ccc} 1&0&0\\ 0&1&0\\ 0&0&1 \end{array}\right |$$
The Attempt at a Solution
Interchange rows 1 and R
R1 <-> R2
$$A =\left | \begin{array}{ccc} 1&0&0\\ 2&1&0\\ 4&1&2 \end{array}\right | =$$
$$I =\left | \begin{array}{ccc} 0&1&0\\ 1&0&0\\ 0&0&1 \end{array}\right |$$
Now R2 - 2R1
$$A =\left | \begin{array}{ccc} 1&0&0\\ 0&1&0\\ 4&1&2 \end{array}\right | =$$
$$I =\left | \begin{array}{ccc} 0&1&0\\ 1&-2&0\\ 0&0&1 \end{array}\right |$$
R3 - 4R1
$$A =\left | \begin{array}{ccc} 1&0&0\\ 0&1&0\\ 0&1&2 \end{array}\right | =$$
$$I =\left | \begin{array}{ccc} 0&1&0\\ 1&-2&0\\ 0&-4&1 \end{array}\right |$$
R3 - R2
$$A =\left | \begin{array}{ccc} 1&0&0\\ 0&1&0\\ 0&0&2 \end{array}\right | =$$
$$I =\left | \begin{array}{ccc} 0&1&0\\ 1&-2&0\\ -1&-4&1 \end{array}\right |$$
R3 / 2
$$A =\left | \begin{array}{ccc} 1&0&0\\ 0&1&0\\ 0&0&1 \end{array}\right | =$$
$$I =\left | \begin{array}{ccc} 0&1&0\\ 1&-2&0\\ -1/2&-2&1/2 \end{array}\right |$$
but multiplying my A and new I together does not give me I? What have I done wrong?
Check the R3-R2 step. (-4)-(-2)=(-2) not (-4).
You made a mistake in the calculation right after what you show as R3 - R2.
The -4 in the last row of I should be -2.
ahh got it working by starting again and doing another combination... made a silly mistake in this jumble!
Thanks for the pointers guys. My other route still took just as long as this one. Do any of you any tricks to improve Guass Elimination efficiency or is it just experience.
Thanks
Tom
I don't know how you are doing the calculations, but your work was difficult to check with two separate matrices. The usual practice is to create an augmented matrix, starting with A in the left half and I in the right half. After you get to I in the left half, your inverse of A will be in the right half. Each augmented matrix will look something like this.
$$\left[ \begin{array}{ccccccc} 2&1&0&|&1&0&0\\ 1&0&0&|&0&1&0\\ 4&1&2&|&0&0&1 \end{array}\right ]$$
An "accounting" tip to make it easier to understand what you did is to use a notation that indicates which row changes when you add a multiple of one row to another. One way to do this is fairly verbose R1 <-- R1 - 3R2. You can abbreviate this to R1 - 3R2 if it's understood that it's always the first row listed that is added to. Obviously if you just switch two rows, it doesn't matter which one you list first, and if you replace a row by a multiple of itself, there's only one row involved, so there shouldn't be any confusion about which rows are involved.
Other than that, you row-reduced your matrices the way I would have, so I don't see anything that you could have done that would have economized your efforts.
What is the Gauss approach for finding inverse matrices?
The Gauss approach, also known as Gaussian elimination, is a method for finding the inverse of a square matrix by performing a series of row operations to transform the matrix into reduced row echelon form.
Why is finding inverse matrices important in scientific research?
Finding inverse matrices is important in scientific research because it allows for the efficient solving of systems of linear equations and can be applied to various mathematical models and simulations.
What are the steps involved in using the Gauss approach to find inverse matrices?
The steps involved in using the Gauss approach to find inverse matrices are:1. Augment the given matrix with the identity matrix2. Use row operations to transform the given matrix into reduced row echelon form3. If the resulting matrix is the identity matrix, then the original matrix is invertible and the transformed matrix is its inverse4. If the resulting matrix is not the identity matrix, then the original matrix is not invertible
What are the limitations of using the Gauss approach to find inverse matrices?
The Gauss approach may not work for matrices that are not square or for matrices that are singular (have a determinant of 0). It also involves a significant amount of computation, especially for larger matrices, which can make it time-consuming.
Are there any alternative methods for finding inverse matrices?
Yes, there are other methods for finding inverse matrices such as using the adjugate matrix, the Cramer's rule, or the LU decomposition method. Each method may have its own advantages and limitations, and the most suitable method may depend on the specific characteristics of the given matrix.
• Calculus and Beyond Homework Help
Replies
5
Views
461
• Calculus and Beyond Homework Help
Replies
6
Views
710
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
9
Views
2K
• Calculus and Beyond Homework Help
Replies
0
Views
412
• Calculus and Beyond Homework Help
Replies
2
Views
463
• Calculus and Beyond Homework Help
Replies
3
Views
4K
• Calculus and Beyond Homework Help
Replies
7
Views
3K
• Linear and Abstract Algebra
Replies
5
Views
1K
• Calculus and Beyond Homework Help
Replies
10
Views
2K | 1,695 | 5,753 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.46875 | 4 | CC-MAIN-2024-33 | latest | en | 0.74703 |
http://mathhelpforum.com/advanced-algebra/165335-please-help-clarify-galois-proof.html | 1,516,227,337,000,000,000 | text/html | crawl-data/CC-MAIN-2018-05/segments/1516084886979.9/warc/CC-MAIN-20180117212700-20180117232700-00540.warc.gz | 208,220,385 | 11,311 | I more or less copied the following notes from my professor:
Find the Galois group of $x^4-2$ over $\mathbb{Q}$.
Solution: $x^4-2$ has roots $\pm\sqrt[4]{2},\pm i\sqrt[4]{2}$. Note that since $G$ acts on the four roots, $G$ is isomorphic to a subgroup of $S_4$. The splitting field is $\mathbb{Q}(\sqrt[4]{2},i)$, so $|G|=8$. By Sylow, any two subgroups of $S_4$ of order $8$ must be isomorphic. So $G\cong D_8$.
Everything about this proof makes perfect sense to me except the part in bold: Why is $G$ a subgroup (isomorphic) of $S_4$? It cannot be simply because $G$ acts on the four roots (because for instance any group of any order acts on any set trivially). All I can tell from that is that there is a homomorphism from $G$ into $S_4$. So, how do I conclude that $G$ is a subgroup (isomorphic) of $S_4$?
Also, I'm a bit confused as to how it can be that $|G|=8$. For each $\alpha\in G$ is completely determined by $\alpha(i)$ and $\alpha(\sqrt[4]{2})$. Now, $1=\alpha(1)=\alpha(-i^2)=-\alpha(i)^2$, which seems to imply $\alpha(i)=i$. But then $\alpha(\sqrt[4]{2})$ must be one of the four roots. So $|G|\leq 4$, a contradiction. Where did I go wrong, I wonder?
Any help would be much appreciated!
2. Originally Posted by hatsoff
I more or less copied the following notes from my professor:
Everything about this proof makes perfect sense to me except the part in bold: Why is $G$ a subgroup (isomorphic) of $S_4$? It cannot be simply because $G$ acts on the four roots (because for instance any group of any order acts on any set trivially). All I can tell from that is that there is a homomorphism from $G$ into $S_4$. So, how do I conclude that $G$ is a subgroup (isomorphic) of $S_4$?
Any help would be much appreciated!
As you remark the part in bold is not completely correct. For it to be it should be that the Galois group acts transitively on
the four roots, and then it'd follow that the group is isomorphic to a subgroup of $S_4$.
There's another way to show that $G\cong D_8$: first, it's easy to see that $G$ cannot be abelian and that
its order is divisible by four and divides 8, so it is either $D_8\,\,or\,\,Q_8$. Nevertheless, this extension has a non-normal
subextension, $\mathbb{Q}(\sqrt[4]{2})/\mathbb{Q}$ , from where it follows that there is a non-normal subgroup of $G$
of index 4, and since all the subgroups $Q_8$ are normal then we're done.
Tonio
3. Originally Posted by hatsoff
I more or less copied the following notes from my professor:
Everything about this proof makes perfect sense to me except the part in bold: Why is $G$ a subgroup (isomorphic) of $S_4$? It cannot be simply because $G$ acts on the four roots (because for instance any group of any order acts on any set trivially). All I can tell from that is that there is a homomorphism from $G$ into $S_4$. So, how do I conclude that $G$ is a subgroup (isomorphic) of $S_4$?
Also, I'm a bit confused as to how it can be that $|G|=8$. For each $\alpha\in G$ is completely determined by $\alpha(i)$ and $\alpha(\sqrt[4]{2})$. Now, $1=\alpha(1)=\alpha(-i^2)=-\alpha(i)^2$, which seems to imply $\alpha(i)=i$. But then $\alpha(\sqrt[4]{2})$ must be one of the four roots. So $|G|\leq 4$, a contradiction. Where did I go wrong, I wonder?
Any help would be much appreciated!
$\alpha(i)^2=-1\Longrightarrow \alpha(i)=\pm i$ , since $i^2=(-i)^2=-1$
Tonio
4. Okay, I see now what is going on. Thanks a bunch for the help! | 1,035 | 3,431 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 58, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.671875 | 4 | CC-MAIN-2018-05 | longest | en | 0.927491 |
https://ru-facts.com/how-does-a-simple-dc-circuit-work/ | 1,719,274,452,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198865490.6/warc/CC-MAIN-20240624214047-20240625004047-00775.warc.gz | 437,798,290 | 12,171 | # How does a simple DC circuit work?
## How does a simple DC circuit work?
A simple DC circuit consists of a current source (e.g. battery) and one or more “loads” (circuit elements). Each “load” absorbs electrical energy, converting it to some other form of energy, e.g. a light bulb emits heat and light energy, an electric motor performs mechanical work and emits heat.
### What are the three common components of a DC circuit?
The resistor is the main component of the DC circuit. A simple DC circuit is shown in the figure below which contains a DC source (battery), a load lamp, a switch, connecting leads, and measuring instruments like ammeter and voltmeter.
#### Can a parallel circuit be DC?
When two or more electrical components are connected in a way that one end of each component is connected to a common point and the other end is connected to another common point, then the electrical components are said to be connected in parallel, and such an electrical DC circuit is referred as a parallel DC circuit.
What is the formula of DC circuit?
Equations
Equation Symbols
P = I Δ V P = I \Delta V P=IΔV P P P is power, I is current, and Δ V \Delta V ΔV is electric potential difference
What are the rules for drawing circuit diagrams?
Circuit diagrams Always try to make the wires straight lines. Do not be tempted to make them wiggly because the whole point is to make it easier to see what is connected to what. If you have to draw a circuit diagram: draw the circuit symbols first, then.
## What is common in a DC circuit?
“Common” is a point of reference shared throughout a circuit, typically the node in reference to which every other node is described. In the case of a single-pole double-throw (SPDT) switch, common (also called the pole) is the terminal which the “normally open” and “normally closed” terminals switch to.
### How does a DC current flow?
Direct current (DC) is an electric current that is uni-directional, so the flow of charge is always in the same direction. As opposed to alternating current, the direction and amperage of direct currents do not change.
#### What is the common wire in a DC circuit?
The black wire in DC is called NEGATIVE or COMMON. In the british system theBLK. is called neutral.
What is the effective resistance of the DC circuit?
The effective resistance is basically the resistance of that single edge. In the study of electrical circuits, one is interested in the behaviour of idealized electrical com- ponents in a circuit in terms of voltage (or potential difference) and current [13, p. 11].
How does a DC Circuit work?
A direct current (DC) electric circuit consists of a source of DC electricity—such as a battery—with a conducting wire going from one of the source terminals to a set of electric devices and then back to the other terminal, in a complete circuit. A DC circuit is necessary for DC electricity to exist.
## How to read a circuit diagram?
In order to learn how to read a circuit diagram, it is necessary to learn what the schematic symbol of a component looks like. It is also necessary to understand how the components are connected together in the circuit.
### What is a direct current (DC) Circuit?
A direct current (DC) electrical circuit consists of a source of DC electricity with a conducting wire going from one of the source terminals to a set of electrical devices and then back to the other terminal, in a complete circuit.
#### What is a circlecircuit diagram?
Circuit diagrams or schematic diagrams show electrical connections of wires or conductors by using a node as shown in the image below. A node is simply a filled circle or dot.
Begin typing your search term above and press enter to search. Press ESC to cancel. | 800 | 3,745 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4 | 4 | CC-MAIN-2024-26 | latest | en | 0.939576 |
https://cracku.in/48-the-present-age-of-bob-is-equal-to-abbys-age-8-yea-x-ibps-po-2015-mock-1 | 1,721,290,069,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514826.9/warc/CC-MAIN-20240718064851-20240718094851-00468.warc.gz | 155,898,500 | 23,067 | Question 48
# The present age of Bob is equal to Abby’s age 8 years ago. Four years hence, the respective ratio between Bob’s age and Abby’s age will be 4 : 5 at that time. What is Bob’s present age?
Solution
Let the present ages of Abby be 'a' and Bob be 'b'
b = a - 8 ---> (1)
$$\frac{b+4}{a+4} = \frac{4}{5}$$
5b + 4 = 4a
5 (a - 8) + 4 = 4a
a = 36
b = 28
Hence, the present age of Bob = 28 years | 152 | 401 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.375 | 4 | CC-MAIN-2024-30 | latest | en | 0.902052 |
http://docplayer.net/10453557-Cost-benefit-analysis-a-methodology-for-sound-decision-making-student-lesson-document.html | 1,545,018,629,000,000,000 | text/html | crawl-data/CC-MAIN-2018-51/segments/1544376828056.99/warc/CC-MAIN-20181217020710-20181217042710-00080.warc.gz | 65,761,147 | 31,134 | # Cost/Benefit Analysis. A Methodology for Sound Decision Making. Student Lesson Document
Save this PDF as:
Size: px
Start display at page:
Download "Cost/Benefit Analysis. A Methodology for Sound Decision Making. Student Lesson Document"
## Transcription
2 Prior to checkout, you remember the sign that appeared in the drugstore s window: Soda pop 12- packs - \$2.50 each. Again, you use math to calculate that this would be a savings of \$1.00 per 12-pack; this would add up to a savings of \$4.00 if you purchase four 12-packs! Ask yourself: Is it worth it to make an extra trip into the drugstore in order to save \$4.00 instead of purchasing the soda at the grocery store? Presumably, \$4.00 is a significant enough savings for exactly the same item. In fact, many people would decide to get their soda pop at the drugstore. Heck, at this price, you could buy a fifth 12-pack and still save \$1.50 over the grocery store purchase! Discussion #2: Cost/Benefit Analysis is a practice that most of us use all the time when making decisions, even when we do not realize we are utilizing the tactic. So, let s define what it is. Cost/Benefit Analysis (CBA) - a comparative analysis of estimates of the costs and the benefits of undertaking a particular action in order to determine if taking the action is worthwhile. The procedure to complete an effective Cost/Benefit Analysis is a four-step process: 1. Try to determine all of the costs associated with taking a particular action. Try to determine all of the benefits associated with taking that action. 2. Try to estimate the monetary value of each cost and total them. 3. Try to estimate the monetary value of each benefit and total them. 4. Finally, compare the total costs versus the total benefits to determine if the action is worth taking. Note: These are seemingly simple steps; however, things are rarely as simple as they seem. The more accurate the monetary estimates of the costs and benefits are, the easier it will be to make a decision. Also, the greater the difference between the totals of the costs and the benefits, the easier the decision will be. Yet, when the costs and benefits totals are closer together, the decision becomes more subjective. Also, while the monetary value of costs may be a relatively easy thing to estimate, many times there are benefits that cannot be ignored and are difficult, if not impossible, to value in monetary terms. Again, this can make a decision subjective. Explanation: In the example of purchasing soda pop 12-packs, the difference in prices between the grocery store and the drugstore was substantial enough to make it a seemingly easy decision regarding where to purchase the soda. However, what if the difference in price per 12-pack was only a nickel instead of a dollar? The savings on purchasing four 12-packs would be merely 2
3 \$0.20. Depending on the individual shopper, this might not be a significant enough savings to make the trip into the drugstore. The decision on where to purchase the soda pop might be different because the CBA results changed the shopper s mind. Would a \$0.20 savings difference, instead of a \$4.00 savings, change your mind? Discussion #3: As previously stated, assigning a monetary value to some benefits may be difficult, if not impossible, to do. Let s take a look at an example that still produces a reasonably clear-cut result. Example: A small community, located in the vicinity of your neighborhood, is host to a troublesome intersection. Most of the time, it seems to be quite busy. Not only does heavy traffic move through the intersection, but it is also frequented by many people, especially small children. Perhaps it is even a school bus drop-off site? In any case, the intersection is clogged with vehicular and pedestrian traffic. It appears to be an accident waiting to happen. Residents continually petition the village council to install a traffic control device. A traffic light is preferred to control the flow of traffic and to help keep pedestrians safe. The village council has agreed and does a Cost/Benefit Analysis on installing the traffic light at this intersection. The costs are straightforward. \$8,500 for the traffic light and equipment needed to control it. \$2,500 for installation. \$1,000 per year to maintain the light and pay for the electricity to run it. Total costs = \$11,000 + \$1,000 annually The benefits are also straightforward, but assigning a monetary value to them is more difficult. A traffic light would almost certainly prevent injury or worse to pedestrians. How does one put a monetary value on a human life? Note: The village council would be hard pressed to find anyone who would disagree with the fact that if the traffic light saved even one life, the \$11,000 cost would be well worth the expense. Placing a monetary value on a human life would be difficult, if not impossible, to do; yet, almost everyone would agree that whatever the monetary value of that life is, it is much greater than \$11,000. Therefore, the project should be completed. Do you agree? 3
4 Discussion #4: Remember, step #1 of conducting a CBA is to: 1. Try to determine all of the costs associated with taking a particular action. Try to determine all of the benefits associated with taking that action. It can be difficult to assign monetary values to benefits; yet, those benefits cannot and should not be ignored. In addition, there may be some obscure costs involved that should also be taken into account. Opportunity Costs Opportunity costs are real and are usually a part of the decision-making process; yet, they are easily missed when completing a CBA. What are Opportunity Costs? Opportunity cost - the cost associated with forgoing the benefit of making a different decision. If there are two choices, Option A and Option B, but only one option can be selected, then the opportunity cost of choosing one option is the benefit of choosing the other option. Example: It s Friday night and you are looking forward to having a nice dinner at your favorite restaurant. Then, some friends contact you to invite to a movie tonight. Sounds like a blast! However, you only have enough money for dinner OR a movie, not both. So, you must make a choice: dinner OR a movie with your friends? Option A: If you choose to go to dinner, you will miss out on going to the movies. Option B: If you choose to go to the movies, you will miss out on dinner at your favorite restaurant. Explanation: The opportunity cost of choosing dinner is the movie with friends. The opportunity cost of choosing the movie is dinner at your favorite restaurant. Opportunity costs can be thought of as the consequences of selecting one action over another. In most CBAs, these opportunity costs are real and should always be taken into account. What would you decide to do? Discussion #5: Reflect back to the CBA completed based on the traffic light scenario. What if there is an opportunity cost wrinkle? As the village council works on the CBA to decide if installing the traffic light project should move forward, the council determines that the funds would need to come from a previously 4
5 approved project. The community park s equipment is in disrepair and needs to be fixed and/or replaced. The jungle gyms, swing sets, etc. have all seen better days and \$11,000 has already been approved and allocated to repair and upgrade the park. The current state of the equipment also presents a potential safety hazard to the people who frequent the park. The park upgrade project and the traffic light project are two choices. The council, however, is aware that there is only enough money to fund one. The opportunity cost of installing the traffic light is that the park upgrade would have to wait. The opportunity cost of upgrading the park is that the traffic light would have to wait. Both costs deal with the value of a human life or injury. The decision to install the traffic light is no longer so clear. Exercise: In order to solve this dilemma, use the four-step CBA process and include the opportunity costs of each choice. Total cost of installing a traffic light Opportunity cost of installing a traffic light Total cost of park repair Opportunity cost of park repair Benefits of installing a traffic light Benefits of repairing the park Respond: How would you rationalize your decision? Discussion #6: Future Value Not all opportunity costs of a decision are subjective and/or difficult to value in monetary terms. Some can have hard and fast monetary values that can be easily calculated if we remember to include them. Example - Mr. and Mrs. Jones have been planning necessary renovations/upgrades to their home for quite a while and have been saving to make them happen. They did their homework and determined the cost for the materials and the manual labor to be \$2,700. They have also determined that the upgrades would have a useful life of 10 years. The benefits they perceive are entirely subjective, but the pride in their home and the increased comfort they will enjoy for the 10-year period far outweigh the \$2,700 price tag. Fair enough. There is an opportunity cost the Jones family is missing. This should be calculated and taken into account. This is what we call the future value of the \$2,700 if they decided not to renovate. The \$2,700 they have set aside for the upgrade could alternatively be placed in a safe investment over 5
6 the same 10-year period. When the 10 years is up, the original \$2,700 investment will have grown. By choosing the renovation, Mr. and Mrs. Jones will be giving up the \$2,700 and the value of their \$2,700 after 10 years of growth. So, the true cost of their renovations is not only the \$2,700 for materials and labor, but also the opportunity cost of the investment growth of the \$2,700. Their upgrade will cost them substantially more than \$2,700 if they include this cost. But, the question remains, how much more? To accurately determine the future value of their \$2,700, we turn to a formula called the Future Value Formula, also known as the Compound Interest Formula. It looks like this: F = P*(1 + r) n Where: F = the future value of an investment P = the present value r = the interest rate n = the time frame (in years) Example: What is the future value of \$1,000 invested at 6% for 8 years? F= Future Value P= \$1,000 r = 6% or.06 n = 8 F = P*(1 + r) n F = 1000*(1 +.06) 8 F = 1000*( ) F = 1, Hands-on Activity #1 Now you try. Calculate the Future Value of the Jones family s \$2,700 if Mr. and Mrs. Jones decide not to renovate their home and instead invest in a fixed, 10-year safe investment paying 2.5%. If Mr. and Mrs. Jones renovate, they will pay \$2,700 and enjoy the upgrade for 10 years but will have \$0.00 in 10 years. Question: If they choose to invest, instead of renovate, how much will their investment be worth 10 years from now? This amount should be added to the \$2,700 as an opportunity cost of finishing the renovations. 6
7 Hands-on Activity #2 CBA Activity Tom and Patty Conley purchased their dream home 10 years ago. They both work, so with two incomes and excellent credit, they qualified for and received a \$200,000 mortgage to purchase their dream home. Now, 10 years later, economic times have changed. The U.S. economy has suffered through tough economic challenges, including a severe housing market collapse. As a result, interest rates on today s mortgages are lower than when the Conleys purchased their house. The Conleys are considering refinancing their mortgage to take advantage of these low interest rates. They believe refinancing will give them a lower monthly payment, which will give them more of their money to use for other things. The refinancing will save lots of interest over the long term. CBA can be used to help the Conleys analyze their opportunity. First, it would be a good idea to define a couple of terms. Refinancing a mortgage means to pay off an existing mortgage with a new one, usually because the terms of the new mortgage are more favorable than the old one. Monthly Mortgage Payment is the amount of money paid each month to pay back a mortgage loan. The pure monthly mortgage payment consists of principal (the money borrowed) and interest (the price paid for borrowing the money), or P & I. Some monthly mortgage payments consist of P & I, plus another amount called escrow, which is an estimate of the property taxes and property insurance that will be needed to be paid on the home. Some lenders require the annual insurance and property tax estimates to be totaled and divided by 12 (monthly amount), then added to the monthly mortgage payment as a payment into an escrow account that the lender will use to pay the insurance and taxes when they become due. In the case of Mr. and Mrs. Conley, the lender did not require a monthly escrow payment and the Conleys are responsible for paying their insurance and taxes when they are due. Therefore, the Conleys monthly mortgage payment consists only of P & I. Amortization Table is a table showing the amount of interest and principal being applied to a mortgage loan each month. There are 360 monthly payments due in a 30-year mortgage (12 x 30 = 360 months). Each payment is the same amount but how the funds are used changes each month. For each payment, the lender calculates the amount of interest the borrower owes on the money he/she has borrowed. The lender subtracts that interest amount from the mortgage payment. Whatever 7
8 money is left over, the lender applies to the outstanding balance, which reduces the amount the borrower now owes. Next month then, when the lender calculates the amount of interest due, it will be a bit less because the borrower owes a bit less. So, the lender takes a smaller amount from the mortgage payment to pay for interest and applies a larger amount to pay back the amount owed. Each and every month then, the borrower owes a bit less than he/she did the previous month; therefore, more of the mortgage payment is being used to pay back the loan, and less of the payment is being used to pay interest. The exact breakdown of the amounts of interest and principal being used out of each payment is presented in a table called an Amortization Table. Discussion #7: To complete the calculations necessary for this activity, we will need the help of a mortgage loan calculator. There are plenty of fine calculators available on the web (any one of which would probably be fine to use). One calculator to use is found on BankRate.com. Exercise: Let s begin by identifying a couple of important numbers. The Conleys original mortgage loan was for \$200,000. It is a 30-year mortgage at 8.625% interest, and their monthly payment consists of only principal and interest. Using the mortgage loan calculator, answer these questions: What is the Conleys current monthly mortgage payment? If the Conleys keep their existing mortgage, how much total interest will they end up paying after 30 years? It has been exactly 10 years since the mortgage was taken out (120 months). How much do the Conleys still owe on their loan? How much interest have they paid so far over the 10 years? If they keep their existing mortgage, how much interest will they pay from now until the end? Cost/Benefit Analysis(CBA) is a comparative analysis of estimates of the costs and the benefits of undertaking a particular action in order to determine if taking the action is worthwhile. 8
9 The procedure to do an effective Cost/Benefit analysis is a four-step process: 1. Try to determine all of the costs associated with taking a particular action. Try to determine all of the benefits associated with taking that action. 2. Try to estimate the monetary value of each cost and total them. 3. Try to estimate the monetary value of each benefit and total them. 4. Finally, compare the total costs versus the total benefits to determine if the action is worth taking. Step 1: 1. Try to determine all of the costs associated with taking a particular action. Try to determine all of the benefits associated with taking that action. Refinancing a mortgage means to pay off an existing mortgage with a new one, usually because the terms of the new mortgage are more favorable than the old one. There are costs associated with refinancing a mortgage. The Conleys are considering refinancing their mortgage to take advantage of lower interest rates. They believe refinancing will give them a lower monthly payment, which will give them more of their money to use for other things. Refinancing will save lots of interest over the long term. Step 2: 2. Try to estimate the monetary value of each cost and total them. Refinancing costs vary from state to state and lender to lender. For our purposes, we will list costs typically found in any refinancing. Also, some of the costs are based on the amount borrowed and some are fixed amounts. Cost Explanation Amount Application fee Initial cost of determining if a borrower qualifies for a loan. Can \$125 include fee for checking credit report. Loan origination fee 1.5% of loan amount Lender s fee for documenting and preparing the loan. Points 2 points 1 point = 1% of loan amount Bank s profit for establishing the loan Appraisal fee A current appraisal of the home s \$350 value done by a licensed appraiser. 9
10 Inspection fee Home inspection by a licensed inspector to determine the home s condition \$200 In the table that follows, we will fill in fixed costs and ask you to calculate those based on the loan amount. The new loan amount is the amount the Conleys still owe on their existing mortgage. Costs of a refinance: Bank attorney fee Conleys attorney fee Title work Survey TOTAL COSTS Bank s attorney reviews and signs off on the loan agreement and may conduct the closing. Borrower s attorney reviews the documents to protect the borrower. Searching the property s history to determine if the ownership is clear. Also, insures against future claims against ownership. An updated survey of the property is required. Total cost of the refinance \$500 \$500 \$800 \$125 Step 3: 3. Try to estimate the monetary value of each benefit and total them. Look at the new loan. How much do the Conleys owe on their existing loan? (You answered this question earlier. This will be the new loan amount.) The new loan will be a 30-year fixed rate mortgage at 4.75% interest. What will the Conleys new monthly mortgage payment be? If the Conleys keep their new mortgage, how much total interest will they end up paying after 30 years? 10
11 Step 4: 4. Finally, compare the total costs versus the total benefits to determine if the action is worth taking. We have information about the existing 30-year mortgage loan. We have information about the cost to refinance this loan into a new loan. We have information about the new loan. Now, analyze the information and determine what the benefit of refinancing the loan will be for Mr. and Mrs. Conley. One benefit the Conleys anticipate from refinancing is a lower monthly mortgage payment. Will the refinanced mortgage free up money in their monthly budget for other things? What is the Conleys current monthly mortgage payment? What will be the Conleys new monthly mortgage payment? Subtract (if positive, this will be the Conleys monthly savings): Were Mr. and Mrs. Conley right? Is the monthly savings a substantial benefit? Break-Even Period The Conleys will be saving money on their mortgage each and every month. However, it cost them a substantial sum to refinance and get a new mortgage. How many months of savings will it take before the Conleys recover the cost of refinancing the loan? Total Costs Monthly Savings equals months Another benefit the Conleys expect is that refinancing will save them a substantial amount of interest over the long term. Will it? If the Conleys keep their existing mortgage, how much interest do they still have left to pay? How much interest will they pay with the new mortgage? - Subtract 11
12 The difference here is how much interest they will save by refinancing. Do you consider it substantial? Answer: It appears the Conleys expectations were correct. They will save a substantial amount of money each and every month on their mortgage payment. They will reimburse themselves for the cost of the refinance in a reasonable amount of time and will ultimately save a substantial amount of interest over the long term. Note: You may have noticed that there is an opportunity cost missing from the analysis: the opportunity cost of investing the money they paid for the refinance, instead of spending it on refinancing. Opportunity costs are the consequences of choosing one course of action over another. The Conleys needed to reduce monthly spending. Saving big on interest over the long term was not a choice, but a necessity. This calculation was not included in the calculation of costs. But, they really should know what it would be, so Please calculate the Future Value of investing the Total Costs figure from the table above in a safe investment yielding 2.5% interest for 30 years. Conclusion: Cost/Benefit Analysis is a practice most of us use all the time when making decisions, even if we do not realize we are utilizing the tactic. Cost/Benefit Analysis (CBA) is a comparative analysis of estimates of the costs and the benefits of undertaking a particular action in order to determine if taking the action is worthwhile. Just because you can do something, does not mean you should do it. Not every idea is a good one. Look before you leap. This is good advice, wouldn t you say? Following a particular course of action can sometimes simply feel right. Subjective benefits can be difficult to quantify; therefore, these make it difficult to prove that feeling like the right thing to do is the right thing to do. 12
13 Extension Activities: Can you think of a situation where you just know what the right course of action is, but cannot prove that it is merely by reducing the action to dollars and cents? Can you think of a situation where what seems to be the right thing to do turns out to be the wrong thing to do? Thank you for your attention. 13
### Cost/Benefit Analysis. Lesson Plan
Cost/Benefit Analysis Lesson Plan Objective Introduce methodologies for determining if a course of action is worth taking; determine the value of items and actions and define opinions regarding subjective
### OBJECTIVES. The BIG Idea MONEY MATTERS. How much will it cost to buy, operate, and insure a car? Paying for a Car
Paying for a Car 4 MONEY MATTERS The BIG Idea How much will it cost to buy, operate, and insure a car? AGENDA Approx. 45 minutes I. Warm Up (5 minutes) II. What Can You Spend? (15 minutes) III. Getting
### Standard 10: The student will explain and compare the responsibilities of renting versus buying a home.
TEACHER GUIDE 10.3 RENTING VERSUS BUYING PAGE 1 Standard 10: The student will explain and compare the responsibilities of renting versus buying a home. Buying a House Priority Academic Student Skills Personal
### Standard 10: The student will explain and compare the responsibilities of renting versus buying a home.
STUDENT MODULE 10.3 RENTING VERSUS BUYING PAGE 1 Standard 10: The student will explain and compare the responsibilities of renting versus buying a home. Buying a House Hank and Peggy decide it is finally
### Chapter F: Finance. Section F.1-F.4
Chapter F: Finance Section F.1-F.4 F.1 Simple Interest Suppose a sum of money P, called the principal or present value, is invested for t years at an annual simple interest rate of r, where r is given
### Math 120 Basic finance percent problems from prior courses (amount = % X base)
Math 120 Basic finance percent problems from prior courses (amount = % X base) 1) Given a sales tax rate of 8%, a) find the tax on an item priced at \$250, b) find the total amount due (which includes both
### Dimes to Riches Money Management for Teens in Grades 7-12
Dimes to Riches Money Management for Teens in Grades 7-12 s e e r t n o row g t n s e o yd e n o M t a! h k t n a w b o n a k t u n By now yo Your parents are & Life is getting busy and you need cash for
### First Timer s Guide PREParing First Time Homebuyers
First Timer s Guide PREParing First Time Homebuyers SO MANY QUESTIONS Maybe you live in the best apartment with a great landlord and don t want to change a thing. Or maybe you ve looked at the rent going
### How to Avoid The Five Biggest First-time Homebuyer Mistakes. Mistake #1
How to Avoid The Five Biggest First-time Homebuyer Mistakes Mistake #1 Failure To Examine And Repair Any Credit Problems Prior To Applying For Your Loan Many potential new home buyers have no idea what
### Outstanding mortgage balance
Using Home Equity There are numerous benefits to owning your own home. Not only does it provide a place to live, where you can decorate as you want, but it also provides a source of wealth. Over time,
### Refinancing. Refinancing WISCONSIN HOMEOWNERSHIP PRESERVATION EDUCATION. Section Overview
People refinance their homes to take advantage of lower interest rates or to decrease their monthly payment. Sometimes it is done to create extra money for purchases (like a car) or for debt repayment.
BETTER YOUR CREDIT PROFILE Introduction What there is to your ITC that makes it so important to you and to everyone that needs to give you money. Your credit record shows the way you have been paying your
### The USA Mortgage Smart-Loan Guide
The USA Mortgage Smart-Loan Guide Page 1 of 8 The USA Mortgage Smart-Loan Guide Hello! Welcome to the USA Mortgage Smart-Loan Guide. Please keep in mind that this simple guide is not intended to be an
### A Home Equity Loan: Is it right for you?
A Home Equity Loan: Is it right for you? 2004 CBR. This document was produced by the CBR Foundation for Financial Education, Inc., in Racine, Wisconsin. CBR is a non-profit organization dedicated to providing
READY TO REFI? A SIMPLE GUIDE TO REFINANCING YOUR HOME INTRODUCTION You ve had your home loan for a few years, but the rate you are currently paying is much higher than the near-historic low rates you
### Designate an A + B, two mortgage lenders, Goldman Sachs and an Authoritative Voice; make signs that they wear.
Economic Crisis Workshop Buying a House Skit Make copies for Mortgage Lender 1, A, B (the couple that buy the house), Authoritative voice; second mortgage lender can take the script from first mortgage
### Rent to Own Housing. What is a Rent to Own Housing contract?
Rent to Own Housing A rent to own housing agreement is not a good idea unless you are in very good financial condition and know that you will be so for the foreseeable future. There are many disadvantages
### MONEY MANAGEMENT WORKBOOK
MICHIGAN ENERGY ASSISTANCE PROGRAM MONEY MANAGEMENT WORKBOOK Life is a challenge. As the saying goes, just when you're about to make ends meet, someone moves the ends. While it can be a struggle to pay
### The 10 Most Costly Mistakes You Can Make When Selling Your Home
The 10 Most Costly Mistakes You Can Make When Selling Your Home When you are getting ready to put your property on the market, there is a myriad of things to think about, to prepare for and to organize.
### Money and the Single Parent. Apprisen. 800.355.2227 www.apprisen.com
Money and the Single Parent Apprisen 800.355.2227 www.apprisen.com Money and the Single Parent Being a single parent is not easy. As a matter of fact, it is really hard. Single parents are stretched in
Questions to Ask Yourself When Buying a New Home When I start visiting homes, what should I be looking for the first time through? The house you ultimately choose to call home will play a major role in
### The Guide to Single Family Home Mortgage Insurance
U.S. Department of Housing and Urban Development Office of Housing Office of Single Family Housing The Guide to Single Family Home Mortgage Insurance www.hud.gov espanol.hud.gov Becoming a Homeowner Many
### Lesson 2: Savings and Financial Institution Knowledge Making the Most of Your Money
Lesson 2: Savings and Financial Institution Knowledge Making the Most of Your Money All the materials and information included in this presentation is provided for educational and illustrative purposes
### 27PercentWeekly. By Ryan Jones. Part II in the Series Start Small and Retire Early Trading Weekly Options
By Ryan Jones Part II in the Series Start Small and Retire Early Trading Weekly Options Important My 27% Option Strategy is one of the best option trading opportunities you will come across. When you see
### Example. L.N. Stout () Problems on annuities 1 / 14
Example A credit card charges an annual rate of 14% compounded monthly. This month s bill is \$6000. The minimum payment is \$5. Suppose I keep paying \$5 each month. How long will it take to pay off the
### K.4 Using Credit Wisely After Bankruptcy
Appx. K.4 K.4 Using Credit Wisely After Bankruptcy Beware of Credit Offers Aimed at Recent Bankruptcy Filers Disguised Reaffirmation Agreement Carefully read any credit card or other credit offer from
### Gr. 6-12 English Language arts Standards, Literacy in History/Social Studies and Technical Studies
Credit Lesson Description Concepts In this lesson, students, through a series of interactive and group activities, will explore the concept of credit and the impact of liabilities on an individual s net
### Borrowing on Home Equity
ABCs of Mortgages Series Borrowing on Home Equity Smart mortgage decisions start here Table of Contents Overview 1 What are the different options? 2 1. Refinancing 3 2. Borrowing amounts you prepaid 4
### Budget Main Window (Single Bank Account) Budget Main Window (Multiple Bank Accounts)
Budget Main Window (Single Bank Account) Budget Main Window (Multiple Bank Accounts) Page 1 of 136 Using Budget Help Budget has extensive help features. To get help use Budget's Help > Budget Help menu
### Managing High Levels of Debt
STUDENT MODULE 13.1 BANKRUPTCY PAGE 1 Standard 13: The student will evaluate the consequences of bankruptcy. Managing High Levels of Debt Montana and Carolina could not wait to graduate from high school
### SPECIAL REPORT: How to Use the 203K Loan Program to Profit From the Foreclosure Crisis
SPECIAL REPORT: How to Use the 203K Loan Program to Profit From the Foreclosure Crisis Presented By: RGR Construction, Ltd. 1-877-289-7231 Welcome, and thank you for ordering this report from RGR Construction
### Chapter 4: Managing Your Money Lecture notes Math 1030 Section D
Section D.1: Loan Basics Definition of loan principal For any loan, the principal is the amount of money owed at any particular time. Interest is charged on the loan principal. To pay off a loan, you must
### A GUIDE TO HOME EQUITY LINES OF CREDIT. Call or visit one of our offices today to see what products in this guide we have to offer you!
A GUIDE TO HOME EQUITY LINES OF CREDIT Call or visit one of our offices today to see what products in this guide we have to offer you! TABLE OF CONTENTS Introduction What is a home equity line of credit
### Mortgages and other money matters
Mortgages and other money matters 2 Buying a Home What about the money? The first thing most people think about when they want to buy a home is how much will it cost and where will I find the money for
### Omaha Mortgages. Omaha Mortgage Rates
Omaha Home Loan My name is Dennis Clark. I want to thank you for taking the time to visit my website, where I guarantee you'll find plenty of helpful information to assist you in obtaining a home loan
### Money Math for Teens. Dividend-Paying Stocks
Money Math for Teens Dividend-Paying Stocks This Money Math for Teens lesson is part of a series created by Generation Money, a multimedia financial literacy initiative of the FINRA Investor Education
### Introduction 4. What is Refinancing? 5. Changing Home Loans 5 Changing Needs 6 Identifying Better Opportunities 6 Additional Home Loan Features 6
Contents Introduction 4 What is Refinancing? 5 Changing Home Loans 5 Changing Needs 6 Identifying Better Opportunities 6 Additional Home Loan Features 6 What are the Advantages of Refinancing? 7 1. Consolidating
### How To Sell Your House Fast and Easily in 2013
Special Seller Report: How To Sell Your House Fast and Easily in 2013 Hello and thank you for downloading this special report. Often times I have people asking us, How do you find buyers for houses so
### Options for Obtaining Cash from Your Home s Equity Last update: May 16, 2014
Summary Options for Obtaining Cash from Your Home s Equity Last update: May 16, 2014 For most people, it is better not to tap your home equity unless you need to, but there are circumstances when it may
### USING CREDIT WISELY AFTER BANKRUPTCY
USING CREDIT WISELY AFTER BANKRUPTCY Copyright April 2011, Legal Aid Society of Hawai`i All rights reserved. These materials may be reproduced. However, under no circumstances are persons receiving these
HOME BUYING MADE EASY Live the dream of owning your own home. sm Getting started For most of us, buying our first home is a dream come true. It s also a lengthy process where potential and sometimes very
### TALKING POINTS COLLEGE STUDENTS PRESENTED BY JEAN CHATZKY AND PASS FROM AMERICAN EXPRESS SM
INTRODUCTION Give your kids a heads-up that tonight you re going to make some time to talk about money. It won t take all night, but they should clear their schedule. If your kid is away at college, and
### usbank.com PARTICIPANT S GUIDE The Budget Zone Saving for Spring Break Without Breaking The Bank
usbank.com PARTICIPANT S GUIDE The Budget Zone Saving for Spring Break Without Breaking The Bank Message to the Participant As a community service, U.S. Bank has developed a series of learning modules
### Reverse Mortgage Guide
Reverse Mortgage Guide Reverse Mortgage Answers, LLC Find out why 25 years of experience makes a difference (800) 420-5515 www.rmanswers.com What is a Reverse Mortgage and How does it Work? A Reverse Mortgage
### Personal Credit. Conserving your wealth. City of Seattle September 12, 2007
Personal Credit Conserving your wealth City of Seattle September 12, 2007 Workshop Handout Alice Jenkins Bellevue Community College [email protected] www.bellevuecollege.edu/financialeducation
### Equity release using your home to get a cash sum
Equity release using your home to get a cash sum for independent information About us We are an independent watchdog set up by the Government to: regulate firms that provide financial services; and help
### Finance 197. Simple One-time Interest
Finance 197 Finance We have to work with money every day. While balancing your checkbook or calculating your monthly expenditures on espresso requires only arithmetic, when we start saving, planning for
### 800.550.9435 www.pwsb.com
800.550.9435 www.pwsb.com Thinking of buying your first home? Bet you re pretty nervous, huh? Maybe even cold sweats nervous. How in the world are you going to afford it? Where do you start? All these
### Selling To A Real Estate Investor
Selling To A Real Estate Investor Learn How to Sell l Your House Fast and in AS IS condition We Buy Houses Fast & in as is Condition Office: 617-860-2782 3 Options to Sell Your Home for Maximum Speed and
### An Easy Guide to Buying a Home
An Easy Guide to Buying a Home Dear Future Homeowner, You have just taken the first step on the road to realizing the American dream. Choosing your home loan is one of the most important decisions you
### WELCOME COURSE OUTLINE
WELCOME COURSE OUTLINE Dear Home Buyer: Thank you for giving us the opportunity to help guide you through your home lending process. It is typically the largest financial transaction you will make and
### CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY
CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY 1. The simple interest per year is: \$5,000.08 = \$400 So after 10 years you will have: \$400 10 = \$4,000 in interest. The total balance will be
### Annuities and Sinking Funds
Annuities and Sinking Funds Sinking Fund A sinking fund is an account earning compound interest into which you make periodic deposits. Suppose that the account has an annual interest rate of compounded
### CHAPTER 1. Compound Interest
CHAPTER 1 Compound Interest 1. Compound Interest The simplest example of interest is a loan agreement two children might make: I will lend you a dollar, but every day you keep it, you owe me one more penny.
### Omaha Mortgage Brokers
Omaha Mortgage Brokers My name is Dennis Clark. I want to thank you for taking the time to visit my website, where I guarantee you'll find plenty of helpful information to assist you in obtaining a home
### Buying a Car. A Car Means Convenience. Which Car is Right for You?
Buying a Car A Car Means Convenience It s Wednesday morning and you are sleeping soundly, dreaming about that Hawaiian vacation you d like to take. Suddenly you hear, instead of ocean waves, a loud buzz.
### Remember the Interest
STUDENT MODULE 7.1 BORROWING MONEY PAGE 1 Standard 7: The student will identify the procedures and analyze the responsibilities of borrowing money. Remember the Interest Mom, it is not fair. If Bill can
### Don t Borrow Trouble! Mortgages, Home Equity Loans and Refinancing
Don t Borrow Trouble! Mortgages, Home Equity Loans and Refinancing Find more easy-to-read legal information at www.ptla.org Five Basic Tips Are you looking for a home equity loan? Or are you trying to
Table of Contents Checking In... 3 Pre-Test... 4 What Is Credit?... 6 Collateral... 6 Types of Loans... 7 Activity 1: Which Loan Is Best?... 8 The Cost of Credit... 9 Activity 2: Borrowing Money Responsibly...
### Before you agree to buy a house, make sure
SPECIAL ALERT Office of the Attorney General, Consumer Protection Division Home Buyers: Beware of Flipping Scams Bought by "flipper" \$24,000 February Before you agree to buy a house, make sure you re not
### A Guide to Student Loan Consolidation
A Guide to Student Loan Consolidation Introduction If you are reading this, you probably have student loan repayment fast approaching. Fortunately, you have options for repaying your student loans. One
Understanding Your Credit Score Whether or not you re aware of it, your credit score (or lack thereof) plays a significant role in your financial well being. Build a great score and you ll enjoy the rewards
### The Knowledge Resource. First-Time Home Buyers FOR. Your Agent Is the Best Guide Save Time, Money, and Frustration
K N O W L E D G E I S P O W E R The Knowledge Resource FOR First-Time Home Buyers Your Agent Is the Best Guide Save Time, Money, and Frustration The Keys to Homeownership Unlock the American Dream Is Buying
### UNIT 6 2 The Mortgage Amortization Schedule
UNIT 6 2 The Mortgage Amortization Schedule A home mortgage is a contract that requires the homeowner to make a fixed number of monthly payments over the life of the mortgage. The duration, or length of
### UNIT AUTHOR: Elizabeth Hume, Colonial Heights High School, Colonial Heights City Schools
Money & Finance I. UNIT OVERVIEW & PURPOSE: The purpose of this unit is for students to learn how savings accounts, annuities, loans, and credit cards work. All students need a basic understanding of how
### Retire Rich. Written By Joshua Sharp Self Directed IRA and 401k Expert. Larry Goins. Real Estate Author, Trainer and Investor
Retire Rich GetOutOfTheVolatileFinancialMarkets AndSuperchargeYourIRAorRetirement AccountUsingDiscountedRealEstate! RentorFlip UsingyourIRAor401ktothefullest.KnowtheRightTimetoUse YourRetirementAccountandGettheMostFromYourRealEstateInvestments.
### Part 7. Capital Budgeting
Part 7. Capital Budgeting What is Capital Budgeting? Nancy Garcia and Digital Solutions Digital Solutions, a software development house, is considering a number of new projects, including a joint venture
### Basics of Budgeting. Ten Steps To Create A Budget. Reviewing:
Basics of Budgeting Reviewing: Ten Steps to creating a budget How to find where your money is going Tips to stay on course Budget format included Ten Steps To Create A Budget Basics of Budgeting A budget
### + What you ll learn from this? Improve your cash flow & learn about Asset- Based Loans
Improve your cash flow & learn about Asset- Based Loans + What you ll learn from this? For businesses in need of funding, one option is the assetbased loan. In this guide, you will learn about asset-based
### Check off these skills when you feel that you have mastered them.
Chapter Objectives Check off these skills when you feel that you have mastered them. Know the basic loan terms principal and interest. Be able to solve the simple interest formula to find the amount of
### During the course of our research on NBA basketball, we found out a couple of interesting principles.
After mining all the available NBA data for the last 15 years, we found the keys to a successful basketball betting system: If you follow this system exactly, you can expect to hit 90% of your NBA bets.
### What Will Keep Me From Getting A Mortgage Right Now?
What Will Keep Me From Getting A Mortgage Right Now? Why can t I get a mortgage? Why was my mortgage application denied? What s keeping me from getting a mortgage right now? Page 2 of 11 What Will Keep
### Financial Ratio Cheatsheet MyAccountingCourse.com PDF
Financial Ratio Cheatsheet MyAccountingCourse.com PDF Table of contents Liquidity Ratios Solvency Ratios Efficiency Ratios Profitability Ratios Market Prospect Ratios Coverage Ratios CPA Exam Ratios to
HOME BUYING MADE EASY Live the dream of owning your own home. HOME buying Made Easy PNC, PNC AgentView and Home Insight are registered service marks of The PNC Financial Services Group, Inc. ( PNC ). PNC
### Getting a Difficult Business Loan Is Easy
Getting a Difficult Business Loan Is Easy Learn How Easy IT Is To Get a Business Loan Even IF You Have Bad Credit And Have Been Turned Down Before Every Winner Has A Coach! John P Fazzio, Your Coach www.winning-advantage.com
### Synthesis of Financial Planning
P 7A R T Synthesis of Financial Planning 1. Tools for Financial Planning Budgeting (Chapter 2) Planned Savings (Chapter 3) Tax Planning (Chapter 4) 2. Managing Your Liquidity Bank Services (Chapter 5)
### CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY
CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY Answers to Concepts Review and Critical Thinking Questions 1. The four parts are the present value (PV), the future value (FV), the discount
### HOME BUYING MADE EASY. Know what you need to get it right.
HOME BUYING MADE EASY Know what you need to get it right. HOME BUYING MADE EASY PNC, PNC AgentView and Home Insight are registered service marks of The PNC Financial Services Group, Inc. ( PNC ). PNC has
### Mortgage Broking Services
Mortgage Broking Services Page 2 2014 Buying property is not just about the dollars and cents, it is often a very emotional and exciting experience. There is no reason why your mortgage experience should
### Chapter 6 Contents. Principles Used in Chapter 6 Principle 1: Money Has a Time Value.
Chapter 6 The Time Value of Money: Annuities and Other Topics Chapter 6 Contents Learning Objectives 1. Distinguish between an ordinary annuity and an annuity due, and calculate present and future values
### Step 1: Decide to Buy
Step 1: Decide to Buy Do you really want to pay someone else's mortgage? If you re renting and have a stable job with some savings, and a credit score in the high 600 range, you can likely qualify for
### Chapter 6. Learning Objectives Principles Used in This Chapter 1. Annuities 2. Perpetuities 3. Complex Cash Flow Streams
Chapter 6 Learning Objectives Principles Used in This Chapter 1. Annuities 2. Perpetuities 3. Complex Cash Flow Streams 1. Distinguish between an ordinary annuity and an annuity due, and calculate present
### Romans 13:8 is an often misunderstood verse because it says, Owe nothing to anyone.
Debt and Credit Introduction We will be discussing the subject of debt from a biblical perspective. But before we begin looking at biblical principles concerning economics and finances, we need to put
### Appendix C: HUD-1 Settlement Statement
Appendix C: HUD-1 Settlement Statement HUD-1 Settlement Statement The Settlement Statement, or HUD-1 Form, details the exact breakdown of all the money paid or received by both the buyer and the seller.
### lesson nine cars and loans overheads
lesson nine cars and loans overheads the cost of owning and operating a motor vehicle ownership (fixed) costs: Depreciation (based on purchase price) Interest on loan (if buying on credit) Insurance Registration
### The. Path. Refinancing. www.totalmortgage.com October. totalmortgage.com 877-868-2503
The Path Refinancing totalmortgage.com 877-868-2503 www.totalmortgage.com October 1 2012 The Path Refinancing Over time, many things change and need adjustment, and the reality is your home financing is
### Welcome to Marketeach! Here is the sample lesson plan that you requested be sent to you. The next page is a summary of the entire set of lesson plans.
Dear Educator, Welcome to! Here is the sample lesson plan that you requested be sent to you. The next page is a summary of the entire set of lesson plans. We at created this program to allow teachers to
### GUIDE TO DEALING WITH DEBT
GUIDE TO DEALING WITH DEBT CONTENTS Page 4 Where do I start? Page 5 Do I have to pay? What can I afford to pay? Page 7 Who to pay first? Page 10 What can I afford to pay on my Credit Debts? Page 12 What
### Financial Literacy. Credit is widely misunderstood and, perhaps even more to the point, widely misused. - Author Unknown. Section V.
Credit is widely misunderstood and, perhaps even more to the point, widely misused. - Author Unknown Section V Credit 113 Credit: Friend or Foe? Families today do not spend much time together talking about
### Jane Londerville on Mortgage Financing
June 2012 Jane Londerville on Mortgage Financing In this instalment of Straight Talk, MLI talks to mortgage finance expert Jane Londerville about government backing for residential mortgages in Canada,
### Real Property for a Real Retirement.
Real Property for a Real Retirement. Rent or Flip - Using your IRA or 401k to the fullest. Know the Right Time to Use Your Retirement Account and Get the Most From Your Real Estate Investments. 2011 CompleteIRA
### CHAPTER 5. Interest Rates. Chapter Synopsis
CHAPTER 5 Interest Rates Chapter Synopsis 5.1 Interest Rate Quotes and Adjustments Interest rates can compound more than once per year, such as monthly or semiannually. An annual percentage rate (APR)
### "Reverse Mortgages are a powerful tool to help eligible homeowners obtain
Dear Friend: Being a senior myself I am constantly looking for ways to improve my quality of life, have more liquid assets, reduce my cost of living and looking for ways to have passive income. My wife
### The following is an article from a Marlboro, Massachusetts newspaper.
319 CHAPTER 4 Personal Finance The following is an article from a Marlboro, Massachusetts newspaper. NEWSPAPER ARTICLE 4.1: LET S TEACH FINANCIAL LITERACY STEPHEN LEDUC WED JAN 16, 2008 Boston - Last week | 10,389 | 47,351 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.9375 | 4 | CC-MAIN-2018-51 | latest | en | 0.922249 |
https://convert-dates.com/days-from/450/2024/06/12 | 1,718,840,269,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861853.72/warc/CC-MAIN-20240619220908-20240620010908-00895.warc.gz | 156,504,598 | 4,324 | ## 450 Days From June 12, 2024
Want to figure out the date that is exactly four hundred fifty days from Jun 12, 2024 without counting?
Your starting date is June 12, 2024 so that means that 450 days later would be September 5, 2025.
You can check this by using the date difference calculator to measure the number of days from Jun 12, 2024 to Sep 5, 2025.
September 2025
• Sunday
• Monday
• Tuesday
• Wednesday
• Thursday
• Friday
• Saturday
1. 1
2. 2
3. 3
4. 4
5. 5
6. 6
1. 7
2. 8
3. 9
4. 10
5. 11
6. 12
7. 13
1. 14
2. 15
3. 16
4. 17
5. 18
6. 19
7. 20
1. 21
2. 22
3. 23
4. 24
5. 25
6. 26
7. 27
1. 28
2. 29
3. 30
September 5, 2025 is a Friday. It is the 248th day of the year, and in the 36th week of the year (assuming each week starts on a Sunday), or the 3rd quarter of the year. There are 30 days in this month. 2025 is not a leap year, so there are 365 days in this year. The short form for this date used in the United States is 09/05/2025, and almost everywhere else in the world it's 05/09/2025.
### What if you only counted weekdays?
In some cases, you might want to skip weekends and count only the weekdays. This could be useful if you know you have a deadline based on a certain number of business days. If you are trying to see what day falls on the exact date difference of 450 weekdays from Jun 12, 2024, you can count up each day skipping Saturdays and Sundays.
Start your calculation with Jun 12, 2024, which falls on a Wednesday. Counting forward, the next day would be a Thursday.
To get exactly four hundred fifty weekdays from Jun 12, 2024, you actually need to count 630 total days (including weekend days). That means that 450 weekdays from Jun 12, 2024 would be March 4, 2026.
If you're counting business days, don't forget to adjust this date for any holidays.
March 2026
• Sunday
• Monday
• Tuesday
• Wednesday
• Thursday
• Friday
• Saturday
1. 1
2. 2
3. 3
4. 4
5. 5
6. 6
7. 7
1. 8
2. 9
3. 10
4. 11
5. 12
6. 13
7. 14
1. 15
2. 16
3. 17
4. 18
5. 19
6. 20
7. 21
1. 22
2. 23
3. 24
4. 25
5. 26
6. 27
7. 28
1. 29
2. 30
3. 31
March 4, 2026 is a Wednesday. It is the 63rd day of the year, and in the 63rd week of the year (assuming each week starts on a Sunday), or the 1st quarter of the year. There are 31 days in this month. 2026 is not a leap year, so there are 365 days in this year. The short form for this date used in the United States is 03/04/2026, and almost everywhere else in the world it's 04/03/2026.
### Enter the number of days and the exact date
Type in the number of days and the exact date to calculate from. If you want to find a previous date, you can enter a negative number to figure out the number of days before the specified date. | 925 | 2,689 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.703125 | 4 | CC-MAIN-2024-26 | latest | en | 0.915864 |
http://mathbitsnotebook.com/Algebra1/StatisticsReg/ST2ScatterPlot.html | 1,542,364,494,000,000,000 | text/html | crawl-data/CC-MAIN-2018-47/segments/1542039743007.0/warc/CC-MAIN-20181116091028-20181116113028-00201.warc.gz | 207,813,002 | 4,417 | Scatter Plots & Line of Best Fit MathBitsNotebook.com Terms of Use Contact Person: Donna Roberts
A scatter plot is a graph of plotted points that shows a relationship between two sets of quantitative data (referred to as bivariate data). Scatter plots are composed of "dots" (points) on a set of coordinate axes. Do NOT connect the dots!
Statisticians and quality control technicians spend a good deal of time gathering sets of data to determine if relationships exist between the sets. Scatter plots are a popular and effective way of graphing data to display patterns, trends, relationships and an occasional extraordinary value located apart from the other values. Let's see an example.
Does studying for that Final Exam really help your score? Does one event really affect the other?
The scatter plot at the right appears to show that the longer students studied, the higher their examination scores.
According to this survey of 15 students studying for the same examination, it appears that the answer to our initial question is "yes", studying does affect your score. At least, the answer is "yes", for this particular group of students.
NOTE: A scatter plot is not necessarily a function. It is often the case where the same x-value may have more than one corresponding y-value, such as (6,70) and (6,80).
Notice how the data in the graph resembles a straight line rising from left to right. When working with scatter plots, if is often useful to represent the data with the equation of a straight line, called a "line of best fit", or a "trend" line. Such a line may pass through some of the points, none of the points, or all of the points on the scatter plot.
To see how to prepare a line of best fit by hand and a line of best fit with a graphing calculator, click the link below.
What is the line of best fit for our "studying affects scoring" problem?
When finding the line of best fit "by hand", different students may arrive at different answers. So who's answer is the best? You will need a graphing calculator to get the "best" answer.
The graphing calculator computed the line of best fit shown at the right. The equation is:
y = 4.722392638x + 50.99539877
Based upon this equation, we can predict scores given any number of hours spent studying.
Interpolate:
If you are making predictions for values that fall within the plotted values, you are said to be interpolating. For this problem, our plotted values range from x = 1 to x = 9.
Example: Predict the final examination score of a student studying for 5½ hours.
(Substitute the number of hours into the equation for x.)
Score: approximately 77
Extrapolate: If you are making predictions for values that fall outside the plotted values, you are said to be extrapolating. Be careful when extrapolating. The further away from the plotted values you go, the less reliable is your prediction. For this problem, outside of the plotted values would be x greater than 9 or x less than 1.
Example: Predict the final examination score of a student studying for 12 hours.
(Substitute the number of hours into the equation for x.)
Score: approximately 108
WOW!!! Great score!!
But is it realistic? It is very likely that the top score is 100.
So, in addition to yielding less reliable predictions, extrapolating
may also give completely unrealistic predictions.
For calculator help with scatter plots click here.
NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation and is not considered "fair use" for educators. Please read the "Terms of Use". | 775 | 3,589 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.09375 | 4 | CC-MAIN-2018-47 | longest | en | 0.947547 |
https://www.coursehero.com/file/73103/Astro-1HW-5-Solutions-Winter-08/ | 1,529,902,802,000,000,000 | text/html | crawl-data/CC-MAIN-2018-26/segments/1529267867424.77/warc/CC-MAIN-20180625033646-20180625053646-00399.warc.gz | 795,092,305 | 64,274 | Astro 1:HW 5 Solutions (Winter 08)
# Astro 1:HW 5 Solutions (Winter 08) - UCSB Winter 2008 Astro...
This preview shows pages 1–2. Sign up to view the full content.
UCSB Winter 2008 Astro 1 - Homework #5 Solutions Kevin Moore 2/9/08 1 How can we use the Moon’s orbital motion, plus the distance an apple falls in one second, to determine the dependence of the gravitational force formula on the distance between two bodies? The basic idea here is to realize that the distance an object falls in one second is directly proportional to acceleration, which is in turn proportional to the force of gravity. You didn’t need to do the actual calculation to find this, but I’ll go through it and show you can deduce that the force of gravity goes as 1 /r 2 . You may know from physics that for constant acceleration, distance is given by d = d o + v o t + 1 2 at 2 Thus the distance an object falls in one second is d Earth = 1 2 a (1 s 2 ), indeed directly proportional to acceleration. We also know that F = ma , so d F for an object falling in one second. Since the force of gravity is directly proportional to mass (see prob. 2), it doesn’t matter whether it’s the Moon or an apple up there - they all fall at the same rate. First we’ll get the distance an object falls on Earth’s surface: here a = 9 . 8 m/s 2 so d = 4 . 9 m Getting the distance the Moon ’falls’ in 1 sec is more involved: First we need to find how far the Moon moves in its orbit in one second: It’s angular speed is (in radians/sec) ω = 2 π 27 . 3 days 1 day 24 hr 1 hr 3600 s = 2 . 66 × 10 - 6 rad sec The Earth-Moon distance is D moon = 384 , 400 km so the Moon travels about (384 , 400)(2 . 66 × 10 - 6 ) = 1 . 02 km in 1 sec.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up to access the rest of the document.
{[ snackBarMessage ]}
### What students are saying
• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.
Kiran Temple University Fox School of Business ‘17, Course Hero Intern
• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.
Dana University of Pennsylvania ‘17, Course Hero Intern
• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.
Jill Tulane University ‘16, Course Hero Intern | 711 | 2,839 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.25 | 4 | CC-MAIN-2018-26 | latest | en | 0.87246 |
https://gmatclub.com/forum/if-a-and-b-are-two-digit-numbers-that-share-the-same-digits-94795.html | 1,558,637,984,000,000,000 | text/html | crawl-data/CC-MAIN-2019-22/segments/1558232257361.12/warc/CC-MAIN-20190523184048-20190523210048-00466.warc.gz | 509,639,646 | 145,550 | GMAT Question of the Day - Daily to your Mailbox; hard ones only
It is currently 23 May 2019, 11:59
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
# If a and b are two-digit numbers that share the same digits,
Author Message
TAGS:
### Hide Tags
Intern
Joined: 12 May 2005
Posts: 29
If a and b are two-digit numbers that share the same digits, [#permalink]
### Show Tags
25 May 2010, 06:47
1
9
00:00
Difficulty:
55% (hard)
Question Stats:
64% (02:05) correct 36% (02:02) wrong based on 276 sessions
### HideShow timer Statistics
If a and b are two-digit numbers that share the same digits, except in reverse order, then what is the sum of a and b?
(1) a-b=45
(2) The difference between the two digits in each number is 5.
Math Expert
Joined: 02 Sep 2009
Posts: 55266
Re: Number Properties - DS [#permalink]
### Show Tags
25 May 2010, 07:04
3
1
srichaks wrote:
Q:If a and b are two-digit numbers that share the same digits, except in reverse order, then what is the sum of a and b?
(1) a-b=45
(2) The difference between the two digits in each number is 5.
OA:E
Any two digit integer $$n$$ can be expressed as $$n=10x+y$$, where $$x$$ and $$y$$ are digits, $$x>0$$.
Given: $$a=10x+y$$ and $$b=10y+x$$. Q: $$a+b=11(x+y)=?$$
(1) $$a-b=10x+y-10y-x=45$$ --> $$x-y=5$$. Multiple choices for x and y: (9,4), (8, 3), (7, 2), (6, 1). Not sufficient.
(2) $$x-y=5$$. Same info as above. Not sufficient.
(1)+(2) No new info. Not sufficient.
_________________
Intern
Joined: 12 May 2005
Posts: 29
Re: Number Properties - DS [#permalink]
### Show Tags
25 May 2010, 18:42
Thanks Bunuel!. Great explanation.
Manager
Joined: 05 Mar 2010
Posts: 165
Re: Number Properties - DS [#permalink]
### Show Tags
29 May 2010, 07:09
I dont understand the "except in reverse order" part???????
_________________
Success is my Destiny
Math Expert
Joined: 02 Sep 2009
Posts: 55266
Re: Number Properties - DS [#permalink]
### Show Tags
29 May 2010, 07:20
hardnstrong wrote:
I dont understand the "except in reverse order" part???????
It means that if digits in $$a$$ are $$xy$$, then in $$b$$ the digits are $$yx$$ (in reverse order).
According to the solution above there are 5 pairs of $$a$$ and $$b$$ possible satisfying both statements (94 and 49, 83 and 38, 72 and 27, 61 and 16). So multiple answer to $$a+b$$. Not sufficient.
Hope it's clear.
_________________
Manager
Joined: 05 Mar 2010
Posts: 165
Re: Number Properties - DS [#permalink]
### Show Tags
30 May 2010, 06:27
Bunuel wrote:
hardnstrong wrote:
I dont understand the "except in reverse order" part???????
It means that if digits in $$a$$ are $$xy$$, then in $$b$$ the digits are $$yx$$ (in reverse order).
According to the solution above there are 5 pairs of $$a$$ and $$b$$ possible satisfying both statements (94 and 49, 83 and 38, 72 and 27, 61 and 16). So multiple answer to $$a+b$$. Not sufficient.
Hope it's clear.
This is the confusing part in this question............ except in reverse order means they cannot be in reverse order (so if a = xy then b cannot be yx)
Dont you think "except in reverse order" takes question is completely different direction, or should it be "in reverse order"
_________________
Success is my Destiny
Math Expert
Joined: 02 Sep 2009
Posts: 55266
Re: Number Properties - DS [#permalink]
### Show Tags
30 May 2010, 06:40
hardnstrong wrote:
Bunuel wrote:
hardnstrong wrote:
I dont understand the "except in reverse order" part???????
It means that if digits in $$a$$ are $$xy$$, then in $$b$$ the digits are $$yx$$ (in reverse order).
According to the solution above there are 5 pairs of $$a$$ and $$b$$ possible satisfying both statements (94 and 49, 83 and 38, 72 and 27, 61 and 16). So multiple answer to $$a+b$$. Not sufficient.
Hope it's clear.
This is the confusing part in this question............ except in reverse order means they cannot be in reverse order (so if a = xy then b cannot be yx)
Dont you think "except in reverse order" takes question is completely different direction, or should it be "in reverse order"
Not a perfect wording - agree. "except in reverse order" here means "but in reverse order".
_________________
Non-Human User
Joined: 09 Sep 2013
Posts: 11005
Re: If a and b are two-digit numbers that share the same digits, [#permalink]
### Show Tags
15 Sep 2018, 10:32
Hello from the GMAT Club BumpBot!
Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).
Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: If a and b are two-digit numbers that share the same digits, [#permalink] 15 Sep 2018, 10:32
Display posts from previous: Sort by | 1,508 | 5,352 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.21875 | 4 | CC-MAIN-2019-22 | latest | en | 0.806459 |
https://www.jiskha.com/display.cgi?id=1251913186 | 1,532,270,944,000,000,000 | text/html | crawl-data/CC-MAIN-2018-30/segments/1531676593302.74/warc/CC-MAIN-20180722135607-20180722155607-00328.warc.gz | 918,525,014 | 5,283 | # Very easy math
posted by Very easy Math
my teacher told me that the inverse of addition was subtraction and that the inverse of subtraction was addition...
could you prove it to me
-x = (-1)x
((-1)x)^-1
I don't see how I'm suppose to get + x by taking the inverse of -x i've always been told in math to just to the opposite of subbtraction which is addition but my teacher is telling me that is a lie and that it's really the inverse of subtraction is addition but I don't see the reasoning behind it
basically can you prove to me that the opposite of subractiion is addition and vise versa??? by taking the inverses????
I don't get it...
like I can prove that the opposite of multiplication is division by taking the inverse and can prove it just by defintion
(5x = 2)5^-1 = x = 5^-1 (2)
that's how you prove that relationship is really just inverses but what about addition and subtraction how are the inverse relationships...???
Thansk
1. Writeacher
Use numbers.
6 + 2 = 8
Therefore, 8 - 6 = 2 or 8 - 2 = 6
How can you state those relationships in abstract terms?
2. Count Iblis
If x is some number and:
x + y = 0
then y is called an inverse (w.r.t. addition) of x
Then it follows from the same definition that x is an inverse of y. Now, what you need to prove is that inverses are unique. I.e. if for some given x
x + y = 0
and also
x + z = 0
you necessarily have y = z.
So, it then follows that the inverse of the inverse of x is x and it can't be anything else than x.
Then, if we denote the inverse of x by
-x, we can prove that:
-x = (-1)*x
THis is because:
x + (-1)*x =
1*x + (-1)*x =
(1 + (-1))*x =
0*x = 0
Here we have used that -1 is the inverse of 1.
So, (-1)*x satisfies the criterium the inverse of x which we always denote as
-x must satisfy and therefore
-x = (-1)*x
Then the fact that taking twice the inverse yields the same number implies that:
(-1)*(-1) = 1
3. Very easy Math
i agree with all of it but still don't see how
X + B = C
we can simply solve for B by simply multiplying the whole equation by B^-1 which we note as -B because????
(X + B = C)B^-1
B cancels out
X = B^-1 C
what allows us to say that B^-1 is equal to -B
4. bobpursley
You are confusing terms:
Inverse is not the reciprocal. You are using reciprocal (B^-1) is reciprocal.
Now it is confusing, because the inverse operation to multiplication is division, and the inverse to division is multiplication
http://www.mathsisfun.com/definitions/inverse-operation.html
Watch the usage to "inverse", a lot of folks really mean reciprocal when they use it.
## Similar Questions
1. ### Another Problem...PLz Help COunt Iblis
Statements Reasons 1. 3x - 7 = -4 2. 3x - 7 + 7 = -4 + 7 3. 3x + 0 = -4 + 7 4. 3x + 0 = 3 5. 3x = 3 6. (1/3) 3x = 3 (1/3) 7. (1/3) 3x = 1 8. 1x = 1 9. x = 1 Is this problem about explaining each step?
2. ### Math
Using the definition of subtraction to determine the subtraction equations that are related to each of the following addition equation: a.8+7=n b.14+x=25 c.r+s=t my answers are this: a. n-7=8 b.x=25-14 c.t-s=r did I do these right?
3. ### MATH 156
What models can be used to help explain the concepts of addition and subtraction of rational numbers?
4. ### math
What inverse operation is needed for the first step in solving the equation 12+5x=22?
5. ### math
An eighth-grade student claims she can prove that subtraction of integers is commutative. She points out that if a and b are integers, then a-b = a+ -b. Since addition is commutative, so is subtraction. What is your response?
6. ### Math--1 question
Name the inverse operations for each of the following operations. a. addition b. division c. multiplication d. subtraction | 1,057 | 3,713 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.28125 | 4 | CC-MAIN-2018-30 | latest | en | 0.969874 |
http://www.us-lotteries.com/Kentucky/Pick_3/Pick_3-Triples.asp | 1,386,827,975,000,000,000 | text/html | crawl-data/CC-MAIN-2013-48/segments/1386164554256/warc/CC-MAIN-20131204134234-00020-ip-10-33-133-15.ec2.internal.warc.gz | 609,294,417 | 6,513 | Frequency and draw dates of all triples (three identical digits) for Kentucky Lottery Pick 3 game. Is time to play triples?
# Kentucky Lottery Pick 3 Triples Data and Statistics
Updated on Wednesday, December 11, 2013 Triples are more appealing to play since the player does not have to worry about in what order to play them. The down side is that they are not drawn that often and, except when they are really late, it is hard to predict which one to play. Many Pick 3 players play all the ten triple numbers when they think that triples are due to come. On this page...
## How often are Kentucky Lottery Pick 3 triples drawn?
The chances of a triple-digit Pick 3 number being drawn are the same as any other straight number, that is, one in one thousand. However, the chances of any one of the ten triples of Pick 3 being drawn are one in 100. This means that we should expect a triple number once every 100 draws. All the winning triple-digit numbers of the Kentucky Lottery Pick 3 game drawn, as well as the draw dates, are listed in the table below. The last column of the list designated by NDAPT (Number of Draws After Previous Triple) represents the number of draws elapsed between triples and is intended to give you an idea of how often the triples are showing up in reality.
## Kentucky Lottery Pick 3 triples never drawn
There are no Kentucky Lottery Pick 3 triples never-drawn. ALL Pick 3 triples are drawn at least once.
## Analysis Results of Kentucky Lottery Pick 3 triples
Here is a summary of the analysis results of Kentucky Lottery Pick 3 triples. Theoretically, since the 10 triples make 1% of the total number of Pick 3 numbers, the percentage of triples drawn is expected to be close to 1%. Also, the average number of draws between triples is expected to be close to 100. Moreover, check if the number of draws since most recent triple number is well over 100. These three results could be an indication of whether or not to expect triples soon.
Kentucky Lottery Pick 3 Triples summary
( As of Wednesday, December 11, 2013)
• Total number of draws = 10238
• Number of triples drawn = 108
• Percent of total draws that triples are drawn = 1.055%
• Average number of draws between triples = 94
• Minimum number of draws between triples = 1
• Maximum number of draws between triples = 526
• Long-time-no-see triple number = 111(Sun Aug 16, 2009)
• Most recent triple number = 999(Wed Nov 13, 2013)
• Draws since most recent triple number = 53
TOP
## Summary of individual Kentucky Lottery Pick 3 triples
The following table summarizes the data for each triple number. It lists, for each Kentucky Lottery Pick 3 triple number, the number of times it is drawn, the date when it was last drawn, and how many draws ago is it since its last appearance.
Triple Number Number of Times Drawn Date Last Drawn Number of draws since last seen 000 13 Sun Jul 1, 2012 981 111 10 Sun Aug 16, 2009 2931 222 9 Sun Jul 22, 2012 942 333 14 Tue Sep 17, 2013 158 444 13 Sat Jun 9, 2012 1022 555 10 Wed Mar 20, 2013 495 666 10 Wed Jul 24, 2013 261 777 11 Wed Sep 8, 2010 2211 888 10 Fri May 20, 2011 1739 999 8 Wed Nov 13, 2013 53
## List of all Kentucky Lottery Pick 3 triples drawn
Number Draw Date NDAPT* 111 Fri Nov 27, 1998 145 333 Thu Jan 7, 1999 41 000 Thu Jan 28, 1999 21 111 Thu Feb 11, 1999 14 444 Mon Mar 29, 1999 51 555 Fri Apr 2, 1999 8 333 Thu Jun 10, 1999 128 999 Wed Sep 22, 1999 184 999 Sat Oct 30, 1999 71 888 Tue Nov 23, 1999 44 222 Wed Aug 30, 2000 522 777 Sat Sep 16, 2000 32 333 Thu Nov 2, 2000 87 999 Fri Jan 5, 2001 118 222 Sun Jan 7, 2001 4 888 Mon Jan 15, 2001 15 888 Wed Feb 7, 2001 42 888 Thu Apr 5, 2001 107 444 Wed Jun 6, 2001 114 444 Tue Jun 19, 2001 24 777 Wed Aug 15, 2001 107 888 Wed Aug 29, 2001 26 000 Thu Sep 6, 2001 14 444 Thu Jan 24, 2002 261 111 Sat Jan 26, 2002 4 333 Tue Feb 26, 2002 56 000 Tue Mar 26, 2002 53 666 Tue Apr 9, 2002 25 111 Fri Jun 28, 2002 150 000 Mon Jul 22, 2002 44 000 Tue Oct 1, 2002 132 666 Fri Nov 29, 2002 110 777 Sat Dec 14, 2002 27 777 Sat Dec 28, 2002 26 444 Sat Jan 11, 2003 26 777 Sat Feb 15, 2003 65 222 Wed Feb 19, 2003 7 222 Mon May 5, 2003 140 333 Thu Oct 2, 2003 279 000 Mon Oct 27, 2003 46 555 Wed Nov 26, 2003 55 666 Wed Dec 3, 2003 13 555 Mon Dec 29, 2003 49 222 Wed Jan 28, 2004 55 666 Thu Mar 11, 2004 80 333 Sat Apr 24, 2004 83 777 Sun Apr 25, 2004 1 000 Wed Jul 14, 2004 148 555 Sun Aug 8, 2004 47 888 Thu Sep 23, 2004 86 777 Mon Oct 18, 2004 45 333 Wed Feb 2, 2005 200 777 Sat Feb 12, 2005 18 000 Fri Feb 25, 2005 25 222 Fri Mar 11, 2005 25 111 Mon Apr 25, 2005 84 444 Wed Apr 27, 2005 3 000 Sun Jun 26, 2005 112 666 Sat Jul 16, 2005 37 999 Wed Feb 15, 2006 398 777 Thu Jun 22, 2006 235 444 Thu Aug 31, 2006 130 444 Fri Sep 1, 2006 2 999 Sat Mar 3, 2007 340 555 Fri Jul 27, 2007 270 777 Mon Aug 13, 2007 32 555 Tue Aug 21, 2007 14 444 Tue Sep 4, 2007 26 333 Mon Sep 17, 2007 24 111 Tue Oct 16, 2007 55 999 Fri Nov 2, 2007 31 333 Sat Feb 2, 2008 172 000 Tue Apr 1, 2008 109 666 Fri Jun 27, 2008 161 222 Sat Jul 5, 2008 16 444 Thu Jul 17, 2008 21 333 Sat Sep 13, 2008 108 111 Fri Oct 24, 2008 76 222 Sun Nov 23, 2008 56 333 Wed Feb 11, 2009 149 111 Mon Mar 9, 2009 48 444 Tue Apr 21, 2009 80 111 Sun Jun 21, 2009 113 888 Wed Jul 15, 2009 44 000 Wed Jul 29, 2009 27 111 Sun Aug 16, 2009 33 555 Mon Sep 7, 2009 40 666 Wed Oct 7, 2009 56 666 Mon Nov 16, 2009 75 888 Tue Dec 8, 2009 41 333 Fri Jun 4, 2010 331 888 Sat Jun 12, 2010 14 444 Sat Sep 4, 2010 156 777 Wed Sep 8, 2010 7 555 Mon Oct 18, 2010 74 888 Fri May 20, 2011 398 000 Mon Feb 27, 2012 526 444 Sat Jun 9, 2012 191 000 Sun Jul 1, 2012 41 222 Sun Jul 22, 2012 39 666 Mon Jul 23, 2012 2 999 Wed Jul 25, 2012 3 555 Thu Jan 17, 2013 328 555 Wed Mar 20, 2013 114 333 Sat Jun 8, 2013 149 666 Wed Jul 24, 2013 85 333 Tue Sep 17, 2013 103 999 Wed Nov 13, 2013 105
* NDAPT = Number of Draws After Previous Triple, where the very first entry is from the starting date
TOP
Kentucky lottery Games Data and Analysis Menu
Pick 3 - Pick 3 Game - Winning Numbers - Number Search - Yearly Calendar - Boxes(Combos) - Triples - Digits Analysis - Straights Analysis - Boxes Analysis - Pairs - Sums - Combinations Generator - Game Info Pick 4 - Pick 4 Game - Winning Numbers - Number Search - Yearly Calendar - Boxes(Combos) - Quadruplets - Digits Analysis - Straights Analysis - Boxes Analysis - Pairs - Sums - Combinations Generator - Game Info Cash Ball - Cash Ball Game - Winning Numbers - Number Search - Yearly Calendar - Numbers Analysis - Pairs Analysis - Number-Ball Pair Analysis - Check Combinations - Full Wheel - Abbreviated Wheel - Game Info Kicker - Kicker Game - Winning Numbers - Yearly Calendar Mega Millions - Mega Millions Game - Winning Numbers - Number Search - Yearly Calendar - Numbers Analysis - Pairs Analysis - Number-Ball Pair Analysis - Check Combinations - Full Wheel - Abbreviated Wheel - Game Info Decades of Dollars - Decades of Dollars Game - Winning Numbers - Number Search - Yearly Calendar - Numbers Analysis - Pairs Analysis - Triads Analysis - Check Combinations - Full Wheel - Abbreviated Wheel - Game Info Powerball - Powerball Game - Winning Numbers - Number Search - Yearly Calendar - Numbers Analysis - Pairs Analysis - Number-Ball Pair Analysis - Check Combinations - Full Wheel - Abbreviated Wheel - Game Info
- Kentucky Lottery Recent Results and Tools - Compact page for quick check - Kentucky Lottery General Information - Kentucky Lottery Games - US-Lotteries Home Page
• WE DO NOT SELL ANY LOTTERY, WE DO NOT BUY LOTTERY NUMBERS ON BEHALF OF ANYONE; this is only an information and guide web site. | 2,589 | 7,590 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.640625 | 4 | CC-MAIN-2013-48 | latest | en | 0.958253 |
https://lucatrevisan.wordpress.com/2009/11/12/the-large-deviation-of-fourwise-independent-random-variables/ | 1,670,195,192,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446710980.82/warc/CC-MAIN-20221204204504-20221204234504-00085.warc.gz | 394,905,545 | 33,637 | # The Large Deviation of Fourwise Independent Random Variables
Suppose ${X_1,\ldots,X_n}$ are mutually independent unbiased ${\pm 1}$ random variables. Then we know everything about the distribution of
$\displaystyle | X_1 + \ldots + X_N | \ \ \ \ \ (1)$
either by using the central limit theorem or by doing calculations by hand using binomial coefficients and Stirling’s approximation. In particular, we know that (1) takes the values ${1,\ldots, 1/\sqrt N}$ with probability ${\Theta(1/\sqrt N)}$ each, and so with constant probability (1) is at most ${O(\sqrt N)}$.
The last statement can be proved from scratch using only pairwise independence. We compute
$\displaystyle \mathop{\mathbb E} \left| \sum_i X_i \right|^2 = N$
so that
$\displaystyle \mathop{\mathbb P} \left[ \left|\sum_i X_i \right| \geq c \cdot \sqrt N \right] = \mathop{\mathbb P} \left[ \left|\sum_i X_i \right|^2 \geq c^2 \cdot N \right] \leq \frac 1 {c^2}$
It is also true that (1) is at least ${\Omega(\sqrt N)}$ with constant probability, and this is trickier to prove.
First of all, note that a proof based on pairwise independence is not possible any more. If ${(X_1,\ldots,X_N)}$ is a random row of an Hadamard matrix, then ${\sum_i X_i = N}$ with probability ${1/N}$, and ${\sum_i X_i =0}$ with probability ${1-1/N}$.
Happily, four-wise independence suffices.
Lemma 1 Suppose ${X_1,\ldots,X_n}$ are four-wise independent ${\pm 1}$ random variables. Then
$\displaystyle \mathop{\mathbb P} \left[ \left| \sum_i X_i \right| \geq \Omega(\sqrt N) \right] \geq \Omega(1)$
This is probably well known to a lot of people, but the only place where I have seen this result is a paper by Andreev et al. that I have mentioned before.
Here is a slightly simplified version of their argument. We compute
$\displaystyle \mathop{\mathbb E} \left| \sum_i X_i \right|^2 = N$
and
$\displaystyle \mathop{\mathbb E} \left| \sum_i X_i \right|^4 = 3N^2 - 2N$
So we have (from here on is our simplified version)
$\displaystyle \mathop{\mathbb E} \left( \left(\sum_i X_i \right)^2 - 2N \right)^2 = 3N^2 - 2N \leq 3N^2$
and
$\displaystyle \mathop{\mathbb P} \left[ \left| \sum_i X_i \right| \leq \frac 13 \sqrt N \right]$
$\displaystyle = \mathop{\mathbb P} \left[ \left| \sum_i X_i \right|^2 \leq \frac 19 N \right]$
$\displaystyle \leq \mathop{\mathbb P} \left[ \left| \left| \sum_i X_i \right|^2 - 2N \right| \geq \frac {17}{9} N \right]$
$\displaystyle = \mathop{\mathbb P} \left[ \left( \left| \sum_i X_i \right|^2 - 2N \right)^2 \geq \frac {289}{81} N^2 \right]$
$\displaystyle \leq \frac 3 {\frac {289}{81}} = .80408$
Thus
$\displaystyle \mathop{\mathbb P} \left[ \left| \sum_i X_i \right| \geq \frac 13 \sqrt N \right] \geq .159$
Following the same proof, but replacing each “${X_i}$” with “${a_i X_i}$” and every “${N}$” with “${\sum_i a_i^2}$”, we get the more general version
Lemma 2 Suppose ${X_1,\ldots,X_N}$ are four-wise independent ${\pm 1}$ random variables and let ${a_1,\ldots,a_N}$ be any real numbers. Then
$\displaystyle \mathop{\mathbb P} \left[ \left| \sum_i a_i X_i \right| \geq \frac 13 \sqrt {\sum_i a_i^2 } \right] \geq .159$
In particular,
$\displaystyle \mathop{\mathbb E} \left| \sum_i a_i X_i \right| \geq \frac 1 {20} \sqrt {\sum_i a_i^2 }$
A related version that was useful in our work on distinguishers for pseudorandom generators is
Corollary 3 Let ${H}$ be a four-wise independent family of functions ${h: S \rightarrow \{-1,1\}}$ and ${g: S \rightarrow {\mathbb R}}$ be any real valued function. Then
$\displaystyle \mathop{\mathbb E}_{h\sim H} \left| \sum_{x\in S} h(x)g(x) \right| \geq \frac 1{20\cdot \sqrt{|S|}} \sum_x | g(x) |$
To briefly show how this relates to our work on distinguishers, suppose ${G: \{ 0,1 \}^{n-1} \rightarrow \{ 0,1 \}^n}$ is a length-increasing function, and that we want to construct a function ${D}$, computable by a circuit as small as possible, such that
$\displaystyle \mathop{\mathbb P}_{z\in \{ 0,1 \}^{n-1}} [ D(G(z)) =1 ] - \mathop{\mathbb P}_{x\in \{ 0,1 \}^n} [D(x)=1] \geq \epsilon$
Then this is equivalent to proving
$\displaystyle \mathop{\mathbb E}_{z \in \{ 0,1 \}^{n-1}} (-1)^{D(G(z))} - \mathop{\mathbb E}_{x\in \{ 0,1 \}^n} (-1)^{D(x)} \geq 2 \epsilon \ \ \ \ \ (2)$
and, if we set
$\displaystyle g(x) := \mathop{\mathbb P}_{z\in \{ 0,1 \}^{n-1}} [G(z) = x] - \frac 1 {2^n}$
then (2) is equivalent to
$\displaystyle \sum_{x\in \{ 0,1 \}^n} (-1)^{D(x)} \cdot g(x) \geq 2 \epsilon \ \ \ \ \ (3)$
Now, fix a simple partition ${\cal P}$ of ${\{ 0,1 \}^n}$ into ${1,600 \epsilon^{2} 2^n}$ subsets each of size ${(40 \epsilon)^{-2}}$, for example based on the value of the first ${\log 1,600 \epsilon^2 2^n}$ bits.
Fix an efficiently computable four-wise independent family ${H}$ of functions ${h: \{ 0,1 \}^n \rightarrow \{ 0,1 \}}$. Then we have, for every set ${S}$ in the partition
$\displaystyle \mathop{\mathbb E}_{h\sim H} \left| \sum_{x \in S} (-1)^{h(x)} g(x) \right| \geq \frac{1}{20\sqrt {|S|}} \sum_{x\in S} |g(x)| = 2\epsilon \sum_{x\in S} |g(x)|$
and if we sum over all sets in the partition we have
$\displaystyle \mathop{\mathbb E}_{h\sim H} \sum_{S\in {\cal P}} \left| \sum_{x \in S} (-1)^{h(x)} g(x) \right| \geq 2\epsilon \sum_{x\in \{ 0,1 \}^n} |g(x)| \geq 2\epsilon$
In particular, there is a function ${h_0 \in H}$ such that
$\displaystyle \sum_{S\in {\cal P}} \left| \sum_{x \in S} (-1)^{h_0(x)} g(x) \right| \geq 2\epsilon$
To finish the argument, define the function ${c: \{ 0,1 \}^n \rightarrow \{ 0,1 \}}$ such that ${c(z)=1}$ iff ${\sum_{x\in S} (-1)^{h_0(x)} g(x) < 0}$, where ${S}$ is the set that ${z}$ belongs to. Then
1. ${c(x)}$ depends only on the first ${\log O(\epsilon^{-2} 2^n)}$ bits of the input, and so it can be computed by a circuit of size ${O(\epsilon^{-2}2^n)}$
2. for every ${S}$ we have
$\displaystyle \left| \sum_{x \in S} (-1)^{h_0(x)} g(x) \right| = \sum_{x\in S} (-1)^{c(x)} \cdot (-1)^{h_0(x)} g(x)$
So, in conclusion
$\displaystyle \sum_{x \in \{ 0,1 \}^n} (-1)^{c(x) + h_0(x)} g(x) \geq 2\epsilon$
meaning that setting ${D(x) := c(x) \oplus h(x)}$ satisfies (3), and gives us a distinguisher obtaining advantage ${\epsilon}$ and having circuit complexity ${O(\epsilon^{-2} 2^n)}$.
## 15 thoughts on “The Large Deviation of Fourwise Independent Random Variables”
1. Similar results have been obtained by Serge Vaudenay using his “decorrelation theory” [1] in the context of the design of secure block ciphers. More precisely, his paper [2] about iterated distinguishers shows that a decorrelation of order $2d$ is sufficient to resist an attack of order $d$. An open question is whether this is a necessary condition.
[1] S. Vaudenay, “Decorrelation: A Theory for Block Cipher Security”, J. Cryptology, 16(4), 249-286, 2003.
[2] S. Vaudenay, “Resistance against general iterated attacks”, Proceedings of Eurocrypt’99, pp. 255-271, LNCS 1592, Springer, 1999.
2. Hi Luca,
Lemma 1 was also proven by Bonnie Berger back in SODA ’91. See her paper, “The Fourth Moment Method”.
3. Yeah, what James said, with Wikipedia’s $Z = (\sum a_i X_i)^2$. Indeed, if the $X_i$‘s in Lemma 2 are any $(2, 4, \eta)$-hypercontractive random variables, then so too is $\sum a_i X_i$; then Paley-Zygmund gives the lower bound (with the RHS depending on $eta$).
4. Thanks, Jelani, and her calculations are much tighter, in Lemma 2 she has a factor of $\sqrt 3$ instead of 20 (Andreev, Clementi and Rolim had 96)
5. Hi Pascal, what is the similarity that you see? Vaudenay shows that if a family of permutations is close to k-wise independent under certain norms then certain classes of attackers have small advantage in distinguishing a function from the family from a truly random permutation. (And some of those conditions are necessary in order to upper bound the advantage of distinguishers from those classes.)
Here we have a completely arbitrary distribution over $n$-bit strings that is far from uniform in total variation distance (for concreteness I have talked about the output of a length-increasing generator) and we show that, within a set S, a distinguisher sampled from a 4-wise independent family of functions has typically advantage about $1/\sqrt {|S|}$ times the best possible advantage within the set S.
(So if we are willing to invest in a look-up table of size $2^n/|S|$ to know the sign of the distinguishing probability in each set of size $|S|$ into which we partition $\{0,1\}^n$, we can get advantage $1/\sqrt{|S|}$ overall. This is not an iterated attack because there are no queries to make, it’s just a case analysis.)
So here it’s the attacker, not the construction that is coming from a $k$-wise independent distribution, and we are proving a lower bound, not an upper bound, on the distinguishing probability.
6. I had a question along similar lines on sum of $k$-wise independent variables. Assume $X_1, \ldots, X_n$ are $k$-wise independent $\pm 1$ random variables. What can be said about the following expression :
$\displaystyle {\mathbb P} \left[\left|\sum_{i} X_i\right| > \frac{n^{0.5}}{100}\right]$
By Berry – Esseen theorem, if $k=n$, then it is $1-o(1)$ and the above discussion shows that $k \geq 4$ implies that it is bigger than a constant. But can we get a better bound if $4?
7. Hi Anindya,
If the $X_i$ are k-wise independent for $k = O(1/ \epsilon^2)$, then
$\displaystyle Pr \left[ \left|\sum_i X_i \right| > \sqrt{n} \right] >= 1 - O(\epsilon). \ \ \ (*)$
Note when $1/\epsilon^2 = n$, this gives a lower bound of $1 - O(1/\sqrt{n} )$, which is what Berry-Esseen would tell you.
(*) follows from Section A.5 of http://web.mit.edu/minilek/www/papers/lp.pdf (last paragraph, which refers back to an earlier argument in the paper ^^;), though I regret that recovering (*) from what’s written there might take a bit of sorting through our notation, since the paper is really about something else. You can also get (*) from “Bounded Independence Fools Halfspaces” (Diakonikolas et al., FOCS ’09) with only a slightly larger k of $O(\log^2(1/\epsilon)/\epsilon^2)$.
8. I don’t understand: if the $X_i$ are mutually independent, then from the Berry-Esseen theorem we have that for every threshold $t$
$\displaystyle \left| {\mathbb P} \left[ \left|\sum_i X_i \right| \geq t \sqrt n \right] - {\mathbb P} \left[ \left| N \right| \geq t \right] \right| \leq O \left( \frac 1 {\sqrt n} \right)$
where $N$ is a normal random variable with expectation zero and variance 1. But the probability that $|N|$ is more than $1/100$ is not $1-o(1)$, it is a constant bounded away from 1 (and from zero).
9. I think Luca is right though I intended to put $n^{0.45}$ instead of $\sqrt{n}$. In that case, Berry-Esseen does guarantee that the sum of $X_i$ exceeds $n^{0.45}$ in absolute value with $1-o(1)$ probability. However, a calculation like above does not give anything beyond a constant probability.
Jelani: Thanks for answering. I assume what you guys prove is that if $k=1/\epsilon^2$, then the difference from the Normal distribution is at most $\epsilon$. That sounds great though I have not yet read your paper.
10. What I should have actually said is, as long as the $X_i$ are $k$-wise independent for $k = \Omega(1/\epsilon^2)$ then
$$\mathrm{Pr}[|\sum_i X_i| > \epsilon \sqrt{n}] \ge 1 – O(\epsilon)$$
I forgot to type that inner $\epsilon$ above. Sorry for the confusion.
11. In general, proving concentration bounds under bounded independence has been known for a while. What I believe was not known until recently, is that bounded independence also suffices for anti-concentration bounds. More generally, the following holds:
The Berry-Esseen version of the CLT also applies under bounded independence.
In particular, in Luca’s formulation, if the X_i’ s in the sum are \Omega(1/eps^2)-wise independent, then the distance between the CDF’s is at most 1/\sqrt{n} + eps. (**)
The degree of independence required for such a statement to hold is actually Omega(1/eps^2), i.e. the above bound is optimal up to constant factors.
As Jelani pointed out, (**) follows as a special case of a result by Gopalan, Jaiswal, Servedio, Viola and myself, but the bound there on the degree of independence required is slightly weaker (up to a log^2 factor).
12. That had been incredibly imformative. I appreciate you all the material. | 3,894 | 12,320 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 112, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.65625 | 4 | CC-MAIN-2022-49 | latest | en | 0.834333 |
https://www.sanfoundry.com/high-voltage-engineering-questions-answers-waveshape-control/ | 1,725,803,738,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651002.87/warc/CC-MAIN-20240908115103-20240908145103-00811.warc.gz | 948,885,339 | 17,404 | # High Voltage Engineering Questions and Answers – Waveshape Control
This set of High Voltage Engineering Multiple Choice Questions & Answers (MCQs) focuses on “Waveshape Control”.
1. An impulse voltage generator has a generator capacitance C1 of 0.02 μF, load capacitance C2 of 0.001 μF, front resistance of R1=400 Ω and tail resistance of R2=2980 Ω. The tail time is __________
a) 49.68 μs
b) 54.26 μs
c) 30.92 μs
d) 76.54 μs
Explanation: Tail time is the discharging time. The capacitances C1 and C2 may be considered to be in parallel and discharging occurs through R1 and R2. The tailing time is given by the equation
t=.7 (R1+R2)(C1+C2)
t=.7(400+2980) (.02+0.001) x 10-6
t=49.68 μs.
2. An impulse voltage generator has a generator capacitance C1 of 0.02 μF, load capacitance C2 of 0.001 μF, front resistance of R1=400 Ω and tail resistance of R2=2980 Ω. The charging time is __________
a) 1.9 μs
b) 1.14 μs
c) 2.0 μs
d) 0.5 μs
Explanation: The time taken for charging is approximately three times the time constant of the circuit.
T=3.0 R1 $$\frac{C_1 C_2}{C_1 + C_2}$$
T=3.0 x 400 x $$\frac{.02 * .001}{0.02+0.001}$$ x 10-6
T=1.14 μs.
3. Wave shape control is flexible and independent.
a) True
b) False
Explanation: Wave shape control is flexible and independent. Wave shape gets changed with the change in test object. But, the circuit is simple and hence is advantageous.
4. The transient voltage may be __________
a) an oscillatory wave
b) a damped oscillatory wave
c) critically damped wave
d) an oscillatory wave or a damped oscillatory wave
Explanation: The transient voltage may be an oscillatory wave or a damped oscillatory wave. The frequency of the waves ranges from few hundred hertz to few kilo hertz. Transient voltage can also be considered as a slow rising impulse.
5. What is the time constant of the wave shaping circuit?
a) $$t=R_1 \frac{C_1 C_2}{C_1 + C_2}$$
b) $$t=2 R_1 \frac{C_1 C_2}{C_1 + C_2}$$
c) $$t=3R_1 \frac{C_1 C_2}{C_1 + C_2}$$
d) $$t=R_1 \frac{C_1 C_2}{2C_1 + C_2}$$
Explanation: The time constant of the wave shaping circuit is $$=R_1 \frac{C_1 C_2}{C_1 + C_2}$$. R1 is the resistance, C1 is the generator capacitance and C2 is the load capacitance. The circuit inductance is very small and hence is not taken into account.
6. What is the time taken for charging the capacitance in the wave shape control circuit?
a) $$t=R_1 \frac{C_1 C_2}{C_1 + C_2}$$
b) $$t=2 R_1 \frac{C_1 C_2}{C_1 + C_2}$$
c) $$t=3R_1 \frac{C_1 C_2}{C_1 + C_2}$$
d) $$t=R_1 \frac{C_1 C_2}{2C_1 + C_2}$$
Explanation: The time taken for charging the capacitance in the wave shape control circuit is $$t=3R_1 \frac{C_1 C_2}{C_1 + C_2}$$. R1 is the resistance, C1 is the generator capacitance and C2 is the load capacitance. The time taken for charging is thrice the time constant of the circuit. The circuit inductance is very small and hence is not taken into account.
7. What is the effective capacitance in wave shaping circuit?
a) $$C=\frac{C_1 C_2}{C_1 + C_2}$$
b) $$C=2 \frac{C_1 C_2}{C_1 + C_2}$$
c) $$C=\frac{C_1 C_2}{2C_1 + 2C_2}$$
d) $$C=3 \frac{C_1 C_2}{C_1 + C_2}$$
Explanation: The effective capacitance in wave shaping circuit is $$C=\frac{C_1 C_2}{C_1 + C_2}$$, C1 is the generator capacitance and C2 is the load capacitance. The circuit inductance is very small and hence is not taken into account.
8. What is the time taken for discharging in the wave shape control circuit?
a) $$t=(R_1+R_2) (C_1 + C_2)$$
b) $$t=.7(R_1+R_2) (C_1 + C_2)$$
c) $$t=3R_1 \frac{C_1 C_2}{C_1 + C_2}$$
d) $$t=R_1 \frac{C_1 C_2}{C_1 + C_2}$$
Explanation: The time taken for discharging in the wave shape control circuit is $$t=.7(R_1 + R_2)(C_1 + C_2)$$. It is also known as the tail time of the circuit. R1 and R2 is the resistance, C1 is the generator capacitance and C2 is the load capacitance. The time taken for charging is thrice the time constant of the circuit. The circuit inductance is very small and hence is not taken into account.
9. The generator capacitance is dependent on the design of the generator.
a) True
b) False
Explanation: The generator capacitance is dependent on the design of the generator. The load capacitance is also fixed for a given design of the generator and test object. So to obtain the desired wave shape only the resistance can be controlled.
10. An impulse voltage generator has a load capacitance C2 of 0.001 μF, front resistance of R1=400 Ω, tail resistance of R2=2980 Ω and tail time 49.68 μs.The generator capacitance C1 of is __________
a) 0.001 μF
b) 0.2 μF
c) 0.02 μF
d) 0.03 μF
Explanation: Tail time is the discharging time. The capacitances C1 and C2 may be considered to be in parallel and discharging occurs through R1 and R2. The tailing time is given by the equation $$t=.7 (R_1 + R_2)(C_1 + C_2)$$
Hence, the generator capacitance C1 can be found out by the following steps.
49.68 = $$.7(400+2980)(C_1 + 0.001)$$
C1 = 0.02 μF.
Sanfoundry Global Education & Learning Series – High Voltage Engineering.
To practice all areas of High Voltage Engineering, here is complete set of Multiple Choice Questions and Answers.
If you find a mistake in question / option / answer, kindly take a screenshot and email to [email protected]
Subscribe to our Newsletters (Subject-wise). Participate in the Sanfoundry Certification contest to get free Certificate of Merit. Join our social networks below and stay updated with latest contests, videos, internships and jobs! | 1,779 | 5,436 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.671875 | 4 | CC-MAIN-2024-38 | latest | en | 0.783956 |
https://gmatclub.com/forum/1-unit-of-x-alcohol-is-mixed-with-3-units-of-y-alcohol-to-give-296986.html?kudos=1 | 1,579,347,581,000,000,000 | text/html | crawl-data/CC-MAIN-2020-05/segments/1579250592565.2/warc/CC-MAIN-20200118110141-20200118134141-00323.warc.gz | 464,956,258 | 152,991 | GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video
It is currently 18 Jan 2020, 04:38
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
# 1 unit of x% alcohol is mixed with 3 units of y% alcohol to give 60%
Author Message
TAGS:
### Hide Tags
General GMAT Forum Moderator
Joined: 15 Jan 2018
Posts: 732
Concentration: General Management, Finance
GMAT 1: 720 Q50 V37
1 unit of x% alcohol is mixed with 3 units of y% alcohol to give 60% [#permalink]
### Show Tags
31 May 2019, 11:00
1
8
00:00
Difficulty:
95% (hard)
Question Stats:
33% (03:28) correct 67% (02:46) wrong based on 55 sessions
### HideShow timer Statistics
1 unit of x% alcohol is mixed with 3 units of y% alcohol to give 60% alcohol. If x > y, how many integer values can x take?
A. 10
B. 20
C. 35
D. 13
E. 30
_________________
General GMAT Forum Senior Moderator | D-Day: March 29, 2020
Should I retake GMAT? Retaking GMAT Strategies!
What Are My Chances - Automated Profile Evaluation Tool is Here!
VP
Joined: 19 Oct 2018
Posts: 1289
Location: India
Re: 1 unit of x% alcohol is mixed with 3 units of y% alcohol to give 60% [#permalink]
### Show Tags
31 May 2019, 12:39
3
x.........y
....60....
1.........3
$$\frac{x-60}{60-y}$$=3/1
x+3y=240...(1)
x=240-3y=3(80-y)
We have 2 constraints
1. x=<100
2. x>y
Also, x will be a multiple of 3
x can be 99,96.....63
At x=60, y=60....Hence x can't take value equal to 60 or lower than 60.
Total number of integer values x can take= [$$\frac{99-63}{3}$$]+1=13
DisciplinedPrep wrote:
1 unit of x% alcohol is mixed with 3 units of y% alcohol to give 60% alcohol. If x > y, how many integer values can x take?
A. 10
B. 20
C. 35
D. 13
E. 30
##### General Discussion
Manager
Joined: 25 Mar 2018
Posts: 89
Location: India
GMAT 1: 640 Q50 V26
GMAT 2: 640 Q50 V26
GMAT 3: 650 Q50 V28
GMAT 4: 690 Q50 V31
GMAT 5: 730 Q50 V40
GPA: 4
WE: Analyst (Manufacturing)
Re: 1 unit of x% alcohol is mixed with 3 units of y% alcohol to give 60% [#permalink]
### Show Tags
31 May 2019, 21:30
2
Question is clear but OA is surprising.
The equation on which question depends is
x + 3y = 4*60
So we can start x with 61, which is integer value. We don't bother about Y. Y can take any value integer or not.
So IMHO answer is (100-61)+1 , which is 40.
Posted from my mobile device
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9983
Location: Pune, India
Re: 1 unit of x% alcohol is mixed with 3 units of y% alcohol to give 60% [#permalink]
### Show Tags
13 Jun 2019, 03:51
1
DisciplinedPrep wrote:
1 unit of x% alcohol is mixed with 3 units of y% alcohol to give 60% alcohol. If x > y, how many integer values can x take?
A. 10
B. 20
C. 35
D. 13
E. 30
1 part x% alcohol + 3 parts y% alcohol -> 60% alcohol
Since x > y, x must be greater than 60% and y must be less than 60% so that they average out to 60.
The maximum value of x can be 100% (the entire solution can be alcohol).
So x can take any integer value from 61 to 100.
But it seems that the intent of the question is that x and y both should be integer values.
y --- 60 --------- x
Since we are mixing 1 part x with 3 parts y, average will be 1 unit away from y and 3 units away from x.
So if y = 59, x = 63
if y = 58, x = 66
and so on...
x will take values from 63% to 99% i.e. (3*21) to (3*33)
This means (33 - 21 = )12 + 1 = 13 multiples of 3
Check this post for the scale method of weighted avgs:
https://www.veritasprep.com/blog/2011/0 ... -averages/
_________________
Karishma
Veritas Prep GMAT Instructor
Manager
Joined: 15 Jan 2019
Posts: 63
Location: India
GMAT 1: 650 Q42 V38
GPA: 4
Re: 1 unit of x% alcohol is mixed with 3 units of y% alcohol to give 60% [#permalink]
### Show Tags
31 May 2019, 20:43
nick1816 wrote:
x.........y
....60....
1.........3
$$\frac{x-60}{60-y}$$=3/1
x+3y=240...(1)
x=240-3y=3(80-y)
We have 2 constraints
1. x=<100
2. x>y
Also, x will be a multiple of 3
x can be 99,96.....63
At x=60, y=60....Hence x can't take value equal to 60 or lower than 60.
Total number of integer values x can take= [$$\frac{99-63}{3}$$]+1=13
DisciplinedPrep wrote:
1 unit of x% alcohol is mixed with 3 units of y% alcohol to give 60% alcohol. If x > y, how many integer values can x take?
A. 10
B. 20
C. 35
D. 13
E. 30
I don't get it. x and y are concentration in percent. 3 and 1 are the value of units.
the 2 equations will be
1x + 3y = ?
x+y=60.
So we have to find to the value of x for which it is greater than y and the first equation gives a valid number.
right?
Manager
Joined: 11 Aug 2017
Posts: 59
Re: 1 unit of x% alcohol is mixed with 3 units of y% alcohol to give 60% [#permalink]
### Show Tags
31 May 2019, 23:14
nick1816 wrote:
x.........y
....60....
1.........3
$$\frac{x-60}{60-y}$$=3/1
x+3y=240...(1)
x=240-3y=3(80-y)
We have 2 constraints
1. x=<100
2. x>y
Also, x will be a multiple of 3
x can be 99,96.....63
At x=60, y=60....Hence x can't take value equal to 60 or lower than 60.
Total number of integer values x can take= [$$\frac{99-63}{3}$$]+1=13
DisciplinedPrep wrote:
1 unit of x% alcohol is mixed with 3 units of y% alcohol to give 60% alcohol. If x > y, how many integer values can x take?
A. 10
B. 20
C. 35
D. 13
E. 30
Why x has to be less than or equ. 100? i could not understand. please guide??
Manager
Joined: 01 Oct 2018
Posts: 112
Re: 1 unit of x% alcohol is mixed with 3 units of y% alcohol to give 60% [#permalink]
### Show Tags
12 Jun 2019, 17:51
nick1816 wrote:
x.........y
....60....
1.........3
$$\frac{x-60}{60-y}$$=3/1
x+3y=240...(1)
x=240-3y=3(80-y)
We have 2 constraints
1. x=<100
2. x>y
Also, x will be a multiple of 3
x can be 99,96.....63
At x=60, y=60....Hence x can't take value equal to 60 or lower than 60.
Total number of integer values x can take= [$$\frac{99-63}{3}$$]+1=13
DisciplinedPrep wrote:
1 unit of x% alcohol is mixed with 3 units of y% alcohol to give 60% alcohol. If x > y, how many integer values can x take?
A. 10
B. 20
C. 35
D. 13
E. 30
Why x has to be less than or equ. 100? i could not understand. please guide??
Maximum value of aclhogol in unit is 100%
Re: 1 unit of x% alcohol is mixed with 3 units of y% alcohol to give 60% [#permalink] 12 Jun 2019, 17:51
Display posts from previous: Sort by | 2,304 | 6,682 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.03125 | 4 | CC-MAIN-2020-05 | latest | en | 0.821736 |
https://www.saralstudy.com/study-eschool-ncertsolution/11th/mathematics/introduction-to-3-dimensional-geometry | 1,721,411,505,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514917.3/warc/CC-MAIN-20240719170235-20240719200235-00652.warc.gz | 832,456,294 | 10,357 | # Chapter 12 Introduction to 3 dimensional Geometry
After the knowledge of 2d geometry, it is time to add a new dimension to the geometry after that it is called 3d geometry. This is just an introduction but details of this chapter is in the next class. This chapter is for basics. This chapter consists of coordinate axis and coordinate planes in three dimensions, coordinates of a point, distance between two points and section formula.
Download pdf of NCERT Solutions for Class Mathematics Chapter 12 Introduction to 3 dimensional Geometry
Download pdf of NCERT Examplar with Solutions for Class Mathematics Chapter 12 Introduction to 3 dimensional Geometry
### Exercise 1
• Q1 A point is on the x-axis. What are its y-coordinates and z-coordinates? Ans: If a point is on the x-axis, then its y-coordinates and z-coordinates are zero. Q2 A point is in the XZ-plane. What can you say about its y-coordinate? Ans: If a point is in the XZ plane, then its y-coordinate is zero. Q3 Name the octants in which the following points lie: (1, 2, 3), (4, –2, 3), (4, –2, –5), (4, 2, –5), (–4, 2, –5), (–4, 2, 5), (–3, –1, 6), (2, –4, –7) Ans: The x-coordinate, y-coordinate, and z-coordinate of point (1, 2, 3) are all positive. Therefore, this point lies in octant I. The x-coordinate, y-coordinate, and z-coordinate of point (4, –2, 3) are positive, negative, and positive respectively. Therefore, this point lies in octant IV. The x-coordinate, y-coordinate, and z-coordinate of point (4, –2, –5) are positive, negative, and negative respectively. Therefore, this point lies in octant VIII. The x-coordinate, y-coordinate, and z-coordinate of point (4, 2, –5) are positive, positive, and negative respectively. Therefore, this point lies in octant V. The x-coordinate, y-coordinate, and z-coordinate of point (–4, 2, –5) are negative, positive, and negative respectively. Therefore, this point lies in octant VI. The x-coordinate, y-coordinate, and z-coordinate of point (–4, 2, 5) are negative, positive, and positive respectively. Therefore, this point lies in octant II. The x-coordinate, y-coordinate, and z-coordinate of point (–3, –1, 6) are negative, negative, and positive respectively. Therefore, this point lies in octant III. The x-coordinate, y-coordinate, and z-coordinate of point (2, –4, –7) are positive, negative, and negative respectively. Therefore, this point lies in octant VIII. | 660 | 2,401 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 2, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.75 | 5 | CC-MAIN-2024-30 | latest | en | 0.774587 |
https://coderanch.com/t/35449/solve-simple-mathematics | 1,477,080,820,000,000,000 | text/html | crawl-data/CC-MAIN-2016-44/segments/1476988718303.21/warc/CC-MAIN-20161020183838-00303-ip-10-171-6-4.ec2.internal.warc.gz | 825,826,580 | 9,242 | Win a copy of Penetration Testing Basics this week in the Security forum!
# solve it - simple mathematics!
Vikrama Sanjeeva
Ranch Hand
Posts: 760
A person was about to die, and before dying he wrote his Will which went as
follows ...
"I have 17 Camels, and I have three sons. Divide my Camels in such a way,
that My eldest son gets half of them, the second one gets 1/3rd of the
total and my youngest son gets 1/9th of the total number of Camels"
How will you divide, so that every son get the right number of camels which father wrote in will?
Bye,
Viki.
Scott Johnson
Ranch Hand
Posts: 518
1st son: 17 * 1/2 = 8.500 ==> round up ==> 9
2nd son: 17 * 1/3 = 5.667 ==> round up ==> 6
3rd son: 17 * 1/9 = 1.889 ==> round up ==> 2
Checking: 9 + 6 + 2 = 17
The fathers will has been satisifed because all sons received the proper fraction (plus a little more.)
Shyju Lakshman
Greenhorn
Posts: 4
Actually i have one camel which i shall add to the total . now there are 17+1 = 18 camels.
1st son ==> 18/2 = 9 camels
2nd son ==> 18/3 = 6 camels
3rd son ==> 18/9 = 2 camels
9+6+2= 17 camels.. one camel is left which is mine.. so i am happy, and the father is happy and the sons are happy
Ryan McGuire
Ranch Hand
Posts: 1083
4
According to the problem statement, that's an invalid solution. The will specifically says, "Divide up my camels in such a way that my eldest son gets half of them..." Adding your camel to the original 17 is going against the father's request.
Besides, the father may have realized that he has allocated only 17/18 of his 17 camels and wants the remaining 17/18 of a camel to be used to pay the executor of his estate (you). Mmmm... a half ton of camel jerky.
[ August 07, 2006: Message edited by: Ryan McGuire ]
Vikrama Sanjeeva
Ranch Hand
Posts: 760
Originally posted by Shyju Lakshman:
Actually i have one camel which i shall add to the total . now there are 17+1 = 18 camels.
1st son ==> 18/2 = 9 camels
2nd son ==> 18/3 = 6 camels
3rd son ==> 18/9 = 2 camels
9+6+2= 17 camels.. one camel is left which is mine.. so i am happy, and the father is happy and the sons are happy
100% | 659 | 2,118 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.0625 | 4 | CC-MAIN-2016-44 | longest | en | 0.957755 |
geoffprewett.com | 1,713,841,324,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296818452.78/warc/CC-MAIN-20240423002028-20240423032028-00481.warc.gz | 13,714,626 | 5,522 | ## Of Bezier Curves and Teapots
Bezier curves always sounded like something super-advanced, something that only mathematically-minded, skilled professionals who wrote Path Routines for operating systems wrote. After much searching and reading, I found out that one of the reasons Bezier curves are so popular is that they are so easy. Unfortunately, a lot of the articles out there are heavy on the math but light on the implementation.
A search on Bezier curves will quickly inform you that two French guys named De Casteljau and Bezier independently invented the Bezier curve in order to design cars, which have all sorts of curves which are not easily represented by normal functions. You can find plenty of discussion on De Castlejau’s algorithm, along with a nice animation cribbed from Wikipedia, but they inevitably end with “but don’t use this in real life.” And you can also find the equation for Bezier’s algorithm. But pseudocode? Nowhere.
Eventually I realized that nobody gives pseudocode because the the equation is the algorithm:
n Pi is the i-th control point B(u) is a Bernstein coefficient P(u) = Σ Bin(u) Pi i = 0
``````
def bezierCurveAt(u, points):
p = [0.0, 0.0] # assuming 2D control points
# Note that n in the summation is len(points) - 1.
# Also note that in Python, xrange(0, m) goes from 0, 1, ... m - 1.
for i in xrange(0, len(points)):
B = bernstein(i, len(points) - 1, u)
p[0] += B * points[i][0]
p[1] += B * points[i][1]
return p
def calcCurve(controlPoints, nSegments):
points = []
for i in xrange(0, nSegments + 1):
u = float(i) / float(nSegments)
points.append(bezierCurveAt(u, controlPoints))
``````
So what are these Bernstein coefficients? Well, from a coding standpoint, they are really easy:
Bin(u) = n! ui (1 - u)n-i i! (n - i)!
``````
def bernstein(i, n, u):
binomial = float(math.factorial(n)) / (float(math.factorial(i)) * float(math.factorial(n - i)))
return binomial * math.pow((1.0 - u), n - i) * math.pow(u, i)
``````
### The Theory
That is not very satisfying from an understanding standpoint, however. Over the years I have observed that there are a bunch of magic functions that show up in math. Can’t integrate a Gaussian function? No problem, use the error function, problem magically solved! When I studied Physics, we were happily deriving the equations of the spherical model of a hydrogen atom (well, the professor was happily deriving it, I was copying Greek letters as fast as possible, and reading the book after the lecture to figure out what was going on). The equations cannot be solved with the methods of solving differential equations familiar to undergraduate students. No problem, Legendre polynomials to the rescue! We spent a week deriving the hydrogen atom, but nobody bothered explaining Legendre polynomials. The book, which was quite good, said something to the effect of “conveniently, it happens that this equation can be solved with Legendre polynomials.” We, however, are not going to take that easy road.
Bernstein coefficients belong to the Bernstein polynomials, which are named after Sergei Bernstein, who created them (or discovered, depending on your philosophical perspective on mathematics) to prove that any curve can be approximated by polynomials. Basically it is the same idea as a Fourier series, which approximates any periodic (cyclical) function as a sum of sines or cosines. A Fourier series approximates a periodic function as
f(x) ≈ A + B sin(αx) + C sin(2αx) + D sin(3αx) + ... Xn sin(nαx)
(where A, B, etc. are appropriate constants). Bernstein polynomials do the same sort of thing, but for any function, not just periodic ones:
f(x) ≈ Ax0 + Bx1 + Cx2 + ... Xn xn
It is only valid over the range [0, 1], but that is not a huge problem, just scale and translate your function appropriately. Approximating a function is pretty much exactly the use case that Bezier was trying to solve—a digital approximation to the smooth curves of cars—so it is not surprising that Bernstein polynomials show up in Bezier curves. Indeed, one would almost expect it.
### Bezier Surfaces
Bezier curves turn out to be so easy that we still have time to look at Bezier surfaces, which are basically a 3D mesh, instead of a 2D line. I recently wanted a model of the famous Utah Teapot for computer graphics (the so-called “teapotahedron”), but it turns out it is defined as a series of Bezier surfaces. I was completely unable to find a teapot model that used vertices that I could just send to glDrawArrays. Since I was intimidated by the fancy name “bezier curve,” I was going to compile Mesa and modify glEvalMesh2 to spit out the vertices that glutSolidTeapot generated. Hah! Too much effort. Let’s just write a short Python script.
n m P(u, v) = Σ Σ Bin(u) Bjm(v) Pij i = 0 j = 0
``````
def bezierSurfaceAt(u, v, points, nUPoints, nVPoints):
p = [0.0, 0.0, 0.0] # assuming 3D control points
for j in xrange(0, nVPoints):
for i in xrange(0, nUPoints):
Bi = bernstein(i, nUPoints - 1, u)
Bj = bernstein(j, nVPoints - 1, v)
cp = points[j * nUPoints + i]
p[0] += Bi * Bj * cp[0]
p[1] += Bi * Bj * cp[1]
p[2] += Bi * Bj * cp[2]
return p
``````
This gives us the vertices, but we still need the normals. The normal is simply the tangent in the u direction crossed with the tangent in the v direction. Since the tangent of a curve is simply its derivative,
N(u, v) = ∂ P(u, v) ⨉ ∂ P(u, v) ∂u ∂v
It turns out that the derivative of a Bernstein polynomial is a sum of lower order Bernstein polynomials:
∂ Bin(t) = n [ Bi-1n-1(t) - Bin-1(t)], where B-1n-1(t) = Bnn-1(t) = 0 ∂t
Thus our normal is:
n m n m N(u, v) = Σ Σ n [Bi-1n-1(u) - Bin-1(u)] Bjm(v) Pij ⨉ Σ Σ Bin(u) m [Bj-1m-1(v) - Bjm-1(v)] Pij i = 0 j = 0 i = 0 j = 0
So we can calculate our normals with a straightforward implementation of the equation:
``````
def bezierNormalAt(u, v, points, nUPoints, nVPoints):
uTangent = [0.0, 0.0, 0.0]
for j in xrange(0, nVPoints):
for i in xrange(0, nUPoints):
Bi1 = bernstein(i - 1, nUPoints - 2, u)
Bi2 = bernstein(i, nUPoints - 2, u)
Bj = bernstein(j, nVPoints - 1, v)
a = nUPoints * (Bi1 - Bi2) * Bj
cp = points[j * nUPoints + i]
uTangent[0] += a * cp[0]
uTangent[1] += a * cp[1]
uTangent[2] += a * cp[2]
vTangent = [0.0, 0.0, 0.0]
for j in xrange(0, nVPoints):
for i in xrange(0, nUPoints):
Bi = bernstein(i, nUPoints - 1, u)
Bj1 = bernstein(j - 1, nVPoints - 2, v)
Bj2 = bernstein(j, nVPoints - 2, v)
a = nUPoints * Bi * (Bj1 - Bj2)
cp = points[j * nUPoints + i]
vTangent[0] += a * cp[0]
vTangent[1] += a * cp[1]
vTangent[2] += a * cp[2]
return normalized(crossProduct(uTangent, vTangent))
``````
I have packaged this up in a short Python script (teapot.py) which will generate the vertices, normals, and vertex ids for the Utah teapot. These are suitable for passing directly to glDrawElements() with GL_VERTICES. | 1,966 | 6,826 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.9375 | 4 | CC-MAIN-2024-18 | latest | en | 0.838371 |
https://www.gradesaver.com/textbooks/math/trigonometry/CLONE-68cac39a-c5ec-4c26-8565-a44738e90952/chapter-5-trigonometric-identities-section-5-6-half-angle-identities-5-6-exercises-page-243/55 | 1,726,790,068,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700652073.91/warc/CC-MAIN-20240919230146-20240920020146-00192.warc.gz | 719,565,180 | 12,959 | ## Trigonometry (11th Edition) Clone
$\frac{sin~x}{1+cos~x} = tan~\frac{x}{2}$
$\frac{sin~x}{1+cos~x}$ When we graph this function, we can see that it looks like the graph of $~~tan~\frac{x}{2}$ We can verify this algebraically: $\frac{sin~x}{1+cos~x} = \frac{sin~(\frac{x}{2}+\frac{x}{2})}{1+cos~(\frac{x}{2}+\frac{x}{2})}$ $\frac{sin~x}{1+cos~x} = \frac{sin~\frac{x}{2}~cos~\frac{x}{2}+cos~\frac{x}{2}~sin~\frac{x}{2}}{1+cos~\frac{x}{2}~cos~\frac{x}{2}-sin~\frac{x}{2}~sin~\frac{x}{2}}$ $\frac{sin~x}{1+cos~x} = \frac{2~sin~\frac{x}{2}~cos~\frac{x}{2}}{1-sin^2~\frac{x}{2}+cos^2~\frac{x}{2}}$ $\frac{sin~x}{1+cos~x} = \frac{2~sin~\frac{x}{2}~cos~\frac{x}{2}}{cos^2~\frac{x}{2}+cos^2~\frac{x}{2}}$ $\frac{sin~x}{1+cos~x} = \frac{2~sin~\frac{x}{2}~cos~\frac{x}{2}}{2~cos^2~\frac{x}{2}}$ $\frac{sin~x}{1+cos~x} = \frac{sin~\frac{x}{2}}{cos~\frac{x}{2}}$ $\frac{sin~x}{1+cos~x} = tan~\frac{x}{2}$ | 447 | 895 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.34375 | 4 | CC-MAIN-2024-38 | latest | en | 0.23313 |
https://www.vedantu.com/jee-main/in-one-dimensional-motion-instantaneous-speed-physics-question-answer | 1,708,663,239,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947474360.86/warc/CC-MAIN-20240223021632-20240223051632-00747.warc.gz | 1,085,942,786 | 27,154 | Courses
Courses for Kids
Free study material
Offline Centres
More
# In one dimensional motion, instantaneous speed satisfies the condition, $0\le v<{{v}_{0}}$, whenA) The displacement in time $T$ must always take non-negative valuesB) The displacement $x$ in time $T$ satisfies $-{{v}_{0}}T< x <{{v}_{0}}T$ C) The acceleration is always a non-negative numberD) The motion has no turning points
Last updated date: 22nd Feb 2024
Total views: 18.3k
Views today: 0.18k
Verified
18.3k+ views
Hint: Let us begin by understanding one-dimensional motion and instantaneous velocity. One dimensional motion applies to objects moving in straight lines. Speed is a measure of how quickly an object is moving along this line; it is a scalar quantity. Velocity is the speed with a direction, making it a vector. If an object’s velocity changes at a constant rate with time, the object is said to be accelerating. When studying motion along one dimension, there are only two possible directions for the velocity and acceleration vectors to point in.
Formula Used:
$v=\dfrac{dx}{dt}$
Complete step by step solution:
We know that velocity is the derivative of displacement.
Mathematically, we can say that $v=\dfrac{dx}{dt}$
Transposing sides, we can say that $dx=v.dt$
Integrating both sides of the above equation, we get
\begin{align} & \int\limits_{0}^{x}{dx}={{v}_{0}}\int\limits_{0}^{T}{dt} \\ & \Rightarrow \left[ x \right]_{0}^{x}={{v}_{0}}\left[ t \right]_{0}^{T} \\ & \Rightarrow x={{v}_{0}}T \\ \end{align}
As discussed in the hint, velocity in one-dimensional motion may be positive or negative. If the velocity is negative, the displacement will be negative as well.
So, displacement cannot always take non-negative values but can vary between the negative and the positive values, like $-{{v}_{0}}T < x < {{v}_{0}}T$.
Hence we can say that option (B) is the correct answer.
Note: We have already seen that option (A) cannot be the correct answer as the displacement can vary between positive and negative values. Similarly, acceleration is the rate of change of velocity and if the velocity is negative, the acceleration would be negative as well. Hence acceleration cannot always be a non-negative number. Whenever the velocity changes from a positive value to a negative value or vice-versa, there is a change in the direction of motion or a turning point is encountered. Hence option (D) is also incorrect. | 600 | 2,411 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 2, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.71875 | 5 | CC-MAIN-2024-10 | longest | en | 0.826387 |
https://richardvigilantebooks.com/what-are-the-conditions-for-paralleling-two-3-phase-transformer/ | 1,685,366,329,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224644855.6/warc/CC-MAIN-20230529105815-20230529135815-00048.warc.gz | 545,092,190 | 10,675 | # What are the conditions for paralleling two 3 phase transformer?
## What are the conditions for paralleling two 3 phase transformer?
Condition for Parallel Operation of Three Phase Transformers The line voltage ratio of the transformers must be same. The transformers should have equal per unit leakage impedance. (You may read per unit system) The ratio of equivalent leakage reactance to equivalent resistance should be same for all the transformers.
Can you connect 2 transformers in parallel?
It is possible to connect transformers that have different polarities in parallel. The supply line polarity determines the primary polarity of the transformer. The primary polarity determines the secondary polarity of the transformer.
### What is parallel operation of transformers?
When we connect the primary windings of two transformers to a common supply voltage and the secondary windings of both the transformers to a common load, this type of connection of transformer is said to be the parallel operation of transformers.
How do two transformers work in parallel?
If two or more transformers are connected in parallel, then load sharing per centage between them is according to their rating. If all are of same rating, they will share equal loads.
#### Which condition for parallel operation of 3 phase transformer is incorrect?
Do not connect the transformers at incorrect polarity in parallel, incorrect polarity results a dead short circuit. Hence, both the transformer will be damaged.
How do you connect two transformers together?
Series Connected Secondary Transformer As the two windings are connected in series, the same amount of current flows through each winding, then the secondary current is the same at 2.5 Amps. So for a series connected secondary, the output in our example above is rated at 24 Volts, 2.5 Amps.
## Which polarity is used for parallel operation of transformer?
In perfect parallel operation of two or more transformers, current in each transformer would be directly proportional to the transformer capacity, and the arithmetic sum would equal one-half the total current. Any combination of positive and negative polarity transformers can be used.
When two transformers operating parallel the following conditions must be satisfied?
When two or more transformers run in parallel, they must satisfy the following conditions for satisfactory performance. These are the conditions for parallel operation of transformers. Same voltage ratio of transformer. Same percentage impedance.
### Can current transformers be connected in series?
A current transformers primary coil is always connected in series with the main conductor giving rise to it also being referred to as a series transformer. The nominal secondary current is rated at 1A or 5A for ease of measurement.
What is the resistance between primary and secondary in transformer?
In a transformer the coils are not electricaly connected therefore the resistance is ideally infinite.
#### Can a parallel transformer be used in a new transformer?
Parallel in a new transformer. It is sometimes more practical to parallel in a new transformer as there is minimal downtime to the operation. Three conditions must be met before you can connect transformers in parallel. 1. The transformers must have the same primary and secondary voltage ratings.
How is the impedance of Transformers in parallel related?
From the above two statements it can be said that impedance of transformers running in parallel are inversely proportional to their MVA ratings. In other words percentage impedance or per unit values of impedance should be identical for all the transformers run in parallel.
## How is CEMF produced in paralleling Transformers?
Even though the voltages induced in the secondaries of the transformers are AC, the same circulating currents flow in each of the secondary windings. Any current flowing in the secondary of the transformer must be matched by a current in the primary so that the proper CEMF is produced in the primary windings.
Why do you parallel a subtractive polarity transformer?
When making the connections, you must observe the terminal polarity of the transformers. This still allows you to parallel a subtractive-polarity transformer with an additive-polarity transformer if you ensure that the connection terminals have the same instantaneous polarity. Figure 10. Circulating currents | 798 | 4,425 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.21875 | 4 | CC-MAIN-2023-23 | latest | en | 0.911427 |
http://www.komal.hu/verseny/feladat.cgi?a=honap&h=201203&t=mat&l=en | 1,532,243,128,000,000,000 | text/html | crawl-data/CC-MAIN-2018-30/segments/1531676593051.79/warc/CC-MAIN-20180722061341-20180722081341-00536.warc.gz | 470,215,987 | 9,098 | Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
Already signed up? New to KöMaL?
# KöMaL Problems in Mathematics, March 2012
Show/hide problems of signs:
## Problems with sign 'K'
Deadline expired on April 10, 2012.
K. 331. The task is to cross a desert with a car. The width of the desert is 600 km. The capacity of the petrol tank of the car is only enough for 400 km, and the car cannot carry fuel in any other way. The car can travel with an average speed of 50--60 km/h. It needs to start out at 8 a.m. and it is to arrive on the other side by 8 p.m. What is the minimum number of such cars needed, for one of them to be able to cross the desert, while the others return back? It is possible to transfer fuel from one car to another.
(6 pont)
solution (in Hungarian), statistics
K. 332. The sum of a few positive integers is 20. Their product is the number X. What is the maximum possible value of X?
(6 pont)
solution (in Hungarian), statistics
K. 333. The three interior angles of a triangle followed by the three exterior angles, in the appropriate order, form six consecutive terms of an arithmetic sequence. Find the measures of the angles of the triangle in degrees. (In an arithmetic sequence, the difference of the consecutive terms is constant, that is, the sequence increases uniformly.)
(6 pont)
solution (in Hungarian), statistics
K. 334. Alma measured and tabulated the distances (in steps) between fruit trees in the garden. How many steps is it from the walnut tree to the pear tree?
Apple Pear Walnut Almond 5 12 35 Apple 0 13 40
(6 pont)
solution (in Hungarian), statistics
K. 335. A peculiar calculator only has four buttons on it: (eight plus root seven), (addition), (reciprocal) and (equals) (see the figure).
The calculator always carries out the operations with exact values, and it can also store the current value as a constant if the button is pressed twice. That is, in subsequent calculations whenever the button is pressed, the number is incremented by that value any number of times. (E.g. if the buttons are pressed, it will display ). Prove that the result of the following sequence of operations is 1:
... (56 times)
... (15 times)
(6 pont)
solution (in Hungarian), statistics
K. 336. The concave 16-sided polygon in the figure is divided into four congruent squares and eight congruent rhombuses. The smaller angle of the rhombuses is 45 degrees. Given that the longer diagonal of the rhombuses is 10.7 cm, find the area of the 16-sided polygon.
(6 pont)
solution (in Hungarian), statistics
## Problems with sign 'C'
Deadline expired on April 10, 2012.
C. 1115. Show that n2(n2-1)(n2-n-2) is divisible by 48 for all natural numbers n.
(5 pont)
solution (in Hungarian), statistics
C. 1116. Each side of a convex quadrilateral is divided into eight equal parts. Each dividing point is connected to the corresponding dividing point on the opposite side, as shown in the figure. The small quadrilaterals obtained in this way are coloured black and white in a chessboard pattern. Prove that the total area of the black quadrilaterals equals the total area of the white quadrilaterals.
(5 pont)
solution (in Hungarian), statistics
C. 1117. We have drawn a rectangle on squared paper, such that (its sides are lattice lines and) it consists of n small lattice squares. Prove that if the half of the number of lattice points on the boundary of the rectangle is added to the number of lattice points in its interior, and 1 is subtracted from the sum then the result will be n.
(5 pont)
solution (in Hungarian), statistics
C. 1118. Solve the equation on the set of real numbers.
(5 pont)
solution (in Hungarian), statistics
C. 1119. Two identical set squares(of 30, 60 and 90 degrees) are fixed together along their shorter legs of length 10 cm, as shown in the figure. Then an isosceles set square is placed on top. Is there enough room for a tennis ball of radius 3.2 cm in the interior of the tetrahedron obtained in this way?
(5 pont)
solution (in Hungarian), statistics
## Problems with sign 'B'
Deadline expired on April 10, 2012.
B. 4432. There is a 98×98 chessboard on the monitor of a computer. With the mouse, you can select a rectangle bounded by lattice lines of the chessboard. If you click on the rectangle, each chessboard field inside it will change to the opposite colour. What is the minimum number of clicks needed to make the whole chessboard have the same colour.
(5 pont)
solution (in Hungarian), statistics
B. 4433. Solve the equation (1+x)8+(1+x2)4=82x4.
(3 pont)
solution (in Hungarian), statistics
B. 4434. Prove that every natural number not divisible by 10 can be multiplied by an appropriate natural number, such that the product is a palindromic number in decimal notation.
(4 pont)
solution (in Hungarian), statistics
B. 4435. T is the foot of the altitude drawn from vertex A of an acute-angled triangle ABC. The midpoint of side BC is F. The centres of the squares drawn over the sides AB and AC on the outside are K and L, respectively. Prove that KTFL is a cyclic quadrilateral.
(Suggested by Sz. Miklós, Herceghalom)
(4 pont)
solution (in Hungarian), statistics
B. 4436. Let x, y, z denote positive integers, such that . Prove that .
(Suggested by J. Mészáros, Jóka)
(4 pont)
solution (in Hungarian), statistics
B. 4437. Given the centres of the circumscribed circle and two escribed circles of a triangle, construct the triangle.
(4 pont)
solution (in Hungarian), statistics
B. 4438. The angle bisectors of a triangle ABC intersect the opposite sides at the points A1, B1 and C1, respectively. For what triangles is it true that ?
(Matlap, Cluj-Napoca -- Kolozsvár)
(3 pont)
solution (in Hungarian), statistics
B. 4439. Given two different points B and C in the plane, determine the locus of those points A for which the altitude drawn from vertex A of the triangle ABC is the geometric mean of the line segments BC+AC and BC-AC.
(3 pont)
solution (in Hungarian), statistics
B. 4440. On a winter's day, an absent minded mathematician went on a walk with his old poodle along a long straight alley. He was so absorbed in his thoughts that when he finally thought of his dog, he had to realize that it was not near him. In the snowfall, he did not see further than 5 metres. He did not see the dog in front of him, he did not see it behind either, and he did not know which direction it had gone. After a little thinking, he went to look for his poodle. The old poodle can only walk half as fast as its master. The mathematician chose a searching strategy, such that the following condition should hold for the smallest possible value of the constant c: if the dog is at a distance x from him then he needs to cover a distance of at most cx to find it. What is this smallest value of c?
(5 pont)
solution (in Hungarian), statistics
B. 4441. The areas of the faces of a tetrahedron are a, b, c, d. The angle enclosed by the faces of areas a and b is , the angle of b and c is , and the angle of c and a is . Prove that d2=a2+b2+c2-2ab.cos -2bc.cos -2ca.cos .
(5 pont)
solution (in Hungarian), statistics
## Problems with sign 'A'
Deadline expired on April 10, 2012.
A. 557. Show that the positive integers can be coloured with three colours in such a way that the equation x+y=z2 has no solution (x,y,z) consisting of distinct numbers with the same colour.
(Kolmogorov's Cup, 2011; a problem by F. Petrov and I. Bogdanov)
(5 pont)
statistics
A. 558. Prove that there exists a constant C>0 for which the following statement holds: if n is a positive integer and are sets such that every two of them has at least two, and every three of them has at most three elements in common, then N<Cn2.
(Proposed by: Z. Gyenes, Budapest)
(5 pont)
statistics
A. 559. The incircle of triangle ABC is k. The circle kA touches k and the segments AB and AC at at A', AB and AC respectively. The circles kB, kC and the points B', C' are defined analogously. The second intersection point of the circles A'B'AB and A'C'AC, other than A', is K. The line A'K meets k at R, other than A'. Prove that R lies on the radical axis of the circles kB and kC.
(Kolmogorov's Cup, 2011; a problem by F. Ivlev)
(5 pont)
statistics | 2,184 | 8,302 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.859375 | 4 | CC-MAIN-2018-30 | latest | en | 0.915814 |
https://www.physicsforums.com/threads/binary-to-hex-conversion-proof.347015/ | 1,513,241,212,000,000,000 | text/html | crawl-data/CC-MAIN-2017-51/segments/1512948542031.37/warc/CC-MAIN-20171214074533-20171214094533-00440.warc.gz | 774,816,176 | 17,775 | # Binary to hex conversion proof
1. Oct 19, 2009
### njama
Hello PF members!
I am interested in proving that the conversion of binary to hex is valid.
Example: 1000100100110111
Breaking into 'quartets' 1000 1001 0011 0111
Now using the fact that 24 = 0...15
1000 = 8, 1001 = 9, 0011 = 3, 0111 = 7. So
10001001001101112=893716
I can create the table starting from 0000 to 1111. I need to prove that this conversion really works.
I would start the proof by stating that the number in binary form is:
$$b_1b_2...b_{n-1}b_n$$
If n mod 4 == 0 (if the remainder of the division of n with 4 is zero) then:
$$b_1b_2b_3b_4 .... .... .... b_{n-3}b_{n-2}b_{n-1}b_n$$
But how to prove afterward?
else we can add m zeroes so that (m+n) mod 4 ==0.
By adding zeroes before the binary number I can easily prove that nothing is changed.
$$(*)=b_1b_2...b_{n-1}b_n=b_n*2^0 + b_{n-1}*2^1+...+b_2*2^{n-2}+b_1*2^{n-1}$$
$$00..00b_1b_2...b_{n-1}b_n=b_n*2^0 + b_{n-1}*2^1+...+b_2*2^{n-2}+b_1*2^{n-1}+0*2^n+0*2^{n+1}...+0*2^{n+m}=(*)$$
Any help would be appreciated.
2. Oct 20, 2009
### DoctorBinary
This is not a proof but an example to show how you might think of it (I reversed the numbering of the coefficients in your example, to match the exponents):
A 12-digit binary number: $$b_{11}2^{11}+b_{10}2^{10}+b_{9}2^{9}+b_{8}2^{8}+b_{7}2^{7}+b_{6}2^{6}+b_{5}2^{5}+b_{4}2^{4}+b_{3}2^{3}+b_{2}2^{2}+b_{1}2^{1}+b_{0}2^{0}$$
Factor out multiples of 16: $$2^8(b_{11}2^{3}+b_{10}2^{2}+b_{9}2^{1}+b_{8}2^{0})+2^4(b_{7}2^{3}+b_{6}2^{2}+b_{5}2^{1}+b_{4}2^{0})+2^0(b_{3}2^{3}+b_{2}2^{2}+b_{1}2^{1}+b_{0}2^{0})$$
Rewrite with powers of 16: $$16^2(b_{11}2^{3}+b_{10}2^{2}+b_{9}2^{1}+b_{8}2^{0})+16^1(b_{7}2^{3}+b_{6}2^{2}+b_{5}2^{1}+b_{4}2^{0})+16^0(b_{3}2^{3}+b_{2}2^{2}+b_{1}2^{1}+b_{0}2^{0})$$
Everything in parentheses is a hex digit, with value 0-15.
3. Oct 20, 2009
### njama
Ohh.... Thanks a lot.
Here is what I've done:
(I've excluded the case which I have already proof with the zeroes before the number)
So lets suppose there are m+n+1 digits, and we can form a quartets.
The binary number is:
$$b_{m+n}2^{m+n} + b_{m+n-1}2^{m+n-1} + ... + b_32^3+b_22^2+b_12^1+b_02^0$$
Now factoring out multiples of 16
$$2^{m+n-3}(b_{m+n}2^3+b_{m+n-1}2^2+...+b_{m+n-3}2^0)+....+2^0(b_32^3+b_22^2+b_12^1+b_02^0)$$
There are m+n / 4 hexadecimal digits. Did I do the proof correctly ? (except for minor mistakes)
4. Oct 21, 2009
### DoctorBinary
That looks pretty good, except I think you need to make it clear that the terms are powers of 16. That will make the notation cumbersome though. For example, $$2^{m+n-3}$$ would be $$(2^{4})^{(m+n+1)/4 - 1}$$. Not too pretty, but a cleaner way doesn't come to mind at the moment.
5. Oct 21, 2009
### zgozvrm
Try thinking about it this way:
Each more significant binary digit is represented by a 0 or 1 multiplied by a power of 2 that is greater than the previous digit by 1. In other words (sticking with whole numbers), the lowest binary digit can be represented by $$b_02^0$$
Remembering, of course that $$b_n=0$$ or $$b_n=1$$ for all n.
The next more significant binary digit can be represented by $$b_12^1$$
and so on... Then, for any binary number with more than 4 digits, you have: $$2^n , 2^{n-1}, ..., 2^3, 2^2, 2^1, 2^0$$
By breaking a large (more than 4 digits) binary number into quartets, and re-assigning them to values of 0..15, you are actually reassigning the appropriate power of 2 (the exponent) for all digits more significant than $$b_32^3$$
Obviously, for the 1st quartet (4 least significant bits), which is $$b_3b_2b_1b_0$$, the value is already correct (there is no reassignment of exponents) and the quartet's value is $$b_32^3+b_22^2+b_12^1+b_02^0$$
For the 2nd quartet, $$b_7b_6b_5b_4$$, you are essentially changing the exponents from $$2^7, 2^6, 2^5, and \phantom{1}2^4$$ to $$2^3, 2^2, 2^1, and\phantom{1}2^0$$
This results in the 2nd quartet having the value $$b_72^3+b_62^2+b_52^1+b_42^0$$ (The actual value is, of course: $$b_72^7+b_62^6+b_52^5+b_42^4$$)
What you've done here is subtracted 4 from each of the exponents in the 2nd quartet: $$b_72^{7-4}+b_62^{6-4}+b_52^{5-4}+b_42^{4-4}$$
Using one of the laws of exponents: $$\frac{X^a}{X^b}=X^{a-b}$$, we can show that 2nd quartet as: $$b_7\frac{2^7}{2^4}+b_6\frac{2^6}{2^4}+b_5\frac{2^5}{2^4}+b_4\frac{2^4}{2^4}$$
We can then factor out $$\frac{1}{2^4}$$, resulting in $$\frac{1}{2^4}(b_72^7+b_62^6+b_52^5+b_42^4)$$
And, since $$2^4=16$$, what you've effectively done, is divided the 2nd quartet by 16!
Extending this same technique out to the next quartet, you'll find that you will have divided that quartet by $$2^8=16^2$$
Remember also, that each more significant digit in hexidecimal (base 16) is a successively larger power of 16, such that you have
$$h_n16^n+h_{n-1}16^{n-1}+...+h_316^3+h_216^2+h_116^1+h_016^0$$
with $$h_n$$ = 0...9, A..F
6. Oct 21, 2009
### zgozvrm
So, for each quartet $$Q_n \phantom{0} (n\geqq1)$$, you obtain the hex digit by dividing the binary value of the quartet by $$16^{n-1}$$
7. Oct 23, 2009
### njama
Thank you for the help all of you.
So do I need to write it like this:
$$16^{(m+n-3)/4)}(b_{m+n}2^3+b_{m+n-1}2^2+...+b_{m+n-3}2^0)+16^{(m+n-3)/4 -1}(b_{m+n-4}2^3+...+b_{m+n-7}2^0)....+16^0(b_32^3+b_22^2+b_12^1+b_02^0)$$
8. Oct 23, 2009
### HallsofIvy
Staff Emeritus
Of course 100002= 16 so while 01112= (01112)(1)= (01112)(160)= 7(160), the second set, "00112", is (00112)(16)= (00112)(161)= 3(161, the third "10012 is (10012)(162)= 9(162), and the fourth "10002" is (10002)(163)= 7(163.
9. Oct 24, 2009
### njama
Ok, I believe to makes things better I should use m+n-1 = 4k
10. Oct 24, 2009
### zgozvrm
I would simplify things a little by first stating that any number (binary, or otherwise) can be led with any number of 0's and still retain the same value. That said, I would FIRST add the number of 0's needed (one, two, or three of them) to make the number of binary digits (bits) evenly divisible by 4. Then use "n" to represent the total number of bits in the binary number (including those added 0's, if any).
Then, you can write it as:
$$16^{\frac{n}{4}-1}(b_n2^3 + b_{n-1}2^2 + b_{n-2}2^1 + b_{n-3}2^0) + 16^{\frac{n}{4}-2}(b_{n-3}2^3 + b_{n-4}2^2 + b_{n-5}2^1 + b_{n-6}2^0) + ... + 16^1(b_72^3 + b_62^2 + b_52^1 + b_42^0) + 16^0(b_32^3 + b_22^2 + b_12^1 + b_02^0)$$ | 2,655 | 6,378 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.15625 | 4 | CC-MAIN-2017-51 | longest | en | 0.686474 |
https://web2.0calc.com/questions/counting_5223 | 1,723,600,044,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722641086966.85/warc/CC-MAIN-20240813235205-20240814025205-00393.warc.gz | 490,673,006 | 5,800 | +0
# counting
0
3
1
+52
The numbers x1, x2, x3, and x4 are chosen at random in the interval [0, 1] Let I be the interval between x1 and x2 and let J be the interval between x3 and x4 Find the probability that intervals I and J overlap.
it is not 1/2 or 1/12
Jul 14, 2024
#1
+739
0
Understanding the Problem
We have four random points on the unit interval [0, 1]. We form two intervals, I and J, from these points. We want to find the probability that these intervals overlap.
Approach
To calculate the probability, we'll consider the complementary event: the probability that the intervals do not overlap. If we can find this probability, we can simply subtract it from 1 to get the desired probability.
Calculating the Probability of Non-Overlapping Intervals
For the intervals to not overlap, one interval must completely lie to the left of the other. There are two possibilities for this:
I is entirely to the left of J:
x1 < x2 < x3 < x4
Probability of this arrangement is 1/4! (since there are 4! equally likely orderings of the four points).
J is entirely to the left of I:
x3 < x4 < x1 < x2
Probability of this arrangement is also 1/4!
The total probability of non-overlapping intervals is the sum of these two probabilities:
P(non-overlapping) = 1/4! + 1/4! = 2/4! = 1/12
Calculating the Probability of Overlapping Intervals
Finally, the probability of overlapping intervals is:
P(overlapping) = 1 - P(non-overlapping) = 1 - 1/12 = 11/12
Therefore, the probability that intervals I and J overlap is 11/12.
Jul 14, 2024 | 426 | 1,551 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.65625 | 5 | CC-MAIN-2024-33 | latest | en | 0.951543 |
https://www.physicsforums.com/threads/how-do-we-know-if-the-coordintes-of-a-point-lie-within-and-outside-a-triangle.170942/ | 1,539,611,123,000,000,000 | text/html | crawl-data/CC-MAIN-2018-43/segments/1539583509196.33/warc/CC-MAIN-20181015121848-20181015143348-00437.warc.gz | 1,029,059,463 | 18,426 | # How do we know if the coordintes of a point lie within and outside a triangle?
1. May 19, 2007
### david07
ex: if the origin points are A = (0, 0)
B = (3,5)
C= (6,3)
How do we know that point (3,1) lies within or outside the triangle using mathematical formulas?
I know there is a similar topic to this somewhere in forums but i don't understand where a & b comes from.
please can anyone help me out?
thanks
2. May 19, 2007
Well, if you're looking at that certain point, the I guess you can find the equation of the line connecting A and C, and simply find the value of that function at x = 3. If it's greater than 1, then (3, 1) doesn't lie in the triangle.
3. May 19, 2007
### david07
Thanks but no not for that particular point.
I mean the coordinates can be different except the origin. (0, 0)
how do we know if we have to find the equation of line of AC or AB or BC??
4. May 19, 2007
### robert Ihnot
If you take that particular problem and draw the triangle, you can determine which sides are where. The picture is important. Of the three points, (where X is the first coordinate, Y the second), (3,5) is the highest point, while (6,3) lies the furthest along the X-axis.
Moving along the X-axis from (0,0) to (6,3) we have a slope of 1/2. The point in question (3,1), connected to (0,0), has a slope of Y/X = 1/3. So that line and point is outside the triangle.
Last edited: May 19, 2007
5. May 19, 2007
### Werg22
Well, I can think of two ways. First, with areas. Say the vertices are A, B and C. We denote the area of that triangle K. A point P, if inside the triangle, satisfies the condition
PAB + PAC + PCB = K.
That is, the sum of the area of all possible triangles formed between two vertices of ABC and P is K. A point that does not satisfy this equality is located outside the triangle.
The second way is equivalent but goes along with the angles. That is (I use < to denote angles):
<APB + <APC + <BPC = 360
PS: Note that points along the sides of the triangle ABC are included.
Last edited: May 19, 2007
6. May 21, 2007
### forrest747
You know the vertices so you can easily get the linear equations [call them y1=, y2= and y3=] for the three lines that contain each side of the triangle.
[Method caveat: begin with a point known to be OUTSIDE the triangle. Find the line equation y4= from that outside point to the point you want to check. Determine if y4 intersects each of y1, y2 or y3. Simple to program this algorithm. Count the number of intersections.
If you begin OUTSIDE the triangle, and you have no intersections, then you are outside the triangle (duh). One intersection, you are inside the triangle. Two intersections, you are outside the triangle.
This trick works for any polygon. If you begin outside the polygon, count the number of intersections and if you have an even number of intersections, you are outside the polygon. If you have an odd number of intersections, you are inside the polygon.
7. May 21, 2007
### Werg22
Certainly a valid method, but it's a shame there is no equivalent analytical method...
8. May 22, 2007
### david07
so u mean first find the area of the triangle using the coordinate points.
and then do PAB+PAC+PCB=are of triangle using the given coordinate points.
what i don't understand is how does PAB+PAC+PCB work??
I mean AB, AC, CB can be found using distance formula but P is suppose some point (0,2) how does that work?
9. May 22, 2007
### mathwonk
wow that is a really cool question. i dont know the NSWER yet.
heres a suggestion, say the vertices are A,B,C, and the extra point is D.
then for each pair of vertices say A,B, try to decide if C and D are on h same side of line joining A,B.
the linear equation which equals zero at both A,B, will hve the same sign at points on the same side of line AB.
does this help?
10. May 22, 2007
### Werg22
I'm afraid I don't quite get the question... maybe I should have been clearer with the expression PAB + PAC + PCB. PAB, PAC, PCB represent the areas of the triangles PAB, PAC and PCB, respectively. So if the sum of the areas of the triangles PAB, PAC and PCB, is equal to the area of ABC, then point P lies inside the triangle. I know it's tedious to calculate all these areas (personally I find Heron's formula very bad looking) but it's still a solution...
Edit: Oh I see what you mean. ABC and P represent the points, not the sides!
Last edited: May 22, 2007
11. May 22, 2007
### Werg22
Though I didn't quite get this part, if it's possible to easily verify if C and D are on the same side, there's yet another solution: if C and D are on the same side, and the x coordinate of C is smaller than that of D or bigger, depending on weather they are on the right or left of segment AB, and if the linear equation that goes through C and D crosses the segment AB, then D is inside the triangle. I know I'm just stating the obvious, but someone must do it.
12. May 22, 2007
### D H
Staff Emeritus
This appears to work: use two of the edges as a basis for the plane. The coordinates of the unknown point in this "edge space" must be non-negative and must sum to one or less.
Algorithm: Place one of the vertices at the origin. Denote the other two vertices as
$$\bmatrix x_i \\ y_i\endbmatrix$$
Denote the point in question as
$$\bmatrix x \\ y\endbmatrix$$
The coordinates of this point are
$$\bmatrix u\\v \endbmatrix = \bmatrix x_1 & x_2 \\y_1 & y_2 \endbmatrix^{-1} \bmatrix x \\ y\endbmatrix$$
The point is inside the triangle if $u,v>=0$ and $u+v<=1$.
Last edited: May 22, 2007
13. May 22, 2007
### mathwonk
the interior of a triangle is the intersection of the three hal planes determiend by the three sides.
so if a point is on the same side of each side as the third vertex, it is in the interior.
e.g. take the triangle with vertices (0,0), (0,1),(1,0).
the three sides determine the three lines x=0, y=0, and x+y-1=0.
now applying the first equation, namely x, to the third vertex (1,0) gives 1, so an interior point shoiuld have x>0.
the second side with vertices (0,0), (1,0), determines the line y=0 and the third vertex has y=1 so the interior points have y>0. then x+y-2 = 0 for the third side, and the third vertex has x+y-1 = -1, so the interior points satisfy x+y-1 < 0.
hence an interior point is one with x>0, y>0 and x+y < 1. e.g. (1/3,1/2).
Last edited: May 22, 2007
14. May 22, 2007
### david07
ok i kinda understand werg22 area method but i still don't get the right answer.
A (0,0)
B (1,5)
C (4,1)
P (2,2)
I am getting the area of ABC as 9.5
PAB+PAC+PCB=0
Technically (2,2) should be in but according to your method if it isn't equal to area of ABC then it is outside.
Don't understand what mistake i am making.
Thanks
technically
15. May 22, 2007
### Werg22
What is the meaning of this?
16. May 22, 2007
### D H
Staff Emeritus
This is a nice canonical triangle. Let (u,v) be the Cartesian coordinates of a point on a plane with a triangle with vertices at (0,0),(1,0),(0,1). As mathwonk pointed out, the conditions that a point (u,v) be in the interior of the canonical triangle are u,v>0 and u+v<1.
This canonical triangle can be mapped to a triangle with vertices at (0,0),(x1,y1),(x2,y2) by the transformation
$$\bmatrix x\\y\endbmatrix = \bmatrix x_1 & x_2 \\y_1 & y_2 \endbmatrix\;\bmatrix u\\v\endbmatrix$$
The inverse of this transformation maps from the generic triangle to the canonical triangle:
$$\bmatrix u\\v \endbmatrix = \bmatrix x_1 & x_2 \\y_1 & y_2 \endbmatrix^{-1} \bmatrix x \\ y\endbmatrix$$
which is what I stated in post #12 (except I did not have strict inequalities).
17. May 23, 2007
### david07
I mean when i find PAB, PAC, PCB i get all of those areas as 0.
which is not equal to the area of ABC.
hence (2,2) does not fall in the triangle ABC although it is supposed to be.
18. May 23, 2007
### VietDao29
How did you come up with all those area equal to 0? It's not true. =.="
An area of a triangle is 0 only when 3 points overlaps each other, or they are co-linear. Your example is neither of the two cases above, so your triangle does have area.
19. Sep 15, 2011
### rdpreet
I got the point of mapping the triangle to a canonical triangle...
But I am hazy about the process, with the all the formulae put in codes. Can you please simplify it a li'l.
20. Sep 15, 2011 | 2,377 | 8,344 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.21875 | 4 | CC-MAIN-2018-43 | latest | en | 0.937637 |
https://eduzip.com/ask/question/an-angle-is-14o-more-than-its-complementary-angle-what-is-the-mea-520180 | 1,642,520,874,000,000,000 | text/html | crawl-data/CC-MAIN-2022-05/segments/1642320300934.87/warc/CC-MAIN-20220118152809-20220118182809-00330.warc.gz | 285,753,334 | 9,023 | Mathematics
An angle is $14^{o}$ more than its complementary angle. What is the measure?
SOLUTION
Let first angle is $={{x}^{o}}$
Then, IInd angle is $=x+{{14}^{o}}$
According to given question,
$x+x+14={{90}^{o}}$
$2x={{90}^{o}}-{{14}^{o}}$
$2x={{76}^{o}}$
$x={{38}^{o}}$
Then, IInd angle is $x+{{14}^{o}}={{38}^{o}}+{{14}^{o}}={{52}^{o}}$
You're just one step away
Subjective Medium Published on 09th 09, 2020
Questions 120418
Subjects 10
Chapters 88
Enrolled Students 86
Realted Questions
Q1 Single Correct Medium
As shown in the given figure, according to corresponding angle theorem which two angles are equal.
• A. $\angle 1$ and $\angle2$
• B. $\angle 1$ and $\angle3$
• C. $\angle 1$ and $\angle4$
• D. $\angle 1$ and $\angle5$
Asked in: Mathematics - Lines and Angles
1 Verified Answer | Published on 09th 09, 2020
Q2 Single Correct Medium
If a line is divided by three parallel lines into two-segments of lengths in the ratio 1 : 3, another line will be divided by these parallel lines into two segments of lengths in the ratio
• A. 3 : 1
• B. 1 : 2
• C. 2 : 1
• D. 1 : 3
Asked in: Mathematics - Lines and Angles
1 Verified Answer | Published on 09th 09, 2020
Q3 Single Correct Medium
If in the figure a || b and c || d, then name the angles that are congruent to $\angle 1$.
• A. $\angle 4, \angle 5, \angle 7, \angle 9, \angle 11, \angle 13, \angle 15$
• B. $\angle 3, \angle 6, \angle 7, \angle 9, \angle 11, \angle 13, \angle 15$
• C. None of these
• D. $\angle 3, \angle 5, \angle 7, \angle 9, \angle 11, \angle 13, \angle 15$
Asked in: Mathematics - Straight Lines
1 Verified Answer | Published on 17th 08, 2020
Q4 Subjective Medium
Two adjacent angles on a straight line are in the ratio $5:4$. Find the measure of each one of these angles.
Asked in: Mathematics - Lines and Angles
1 Verified Answer | Published on 09th 09, 2020
Q5 Subjective Medium
Two supplementary angles differ by $48^{o}$. Find the angles.
Asked in: Mathematics - Lines and Angles
1 Verified Answer | Published on 23rd 09, 2020 | 681 | 2,044 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.21875 | 4 | CC-MAIN-2022-05 | latest | en | 0.783617 |
https://zh.m.wikipedia.org/wiki/%E9%9B%99%E5%80%8D%E7%B2%BE%E7%A2%BA%E6%B5%AE%E9%BB%9E%E6%95%B8 | 1,670,040,212,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446710918.58/warc/CC-MAIN-20221203011523-20221203041523-00706.warc.gz | 1,138,191,389 | 14,877 | # 雙精度浮點數
（重定向自雙倍精確浮點數
## 格式
sign bit（符號）：用來表示正負號
exponent（指數）：用來表示次方數
mantissa（尾數）：用來表示精確度
0代表數值為正，1代表數值為負。
### 指數
1. 「11個位元皆為0」
2. 「11個位元皆為1」
000000000002 = 00016當尾數為0時為±0，尾數不為0時為非正規形式的浮點數
111111111112 = 7ff16當尾數為0時為∞，尾數不為0時為NaN
### 尾數
${\displaystyle {\text{1.mantissa}}\times {\text{2}}^{\text{exponent}}}$
二進位制的 ${\displaystyle {\text{11.101}}\times {\text{2}}^{\text{1001}}}$ 可以規格化為 ${\displaystyle {\text{1.1101}}\times {\text{2}}^{\text{1010}}}$ ，儲存時尾数只需要儲存1101即可
二進位制的 ${\displaystyle {\text{0.00110011}}\times {\text{2}}^{\text{-1001}}}$ 可以規格化為 ${\displaystyle {\text{1.10011}}\times {\text{2}}^{\text{-1100}}}$ ，儲存時尾數只需要儲存10011即可
### 小結
${\displaystyle (-1)^{\text{sign}}\times 2^{\text{exponent}}\times 1.{\text{mantissa}}}$
## 例子
0 01111111111 00000000000000000000000000000000000000000000000000002 ≙ 3FF0 0000 0000 000016 ≙ +20 × 1 = 1 0 01111111111 00000000000000000000000000000000000000000000000000012 ≙ 3FF0 0000 0000 000116 ≙ +20 × (1 + 2−52) ≈ 1.0000000000000002, the smallest number > 1 0 01111111111 00000000000000000000000000000000000000000000000000102 ≙ 3FF0 0000 0000 000216 ≙ +20 × (1 + 2−51) ≈ 1.0000000000000004 0 10000000000 00000000000000000000000000000000000000000000000000002 ≙ 4000 0000 0000 000016 ≙ +21 × 1 = 2 1 10000000000 00000000000000000000000000000000000000000000000000002 ≙ C000 0000 0000 000016 ≙ −21 × 1 = −2
0 10000000000 10000000000000000000000000000000000000000000000000002 ≙ 4008 0000 0000 000016 ≙ +21 × 1.12 = 112 = 3 0 10000000001 00000000000000000000000000000000000000000000000000002 ≙ 4010 0000 0000 000016 ≙ +22 × 1 = 1002 = 4 0 10000000001 01000000000000000000000000000000000000000000000000002 ≙ 4014 0000 0000 000016 ≙ +22 × 1.012 = 1012 = 5 0 10000000001 10000000000000000000000000000000000000000000000000002 ≙ 4018 0000 0000 000016 ≙ +22 × 1.12 = 1102 = 6 0 10000000011 01110000000000000000000000000000000000000000000000002 ≙ 4037 0000 0000 000016 ≙ +24 × 1.01112 = 101112 = 23 0 01111111000 10000000000000000000000000000000000000000000000000002 ≙ 3F88 0000 0000 000016 ≙ +2−7 × 1.12 = 0.000000112 = 0.01171875 (3/256)
0 00000000000 00000000000000000000000000000000000000000000000000012 ≙ 0000 0000 0000 000116 ≙ +2−1022 × 2−52 = 2−1074 ≈ 4.9406564584124654 × 10−324 (Min. subnormal positive double) 0 00000000000 11111111111111111111111111111111111111111111111111112 ≙ 000F FFFF FFFF FFFF16 ≙ +2−1022 × (1 − 2−52) ≈ 2.2250738585072009 × 10−308 (Max. subnormal double) 0 00000000001 00000000000000000000000000000000000000000000000000002 ≙ 0010 0000 0000 000016 ≙ +2−1022 × 1 ≈ 2.2250738585072014 × 10−308 (Min. normal positive double) 0 11111111110 11111111111111111111111111111111111111111111111111112 ≙ 7FEF FFFF FFFF FFFF16 ≙ +21023 × (1 + (1 − 2−52)) ≈ 1.7976931348623157 × 10308 (Max. Double)
0 00000000000 00000000000000000000000000000000000000000000000000002 ≙ 0000 0000 0000 000016 ≙ +0 1 00000000000 00000000000000000000000000000000000000000000000000002 ≙ 8000 0000 0000 000016 ≙ −0 0 11111111111 00000000000000000000000000000000000000000000000000002 ≙ 7FF0 0000 0000 000016 ≙ +∞ (positive infinity) 1 11111111111 00000000000000000000000000000000000000000000000000002 ≙ FFF0 0000 0000 000016 ≙ −∞ (negative infinity) 0 11111111111 00000000000000000000000000000000000000000000000000012 ≙ 7FF0 0000 0000 000116 ≙ NaN (sNaN on most processors, such as x86 and ARM) 0 11111111111 10000000000000000000000000000000000000000000000000012 ≙ 7FF8 0000 0000 000116 ≙ NaN (qNaN on most processors, such as x86 and ARM) 0 11111111111 11111111111111111111111111111111111111111111111111112 ≙ 7FFF FFFF FFFF FFFF16 ≙ NaN (an alternative encoding of NaN)
0 01111111101 01010101010101010101010101010101010101010101010101012 = 3fd5 5555 5555 555516 ≙ +2−2 × (1 + 2−2 + 2−4 + ... + 2−52) ≈ 1/3
0 10000000000 10010010000111111011010101000100010000101101000110002 = 4009 21fb 5444 2d1816 ≈ pi | 1,822 | 3,865 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 7, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.9375 | 4 | CC-MAIN-2022-49 | latest | en | 0.181083 |
https://gmatclub.com/forum/suggestions-for-someone-needing-lots-of-quantitative-help-67164.html | 1,495,825,086,000,000,000 | text/html | crawl-data/CC-MAIN-2017-22/segments/1495463608676.72/warc/CC-MAIN-20170526184113-20170526204113-00501.warc.gz | 934,358,646 | 54,660 | Check GMAT Club Decision Tracker for the Latest School Decision Releases https://gmatclub.com/AppTrack
It is currently 26 May 2017, 11:58
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
# Events & Promotions
###### Events & Promotions in June
Open Detailed Calendar
# Suggestions for someone needing lots of quantitative help
Author Message
TAGS:
### Hide Tags
Intern
Joined: 14 Jul 2008
Posts: 4
Followers: 0
Kudos [?]: 0 [0], given: 0
Suggestions for someone needing lots of quantitative help [#permalink]
### Show Tags
14 Jul 2008, 21:08
After reading many revews I am under the impression that the best book for someone who hasn't practiced any sort of math in over ten years would be the EZ Gmat-math strategies book. Any input would be greatly appreciated. Thanks!
VP
Joined: 22 Oct 2006
Posts: 1438
Schools: Chicago Booth '11
Followers: 9
Kudos [?]: 192 [0], given: 12
Re: Suggestions for someone needing lots of quantitative help [#permalink]
### Show Tags
14 Jul 2008, 21:32
not sure, but be an active participant of this forum and post any questions you may have.
Intern
Joined: 04 Sep 2012
Posts: 8
Followers: 0
Kudos [?]: 0 [0], given: 0
Re: Suggestions for someone needing lots of quantitative help [#permalink]
### Show Tags
06 Sep 2012, 03:07
I am also this Question ?who can help me in math test ?
Manager
Joined: 13 Feb 2012
Posts: 146
Location: Italy
Concentration: General Management, Entrepreneurship
GMAT 1: 560 Q36 V34
GPA: 3.1
WE: Sales (Transportation)
Followers: 4
Kudos [?]: 7 [1] , given: 85
Re: Suggestions for someone needing lots of quantitative help [#permalink]
### Show Tags
06 Sep 2012, 05:50
1
KUDOS
malaka wrote:
I am also this Question ?who can help me in math test ?
Depending on your starting level you might need more or less study material; even before that, thou, you might start exploring the Gmat Club and the quantitative section.
If you are following the OG12 you might look up answers explanations here and then figure out you weaknesses and strenghts.
Basically everything you might need you can find it here, including theory:
For example math-number-theory-88376.html
There are a lot of great guys here who might help you.
Good Luck!
_________________
"The Burnout" - My Debrief
Kudos if I helped you
Andy
Re: Suggestions for someone needing lots of quantitative help [#permalink] 06 Sep 2012, 05:50
Similar topics Replies Last post
Similar
Topics:
1 Need serious help/suggestions here. 7 13 Oct 2015, 08:37
Need help with Verbal. Any suggestions? 1 31 Aug 2013, 12:30
Need some help Suggestions 1 15 Jan 2011, 08:08
Need Help With Quantitative 2 18 Jul 2008, 15:50
Need Suggestions 7 15 May 2008, 23:33
Display posts from previous: Sort by | 851 | 3,160 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.578125 | 4 | CC-MAIN-2017-22 | longest | en | 0.859991 |
https://socratic.org/questions/how-do-you-simplify-x-3-x-2-y-2-y-3-y-2-x-2 | 1,638,967,321,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964363510.40/warc/CC-MAIN-20211208114112-20211208144112-00302.warc.gz | 573,592,505 | 5,770 | # How do you simplify x^3/ (x^2-y^2) + y^3/( y^2-x^2)?
##### 1 Answer
Oct 11, 2015
$\frac{{x}^{2} + x y + {y}^{2}}{x + y}$
#### Explanation:
${x}^{2} / \left({x}^{2} - {y}^{2}\right) + {y}^{3} / \left({y}^{2} - {x}^{2}\right)$
$\textcolor{w h i t e}{\text{XXX}} = \frac{{x}^{3} - {y}^{3}}{{x}^{2} - {y}^{2}}$
color(white)("XXX")=((x-y)(x^2+xy+y^2))/((x-y)(x+y)
$\textcolor{w h i t e}{\text{XXX}} = \frac{{x}^{2} + x y + {y}^{2}}{x + y}$ | 229 | 443 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 5, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.21875 | 4 | CC-MAIN-2021-49 | latest | en | 0.372586 |
https://cs.stackexchange.com/questions/4707/can-two-neighbors-in-a-graph-be-at-the-same-depth-in-a-dfs-tree | 1,719,226,426,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198865348.3/warc/CC-MAIN-20240624084108-20240624114108-00590.warc.gz | 153,763,538 | 40,733 | # Can two neighbors in a graph be at the same depth in a DFS tree?
In an undirected graph, can two nodes at an identical distance n from the root of a DFS tree be neighbors in the original graph? I'm thinking no, but I'm not sure (because of back edges)
I think you are correct:
All neighbors of a node will be of rank one less (the node we got here from) or one more (or two more, etc) because we will rank them before we back out of the node.
• DFS and BFS visit edges in the same order under some constraints! In such cases neighbors in the tree are also neighbors in the graph. Correct me if I'm wrong.
– mrk
Commented Sep 24, 2012 at 16:54
• @saadtaame You are wrong. Commented Jan 1, 2016 at 5:19
The wikipedia article states that "it can be shown that if the graph is undirected then all of its edges are tree edges or back edges."
If you find out how this can be shown, you should be quite close to your solution.
• If the edge is not a back edge (an edge you walk back in on) than it will be a tree edge b/c you will walk out it.
– BCS
Commented Jan 6, 2009 at 0:21
Your question does not stipulate how the DFS tree is generated. Naive implementations simply list nodes in the order they are visited, but most practical implementations rebuild the output tree as the search progresses and the ranks of the nodes are adjusted.
Consider three nodes, A B C, connected to each other. A naive DFS will visit A B C, with ranks of 0 1 2, and the tree will have a back edge from C to A. A more useful implementation will backtrack from C to B to A, and then from A will travel to C again and assign it a rank of 1, then decline to travel to B at rank 2 because B is already rank 1, this tree will have both B and C at rank 1 below A, with a cross edge between B and C.
• this is not correct
– Joe
Commented Sep 25, 2012 at 0:41
• @Joe could you elaborate on that? In re-reading my example I see that it's not as clear as it might be, so I'd be happy to rewrite it if it is confusing to you. Commented Sep 26, 2012 at 16:51
• What you describe is more like a DFS followed by a BFS. Nodes that have already been visited are not re-visited in any standard DFS.
– Joe
Commented Sep 27, 2012 at 18:06 | 587 | 2,206 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.53125 | 4 | CC-MAIN-2024-26 | latest | en | 0.944043 |
https://www.teacherspayteachers.com/Product/Fun-with-Sudoku-Guy-Gr-4-6-LESSON-4-How-to-work-with-small-numbers-4337110 | 1,603,146,904,000,000,000 | text/html | crawl-data/CC-MAIN-2020-45/segments/1603107866404.1/warc/CC-MAIN-20201019203523-20201019233523-00682.warc.gz | 946,327,766 | 35,705 | # Fun with Sudoku Guy. (Gr 4-6, LESSON 4): How to work with small numbers
Created BySudoku Guy
Subject
Grade Levels
Resource Type
Format
Presentation (Powerpoint) File (702 MB|10+1 video)
Standards
\$5.00
Digital Download
Share this resource
More products from Sudoku Guy
\$5.00
Digital Download
#### Also included in
1. This is a 5-resource bundle for grades 4-6. It assumes that students have not done the Fun with Sudoku Guy bundle for K-3. Consequently, the first resource is for those have not seen the K-3 bundle. This bundle takes students step by step to the stage where they can solve simple sudoku puzzles using
\$22.50
\$25.00
Save \$2.50
### Description
What to do when there are 2 empty numbers in a row, column, or block. This is a very important lesson as it leads students to another step in the solving of sudoku puzzles.. It is an easy procedure once you get into the habit.
Robin the Sudoku Guy has a video to show how it is done. Once students have this technique learnt, the next step, which is covered in lesson number 5, becomes very helpful, and exciting as many numbers can be solved with this one technique and using TMB and LCR.
This is the 4th of 5 lessons showing students how to solve simple sudoku puzzles. It is a step-by-step procedure using PowerPoint, activities and videos where Robin the Sudoku Guy demonstrates the procedures, rules, and skills needed to solve puzzles confidently. The lessons can be purchased individually or as a bundle.
5 lessons for Gr 4-6 Printable activities, worksheets an demo videos.
FOR TEACHERS
• puzzles for students are provided to solve, along with suggestions for teachers.
• students learn the importance of mastering procedures that make them successful puzzle solvers
• watching the videos is fun
• help students look at 3 horizontal blocks and spot where a number is missing in 1 of the 3 blocks using TMB Robin the Sudoku guy will show how this is done in the video
• help students look at 3 vertical blocks and spot where a number is missing in 1 of the 3 blocks using LCR. Robin the Sudoku guy will show how this is done in the video
• teachers can print off puzzles from the PowerPoint slides
• go through slides and videos as a class or use it as a math work station (Your choice... I'm pretty entertaining!)
• individual or group activities and worksheets include puzzles to solve
• videos are to watch as a class or for teacher to form lesson plan
• teacher guide in presentation notes (under the slides)
• answer key when approriate
EXPLORE
• logical thinking, spacial relationships, and number sequenceing
• new vocabulary e.g. row, column, block, cell, grid horizontal, vertical.
• the videos are fun to watch. They teach a step by step approach.
RESULTS
• Students will know what to do when a row, or column has 2 empty cells when using TMB and LCR.
• Students wil be able to solve horizontal and vertical blocks with 1 number missing in 1 of 3 blocks.
• Students will be able to observe or spot a number located in 2 out of 3 horizontal and vertical blocks.
• Students will learn a procedure. i.e. TMB, then LCR, then a number missing in a row, column or block.
• Students will know the rules of sudoku, particularly that no row column or block can have a repeated number.
• Students will learn new vocabulary, e.g. row, column, block, grid, horizontal, and vertical.
• By the time students reach the 5th lesson they will be able to solve simple sudoku puzzles using TMB and LCR with one number or two numbers missing in a row, column.
• Above all, I wish students to have fun.
DON'T FORGET!
There are 12 lessons in total. and 2 bundles. (K-gr3 and Gr 4-6)
Robin the Sudoku Guy
After 15 years doing sudoku puzzles, and having created the popular Sudoku Guy series, I saw a need for online tutorials designed for the young. The lessons and tutorials start with simple exercises for beginners and move through tips for those ready to tackle more difficult puzzles. Thanks to a teaching career that spanned kindergarten through university, and with a background in theatre and television, I developed a fun, easy-to-understand, step-by-step series. Each session involves a video and printables. The enthusiasm of the young people who field tested them confirmed my intuition, that sudoku is a terrific aid for assisting them to learn more about thinking skills and spatial relationships.
Here is a recent testimonial.
Hi Sudoku Guy, Thankyou for making such good videos with great tips. My son Mohammed" of 4th grade had participated in a national sudoku competition. He watched many of your videos and used all of the tips and tricks from the videos. And the results were out. We were happy to see that my son has bagged first place in the competition. So, we would like to thank you for all the good videos
Total Pages
10+1 video
Answer Key
Included
Teaching Duration
45 minutes
Report this Resource to TpT
Reported resources will be reviewed by our team. Report this resource to let us know if this resource violates TpT’s content guidelines.
### Standards
to see state-specific standards (only available in the US).
Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and-if there is a flaw in an argument-explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
Determine or clarify the meaning of unknown and multiple-meaning words and phrases by using context clues, analyzing meaningful word parts, and consulting general and specialized reference materials, as appropriate.
Apply knowledge of language to understand how language functions in different contexts, to make effective choices for meaning or style, and to comprehend more fully when reading or listening.
### Questions & Answers
Teachers Pay Teachers is an online marketplace where teachers buy and sell original educational materials.
More About Us | 1,868 | 9,224 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.640625 | 4 | CC-MAIN-2020-45 | latest | en | 0.913738 |
https://web2.0calc.com/questions/algebra_21342 | 1,726,771,955,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700652055.62/warc/CC-MAIN-20240919162032-20240919192032-00182.warc.gz | 572,141,665 | 5,945 | +0
# Algebra
0
20
1
+523
The function \$f(x)\$ is defined only on domain \$[-1,2]\$, and is defined on this domain by the formula
\$\$f(x) = 2x^2-8x+1.\$\$
What is the range of \$f(x)\$? Express your answer as an interval or as a union of intervals.
Apr 15, 2024
#1
+2651
0
We can find the range of the function f(x)=2x2−8x+1 by completing the square and analyzing its extreme points.
Completing the Square:
We can rewrite the function as:
f(x)=2(x2−4x)+1
To complete the square, we take half of the coefficient of our x term, square it, and add it to both sides of the equation:
Half of the coefficient of our x term is -4 / 2 = -2.
Squaring -2 gives us 4.
Therefore:
f(x)=2(x2−4x+4)+1−8
f(x)=2(x−2)2−7
Finding the Vertex and Analyzing Extremes:
The expression inside the parentheses is squared, so it's always non-negative (0 or greater). Multiplying by 2 doesn't change this property. Therefore, the minimum value of f(x) is achieved when the squared term is 0, which happens when x=2. In this case, f(2)=2(2−2)2−7=−7.
Since the coefficient of the squared term (2) is positive, the parabola opens upwards. This means the function has a minimum point at x=2 and increases as we move away from it (either to the left or right on the interval).
Range of the Function:
The function is defined only on the interval [−1,2]. We saw that the minimum value within this interval is f(2)=−7. As x approaches either end of the interval, the function increases without bound because the squared term keeps getting larger. Therefore, the range of f(x) is the single interval:
(−7,∞)
Apr 15, 2024 | 470 | 1,607 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.71875 | 5 | CC-MAIN-2024-38 | latest | en | 0.893032 |
http://www.askiitians.com/iit-jee-mathematics/trigonometry/trigonometric-identities-and-equation/trigonometric-functions.aspx | 1,516,117,528,000,000,000 | text/html | crawl-data/CC-MAIN-2018-05/segments/1516084886437.0/warc/CC-MAIN-20180116144951-20180116164951-00458.warc.gz | 399,709,435 | 43,935 | Click to Chat
1800-2000-838
+91-120-4616500
CART 0
• 0
MY CART (5)
Use Coupon: CART20 and get 20% off on all online Study Material
ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: Rs.
There are no items in this cart.
Continue Shopping
```Trigonometric Functions
Table of Content
Some Basic Results
Trigonometric Ratios of any Angle
Basic Formulae
In a right angled triangle ABC, ∠CAB = A and ∠BCA = 90° = p/2. AC is the base, BC the altitude and AB is the hypotenuse. We refer to the base as the adjacent side and to the altitude as the opposite side. There are six trigonometric ratios, also called trigonometric functions or circular functions. With reference to angle A, the six ratios are:
Obviously, .
The reciprocals of sine, cosine and tangent are called the cosecant, secant and cotangent of A respectively. We write these as cosec A, sec A, cot A respectively.
Important Notes:
Since the hypotenuse is the greatest side in a right angle triangle, sin A and cos A can never be greater than unity and cosec A and sec A can never be less than unity.
All the six trigonometric functions have got a very important property in common that is periodicity. Remember that the trigonometrical ratios are real numbers and remain same as long as the angle remains same.
Some Basic Results
cos2A + sin2A = 1
From this formula, we can also derive cos2A = 1 - sin2A or sin2A = 1 - cos2A
1 + tan2A = sec2A
So, from this it follows that sec2A - tan2A = 1
cot2A + 1 = cosec2A or cosec2A - cot2A = 1
Fundamental inequalities: For 0 < A <, 0 < cosA < <.
It is possible to express trigonometrical ratios in terms of any one of them
e.g.
Trigonometric Ratios of any Angle
Consider the system of rectangular co-ordinate axes dividing the plane into four quadrants. A line OP makes angle q with the positive x-axis. The angle q is said to be positive if measured in counter clockwise direction from the positive x-axis and is negative if measured in clockwise direction. The positive values of the trigonometric ratios in the various quadrants are shown, the signs of the other ratios may be derived. Note that ∠xoy = p/2, ∠xox' = p , ∠xoy' = 3p/2
PiQi is positive if above the x-axis, negative if below the x-axis, OPi is always taken positive. OQi is positive if along positive x-axis, negative if in opposite direction.
(Where i = 1, 2, 3, 4)
Thus depending on signs of OQi and PiQi the various trigonometrical ratios will have different signs.
TABLE
a equals
sin a
cos a
tan a
cot a
Sec a
cosec a
–q
– sinq
cosq
–tanq
– cotq
secq
–cosec q
90° – q
cosq
sinq
cotq
tanq
cosecq
secq
90° + q
cosq
– sinq
–cotq
– tanq
–cosecq
secq
180°– q
sinq
– cosq
– tanq
– cotq
– secq
cosecq
180°+ q
– sinq
– cosq
tanq
cotq
– secq
–cosecq
360°– q
– sinq
cosq
– tanq
– cotq
secq
–cosecq
360°+ q
sinq
cosq
tanq
cotq
secq
cosecq
Note:
Angle q and 90°–q are complementary angles, q and 180°–q are supplementary angles
sin(np + (–1)nq) = sinq, n ÎI
cos(2np ± q) = cosq, n ÎI
tan(np + q) = tanq, n ÎI
i.e. sine of general angle of the form np + (–1)nq will have same sign as that of sine of angle q and so on. The same is true for the respective reciprocal functions also.
Basic Formulae
Table 1
sin (A + B) = sin A cos B + cos A sin B
sin (A – B) = sin A cos B – cos A sin B
cos (A + B) = cos A cos B – sin A sin B
cos (A – B) = cos A cos B + sin A sin B
sin (A + B) sin (A – B) = sin2 A – sin2 B = cos2 B – cos2 A
cos (A + B) cos (A – B) = cos2 A – sin2 B = cos2 B – sin2 A
sin2A = 2sinA cosA =
cos2A = cos2A - sin2A = 1-2 sin2A = 2cos2A-1 =
tan2A =
sin3A = 3sinA - 4sin3A = 4sin(60° - A) sinAsin(60° + A)
cos3A = 4cos3A - 3cosA = 4cos(60° - A) cosAcos(60°+A)
Table 2
tanA + tanB =
2sinAcosB = sin(A + B) + sin (A - B)
2cosAsinB = sin(A + B) - sin (A - B)
2cosAcosB = cos(A + B) + cos(A - B)
2sinAsinB = cos(A - B) - cos (A + B)
Illustration : If in a DABC, cos3A + cos3B + cos3C = 3cosA cosB cosC, then prove that the triangle is equilateral.
Solution: Given that cos3A + cos3B + cos3C – 3cosA cosB cosC = 0
Þ (cosA + cosB + cosC) (cos2A + cos2B + cos2C – cosAcosB – cosB cosC – cosCcosA) = 0
Þ cos2A + cos2B + cos2C – cosAcosB – cosBcosC – cosCcosA = 0
(as cosA + cosB + cosC = 1 + 4 sinA/2 sinB/2 sinC/2 ¹ 0)
Þ (cosA – cosB)2 + (cosB – cosC)2 + (cosC – cosA)2 = 0
Þ cosA = cosB = cosC
Þ A = B = C, Q 0 < A, B, C < p.
ÞDABC is equilatral.
Illustration: If , prove that tan2q = 2tan(3q + a).
Solution: = k
= =
Again
= \
=\ tan2q = 2 tan(3q + a).
Illustration : For any real q , find the maximum value of cos2( cosq) + sin2(sinq) .
Solution: The maximum value of cos2( cosq) is 1 and that of sin2( sinq) is sin21, both exists for q = p/2. Hence maximum value is 1+ sin21.
Illustration 5: If .
Solution: Þ tan2q = 1/11
now, = 3 – 4 sin2q = 3 - 4
Illustration : If A, B,C and D are angles of a quadrilateral and sin A/2 sin B/2. sin C/2. sin D/2 = prove that A = B = C = D = π/2.
Solution: Given
Since, A + B = 2 - (C + D), the above equation becomes,
= 1
= 0.
This is quadratic equation in cos which has real roots.
0
4
Now both cos and cos 1
A = B, C = D.
Similarly, A = C, B = D A = B = C = D = /2.
Illustration : If A, B and C are angles of a triangle, prove that
Solution: Since A + B + C =
=
=
=
as A, B, C are angles of 0 < A, B, C <
sin A, sin B, sin C > 0
E 2 + 2 + 2
E 6
Watch this Video for more reference
Illustration : If cos(A + B) sin (C + D) = cos(A – B) sin(C – D),
prove that cotA cotB cotC = cotD.
Solution: We have cos (A + B) sin(C + D) = cos(A – B) sin(C – D)
i.e.
or
cotA cotB = tanC cotD
or cotA cotB cotC = cotD.
Illustration : Show that
Solution:
LHS =
Let = a
= cos a cos 2a cos 3a cos 4a cos 5a
= – cos a cos 2a cos 4a cos 8a cos 5a
= – cos 20a cos 21a cos 22a cos 23a cos 5a
= – cos 5a = –
=.
Illustration : Prove that cot 7 =
Solution: Let q = 7 Þ 2q = 150
Now cot q =
= = .
For more, you can also refer the papers of previous years.
Illustration : If 2tan2 a tan2 b tan2 g + tan2 a tan2 b + tan2 b tan2 g + tan2 g tan2 a = 1, prove that sin2 a + sin2 b + sin2 g = 1.
Solution: ÞWe have, 2tan2a tan2btan2g + tan2atan2b + tan2btan2g + tan2gtan2a = 1
Þ 2 + cot2g + cot2a + cot2b = cot2a cot2b× cot2g
Þ cosec2a + cosec2b + cosec2g – 1
= (cosec2a – 1) (cosec2b – 1) (cosec2g – 1)
Þ cosec2a + cosec2b + cosec2g – 1
= – 1 + cosec2a + cosec2b + cosec2g – (cosec2a cosec2b + cosec2b cosec2g + cosec2gcosec2a + cosec2a× cosec2b× cosec2g
cosec2a cosec2b + cosec2b× cosec2g + cosec2g cosec2a
= cosec2a cosec2b× cosec2gÞ sin2a + sin2b + sin2g = 1
```
• Complete JEE Main/Advanced Course and Test Series
• OFFERED PRICE: Rs. 15,900
• View Details | 2,621 | 6,774 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.125 | 4 | CC-MAIN-2018-05 | latest | en | 0.887572 |
https://philoid.com/question/7704-a-chord-of-a-circle-of-radius-10-cm-subtends-a-right-angle-at-the-centre-find-the-area-of-the-corresponding-i-minor-segment-ii-m | 1,720,831,163,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514459.28/warc/CC-MAIN-20240712224556-20240713014556-00605.warc.gz | 389,110,457 | 8,307 | A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the corresponding:(i) Minor segment(ii) Major sector (Use π = 3.14)
Let AB be the chord of the circle subtending 90° angle at centre O of the circle.
Area of major sector OADB = () * r2
= () r2
= * 3.14 * 10 * 10
= 235.5 cm2
Area of minor sector OACB = * r2
= * 3.14 * 10 * 10
= 78.5 cm2
Area of ΔOAB = * OA * OB
= * 10 * 10
= 50 cm2
Area of minor segment ACB = Area of minor sector OACB -Area of ΔOAB
= 78.5 - 50
= 28.5 cm2
9 | 200 | 532 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.84375 | 4 | CC-MAIN-2024-30 | latest | en | 0.735394 |
https://www.jiskha.com/questions/549272/How-to-do-a-table-with-fractions-decimals-and-percents | 1,558,930,815,000,000,000 | text/html | crawl-data/CC-MAIN-2019-22/segments/1558232260658.98/warc/CC-MAIN-20190527025527-20190527051527-00089.warc.gz | 817,745,350 | 4,778 | # Tables
How to do a table with fractions,decimals,and percents
1. 👍 0
2. 👎 0
3. 👁 33
1. Try some of the following links:
http://search.yahoo.com/search?fr=mcafee&p=how+to+do+a+table+with+fractions%2C+decimlas+%26+percents
Sra
1. 👍 0
2. 👎 0
## Similar Questions
1. ### math
0k i don't know the answer for these percents, decimals, and fractions. You have to change decimals to percents, fractions to decimals, and percents to fractions. 0.23 3/100 32 1/2% 0.25 3/5 75% 1/8 0.835 10% 95% 4% 120%
asked by Kenya on March 14, 2008
2. ### Math
Well in school we were reviewing percents, and one of the main things being discussed was how to convert between fractions, decimals, and percents. However I didn't understand how she explained converting fractions to decimals.
asked by Jess on February 25, 2009
3. ### math
how do you diverse fractions to decimals and visa versa Just take the top number and divide it by the bottom one. For example, if you have 3/10, take 3 and divide it by 10, giving you .30 and this is 30%. Oh, the the other half,
asked by CHRIS on February 6, 2007
4. ### math
how do i convert fractions to decimals and fractions to percents?
asked by Bobby on December 1, 2014
5. ### math
how do you convert fractions to decimals and percents
asked by harley on April 7, 2009
6. ### fractions, decimals,and percents
What are some uses for fractions, decimals, and percents? FIND AT LEAST 2 AND TELL WHY!
asked by jj on November 17, 2008
7. ### arithmetic
how do u use fractions percents an decimals in real life
asked by genesis on October 12, 2010
8. ### math
where can i find information about fractions decimals and percents
asked by jaansia carswell on January 17, 2011
9. ### Math
Identify the group that does not contain equivalent fractions, decimals, and percents
asked by Sydney on August 29, 2011
10. ### Math
Describe two ways to determine the larger of two fractions. A. Compare fractions by writing fractions as mixed numbers B. Compare fractions with the same denominator by comparing the numerators and compare fractions with different
asked by Tina on October 25, 2017
More Similar Questions | 619 | 2,150 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.78125 | 4 | CC-MAIN-2019-22 | latest | en | 0.926343 |
https://vitoblafgan.by/article.php?1c5e39=finding-missing-side-length-of-rectangle-with-fractions | 1,606,968,520,000,000,000 | text/html | crawl-data/CC-MAIN-2020-50/segments/1606141718314.68/warc/CC-MAIN-20201203031111-20201203061111-00381.warc.gz | 429,027,608 | 4,441 | In this equation, C is the length of the hypotenuse while A and B represent the length of the other two sides. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. A B Divide into 7 segments. Find area of rectangle with fractional side lengths (5.NF.B.4b) Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Title: Finding Missing Side Length of Rectangle with Fractions Author: Robert Smith Created Date: 20170513091605+02'00' In the basic form above, you are required to know the length of Side A and the length of Side B, to calculate the length of Side C. Counting unit squares to find area formula. We tile the rectangle with unit rectangles, and show that the area is the same as would be found by multiplying the side lengths. 2. When two fractions that are both less than 1 are multiplied together, their product is smaller than either fraction. This fifth grade lesson explores the area of a rectangle with fractional side lengths. In this lesson, we find areas of rectagular figures that have fractional lengths and widths. This set of Area of Rectangles with Fractional Side Lengths task cards has 32 cards. How to Find the Perimeter of a Shape. Improve your math knowledge with free questions in "Find the area or missing side length of a rectangle" and thousands of other math skills. There are 12 "given area" problems and 12 "given perimeter" problems.
1-10 93 87 80 73 67 60 53 47 40 33 11-15 27 20 13 7 0 1) Area = 6¤ 24 cm 2 2/4 X 2) Area = 3¤ 24 cm 2 1/6 X 3) Area = 21¤ 40 cm 2 3/5 X 4) Area = 1¤ 10 cm 2 X … Worksheet 1 Finding the Area of a Rectangle with Fractional Side Lengths Complete.
Third grade FF.12 Find the missing side length of a rectangle X66. This game is used to practice finding the missing sides of rectangles given the area or perimeter. Perimeter is the length of the outer edge or boundary of a 2-dimensional shape and is expressed in units of length, such as inches or feet. Mark _4 7 on the line. In addition to it’s standard form, this theorem can also be rearranged and solved in other ways to compute any missing side of a right triangle.
Create your free account Teacher Student. In this lesson you will learn how to find the area of a rectangle with fractional side lengths by tiling. If you're seeing this message, it means we're having trouble loading external resources on our website. Worokshet 177 Practice computing the areas of rectangles that have fractional side lengths. We multiply fractional side lengths to find areas of rectangles. Mark _3 … Worksheet 1 Finding the Area of a Rectangle with Fractional Side Lengths Complete. C D Divide into 10 segments. Fraction Multiplication and Area. Find area, perimeter, or the missing side length (grades 3-4) View in browser Create PDF Write an expression for the area of a two-part rectangle in two ways, thinking of one rectangle … ... Finding missing side when given area.
Title: Finding Missing Side Length of Rectangle with Fractions Author: Robert Smith Created Date: 20170513091605+02'00'
Use this calculator if you know 2 values for the rectangle, including 1 side length, along with area, perimeter or diagonals and you can calculate the other 3 rectangle variables. Mark _3 10 on the line. Label each segment as a unit fraction.
How To Increase Melanin, Margaret O'brien Movies, Church Discipline Procedures, Ashtadhatu Idols Manufacturers In Kolkata, Unreal Tournament Directx 10, What Do Triceratops Eat, We Die Too, Danny Phantom Sam Age, E Type Music Videos, 7 Card Stud Hi-lo, What Type Of Government Does Canada Have 2019, | 901 | 3,780 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.15625 | 4 | CC-MAIN-2020-50 | latest | en | 0.903414 |
http://math.stackexchange.com/questions/tagged/determinant?page=17&sort=newest&pagesize=15 | 1,466,942,029,000,000,000 | text/html | crawl-data/CC-MAIN-2016-26/segments/1466783395166.84/warc/CC-MAIN-20160624154955-00125-ip-10-164-35-72.ec2.internal.warc.gz | 194,872,868 | 25,711 | # Tagged Questions
Question about determinants, computation or theory. If $E$ is a vector space of dimension $d$, then we can compute the determinant of a $d$-uple $(v_1,\ldots,v_d)$ with respect to a basis.
76 views
### Verifying whether a number is the determinant of a matrix
What is the (computationally) fastest way to determine whether a number is the determinant of a given real matrix? I am wondering if I have an upper bound on the absolute value of the determinant of ...
34 views
### How to find the determinant of a parallelogram using the vertices. (Using a matrix)
What it says on the tin. I already know how to find the determinant of a parallelogram using the vector components in a matrix, however, I am curious if there is a way to do it simply through the ...
370 views
77 views
### If $BA$ has $-1$ as an eigenvalue, then so does $AB$?
I was just encountered with a rather tough problem as follows: Suppose $A,B\in M_n(\mathbb R)$, prove: $$\det(I_n+AB)\ne0\Rightarrow\det(I_n+BA)\ne0$$ Although at this moment I am still at a ...
6k views
144 views
### Finding determinant of a 4x4 matrix
I am trying to find the determinant of this matrix but was told by my teacher that we wouldn't need to find the determinant of more than $3\times 3$ matrices so I am guessing there is a way of solving ...
25 views
57 views
161 views
### Can you prove My conjecture about Invertiblity of the Derivative Matrix ?! (to use Inverse function Theorem)
In the Analysis2 midterm exam, we had the following problem: Let the equation $a_nx^n+\cdots+a_1x+a_0=0$ has $n$ simple real roots (distinct) $\{\alpha_1,\cdots,\alpha_n\}$. Prove that the above ...
92 views
### How do you find the determinant of this $(n-1)\times (n-1)$ matrix?
It's for a proof of Cayley's Formula, I know I'm being dumb and can't see it, how do I find the determinant of this $(n-1)\times (n-1)$ matrix where the diagonal entries are $n-1$ and the off diagonal ...
46 views
### Value of determinant using given conditions.
Let $A$ be a $2$ x $2$ matrix with real entries and $det(A)$ is equal to $d$ which is non-zero. It is given that $det(A +d(adjA))=0$ where $adj$ stands for the adjoint of the matrix. We have to find ...
25 views
50 views
### Problems with the inverse of a banded matrix: not invertible?
I am creating with a software a banded matrix, which is also symmetric. In fact, its definition comes from an array, Array[q], whose length is ...
61 views
43 views
98 views
### Why is determinant called volume of the fundamental parallelepiped in geometry of numbers?
Let $v_1, ..., v_n$ be $n$ linearly independent vectors in $\mathbb{R}^n$. Then they form a lattice $\Lambda \subseteq \mathbb{R}^n$ and the volume of the fundamental domain is $|\det A|$, where $A$ ...
178 views | 745 | 2,794 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.53125 | 4 | CC-MAIN-2016-26 | latest | en | 0.876737 |
https://aptitude.gateoverflow.in/8033/cat-2020-set-2-question-67 | 1,669,573,351,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446710417.25/warc/CC-MAIN-20221127173917-20221127203917-00718.warc.gz | 139,060,163 | 20,575 | 125 views
If $\textsf{x}$ and $\textsf{y}$ are non-negative integers such that $\textsf{x+9=z, y+1=z}$ and $\textsf{x+y<z+5},$ then the maximum possible value of $\textsf{2x+y}$ equals
Given that, $x$ and $y$ are non-negative integers. That means $x \geq 0, y \geq 0.$
And,
• $x + 9 = z \Rightarrow x = z – 9 \; \longrightarrow (1)$
• $y + 1 = z \Rightarrow y = z – 1 \; \longrightarrow (2)$
• $x + y < z + 5 \; \longrightarrow (3)$
Put the value of $x,$ and $y$ in the equation $(3),$ we get.
$x + y < z + 5$
$\Rightarrow (z – 9) + (z – 1) < z + 5$
$\Rightarrow z – 10 < 5$
$\Rightarrow \boxed{z < 15}$
Maximum value of $z$ can be $14.$
So,
• $x_{\textsf{max}} = 14 – 9 = 5$
• $y_{\textsf{max}} = 14 – 1 = 13$
Thus, the value of $2x+y = 2(5) + 13 = 23.$
$\therefore$ The maximum possible value of $2x+y$ is $23.$
Correct Answer$: 23$
10.3k points
1
74 views
1 vote | 352 | 882 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.21875 | 4 | CC-MAIN-2022-49 | latest | en | 0.326208 |
https://howkgtolbs.com/convert/48.64-kg-to-lbs | 1,685,422,039,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224645089.3/warc/CC-MAIN-20230530032334-20230530062334-00069.warc.gz | 355,362,923 | 12,230 | # 48.64 kg to lbs - 48.64 kilograms to pounds
Do you want to learn how much is 48.64 kg equal to lbs and how to convert 48.64 kg to lbs? You are in the right place. You will find in this article everything you need to make kilogram to pound conversion - both theoretical and practical. It is also needed/We also want to point out that all this article is dedicated to only one amount of kilograms - this is one kilogram. So if you want to learn more about 48.64 kg to pound conversion - keep reading.
Before we move on to the more practical part - this is 48.64 kg how much lbs calculation - we are going to tell you few theoretical information about these two units - kilograms and pounds. So let’s start.
How to convert 48.64 kg to lbs? 48.64 kilograms it is equal 107.2328442368 pounds, so 48.64 kg is equal 107.2328442368 lbs.
## 48.64 kgs in pounds
We are going to begin with the kilogram. The kilogram is a unit of mass. It is a basic unit in a metric system, formally known as International System of Units (in short form SI).
Sometimes the kilogram is written as kilogramme. The symbol of the kilogram is kg.
The kilogram was defined first time in 1795. The kilogram was defined as the mass of one liter of water. First definition was not complicated but difficult to use.
Then, in 1889 the kilogram was defined using the International Prototype of the Kilogram (in short form IPK). The International Prototype of the Kilogram was made of 90% platinum and 10 % iridium. The International Prototype of the Kilogram was in use until 2019, when it was substituted by another definition.
Today the definition of the kilogram is based on physical constants, especially Planck constant. The official definition is: “The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed numerical value of the Planck constant h to be 6.62607015×10−34 when expressed in the unit J⋅s, which is equal to kg⋅m2⋅s−1, where the metre and the second are defined in terms of c and ΔνCs.”
One kilogram is exactly 0.001 tonne. It is also divided into 100 decagrams and 1000 grams.
## 48.64 kilogram to pounds
You know some information about kilogram, so now we can move on to the pound. The pound is also a unit of mass. It is needed to highlight that there are more than one kind of pound. What are we talking about? For example, there are also pound-force. In this article we are going to to concentrate only on pound-mass.
The pound is used in the British and United States customary systems of measurements. To be honest, this unit is used also in another systems. The symbol of the pound is lb or “.
The international avoirdupois pound has no descriptive definition. It is exactly 0.45359237 kilograms. One avoirdupois pound is divided into 16 avoirdupois ounces or 7000 grains.
The avoirdupois pound was implemented in the Weights and Measures Act 1963. The definition of this unit was written in first section of this act: “The yard or the metre shall be the unit of measurement of length and the pound or the kilogram shall be the unit of measurement of mass by reference to which any measurement involving a measurement of length or mass shall be made in the United Kingdom; and- (a) the yard shall be 0.9144 metre exactly; (b) the pound shall be 0.45359237 kilogram exactly.”
### How many lbs is 48.64 kg?
48.64 kilogram is equal to 107.2328442368 pounds. If You want convert kilograms to pounds, multiply the kilogram value by 2.2046226218.
### 48.64 kg in lbs
The most theoretical section is already behind us. In next section we want to tell you how much is 48.64 kg to lbs. Now you learned that 48.64 kg = x lbs. So it is time to get the answer. Just see:
48.64 kilogram = 107.2328442368 pounds.
That is an exact outcome of how much 48.64 kg to pound. You may also round off this result. After it your result will be as following: 48.64 kg = 107.008 lbs.
You know 48.64 kg is how many lbs, so have a look how many kg 48.64 lbs: 48.64 pound = 0.45359237 kilograms.
Naturally, this time it is possible to also round off the result. After it your result will be exactly: 48.64 lb = 0.45 kgs.
We also want to show you 48.64 kg to how many pounds and 48.64 pound how many kg results in tables. Let’s see:
We want to begin with a table for how much is 48.64 kg equal to pound.
### 48.64 Kilograms to Pounds conversion table
Kilograms (kg) Pounds (lb) Pounds (lbs) (rounded off to two decimal places)
48.64 107.2328442368 107.0080
Now look at a table for how many kilograms 48.64 pounds.
Pounds Kilograms Kilograms (rounded off to two decimal places
48.64 0.45359237 0.45
Now you learned how many 48.64 kg to lbs and how many kilograms 48.64 pound, so it is time to go to the 48.64 kg to lbs formula.
### 48.64 kg to pounds
To convert 48.64 kg to us lbs you need a formula. We are going to show you two formulas. Let’s begin with the first one:
Amount of kilograms * 2.20462262 = the 107.2328442368 outcome in pounds
The first version of a formula will give you the most accurate outcome. Sometimes even the smallest difference can be significant. So if you want to get an accurate result - this formula will be the best for you/option to calculate how many pounds are equivalent to 48.64 kilogram.
So let’s go to the shorer formula, which also enables conversions to know how much 48.64 kilogram in pounds.
The another formula is down below, look:
Amount of kilograms * 2.2 = the outcome in pounds
As you can see, the second formula is simpler. It can be better choice if you need to make a conversion of 48.64 kilogram to pounds in easy way, for instance, during shopping. You only need to remember that your result will be not so accurate.
Now we want to show you these two versions of a formula in practice. But before we are going to make a conversion of 48.64 kg to lbs we are going to show you another way to know 48.64 kg to how many lbs without any effort.
### 48.64 kg to lbs converter
Another way to know what is 48.64 kilogram equal to in pounds is to use 48.64 kg lbs calculator. What is a kg to lb converter?
Converter is an application. It is based on first formula which we showed you above. Thanks to 48.64 kg pound calculator you can quickly convert 48.64 kg to lbs. You only need to enter number of kilograms which you need to convert and click ‘convert’ button. The result will be shown in a flash.
So let’s try to calculate 48.64 kg into lbs with use of 48.64 kg vs pound calculator. We entered 48.64 as an amount of kilograms. This is the outcome: 48.64 kilogram = 107.2328442368 pounds.
As you see, our 48.64 kg vs lbs converter is so simply to use.
Now we can move on to our chief issue - how to convert 48.64 kilograms to pounds on your own.
#### 48.64 kg to lbs conversion
We are going to begin 48.64 kilogram equals to how many pounds calculation with the first version of a formula to get the most correct result. A quick reminder of a formula:
Number of kilograms * 2.20462262 = 107.2328442368 the outcome in pounds
So what need you do to know how many pounds equal to 48.64 kilogram? Just multiply number of kilograms, in this case 48.64, by 2.20462262. It is exactly 107.2328442368. So 48.64 kilogram is exactly 107.2328442368.
It is also possible to round it off, for instance, to two decimal places. It is 2.20. So 48.64 kilogram = 107.0080 pounds.
It is time for an example from everyday life. Let’s calculate 48.64 kg gold in pounds. So 48.64 kg equal to how many lbs? And again - multiply 48.64 by 2.20462262. It is equal 107.2328442368. So equivalent of 48.64 kilograms to pounds, when it comes to gold, is exactly 107.2328442368.
In this case you can also round off the result. Here is the outcome after rounding off, in this case to one decimal place - 48.64 kilogram 107.008 pounds.
Now we can move on to examples calculated using a short version of a formula.
#### How many 48.64 kg to lbs
Before we show you an example - a quick reminder of shorter formula:
Number of kilograms * 2.2 = 107.008 the result in pounds
So 48.64 kg equal to how much lbs? As in the previous example you need to multiply amount of kilogram, this time 48.64, by 2.2. See: 48.64 * 2.2 = 107.008. So 48.64 kilogram is exactly 2.2 pounds.
Let’s do another conversion with use of this version of a formula. Now convert something from everyday life, for instance, 48.64 kg to lbs weight of strawberries.
So let’s calculate - 48.64 kilogram of strawberries * 2.2 = 107.008 pounds of strawberries. So 48.64 kg to pound mass is equal 107.008.
If you learned how much is 48.64 kilogram weight in pounds and can convert it using two different versions of a formula, we can move on. Now we want to show you these outcomes in charts.
#### Convert 48.64 kilogram to pounds
We are aware that outcomes presented in tables are so much clearer for most of you. It is totally understandable, so we gathered all these results in charts for your convenience. Thanks to this you can easily compare 48.64 kg equivalent to lbs outcomes.
Let’s start with a 48.64 kg equals lbs chart for the first version of a formula:
Kilograms Pounds Pounds (after rounding off to two decimal places)
48.64 107.2328442368 107.0080
And now have a look at 48.64 kg equal pound table for the second formula:
Kilograms Pounds
48.64 107.008
As you can see, after rounding off, if it comes to how much 48.64 kilogram equals pounds, the outcomes are not different. The bigger amount the more considerable difference. Keep it in mind when you want to do bigger amount than 48.64 kilograms pounds conversion.
#### How many kilograms 48.64 pound
Now you learned how to convert 48.64 kilograms how much pounds but we want to show you something more. Are you curious what it is? What about 48.64 kilogram to pounds and ounces calculation?
We are going to show you how you can convert it step by step. Let’s start. How much is 48.64 kg in lbs and oz?
First thing you need to do is multiply number of kilograms, this time 48.64, by 2.20462262. So 48.64 * 2.20462262 = 107.2328442368. One kilogram is exactly 2.20462262 pounds.
The integer part is number of pounds. So in this case there are 2 pounds.
To calculate how much 48.64 kilogram is equal to pounds and ounces you have to multiply fraction part by 16. So multiply 20462262 by 16. It is 327396192 ounces.
So your outcome is exactly 2 pounds and 327396192 ounces. It is also possible to round off ounces, for example, to two places. Then your result is 2 pounds and 33 ounces.
As you see, conversion 48.64 kilogram in pounds and ounces simply.
The last conversion which we will show you is conversion of 48.64 foot pounds to kilograms meters. Both foot pounds and kilograms meters are units of work.
To calculate it it is needed another formula. Before we give you this formula, see:
• 48.64 kilograms meters = 7.23301385 foot pounds,
• 48.64 foot pounds = 0.13825495 kilograms meters.
Now look at a formula:
Amount.RandomElement()) of foot pounds * 0.13825495 = the result in kilograms meters
So to convert 48.64 foot pounds to kilograms meters you need to multiply 48.64 by 0.13825495. It is equal 0.13825495. So 48.64 foot pounds is exactly 0.13825495 kilogram meters.
It is also possible to round off this result, for instance, to two decimal places. Then 48.64 foot pounds is equal 0.14 kilogram meters.
We hope that this calculation was as easy as 48.64 kilogram into pounds calculations.
This article was a big compendium about kilogram, pound and 48.64 kg to lbs in calculation. Due to this calculation you know 48.64 kilogram is equivalent to how many pounds.
We showed you not only how to make a conversion 48.64 kilogram to metric pounds but also two other calculations - to check how many 48.64 kg in pounds and ounces and how many 48.64 foot pounds to kilograms meters.
We showed you also other way to do 48.64 kilogram how many pounds calculations, it is with use of 48.64 kg en pound converter. This will be the best option for those of you who do not like converting on your own at all or need to make @baseAmountStr kg how lbs conversions in quicker way.
We hope that now all of you are able to do 48.64 kilogram equal to how many pounds calculation - on your own or with use of our 48.64 kgs to pounds converter.
Don’t wait! Calculate 48.64 kilogram mass to pounds in the way you like.
Do you need to do other than 48.64 kilogram as pounds conversion? For instance, for 10 kilograms? Check our other articles! We guarantee that conversions for other numbers of kilograms are so simply as for 48.64 kilogram equal many pounds.
### How much is 48.64 kg in pounds
At the end, we are going to summarize the topic of this article, that is how much is 48.64 kg in pounds , we gathered answers to the most frequently asked questions. Here you can see all you need to remember about how much is 48.64 kg equal to lbs and how to convert 48.64 kg to lbs . Let’s see.
What is the kilogram to pound conversion? To make the kg to lb conversion it is needed to multiply 2 numbers. How does 48.64 kg to pound conversion formula look? . Have a look:
The number of kilograms * 2.20462262 = the result in pounds
How does the result of the conversion of 48.64 kilogram to pounds? The accurate answer is 107.2328442368 pounds.
It is also possible to calculate how much 48.64 kilogram is equal to pounds with another, easier version of the formula. Have a look.
The number of kilograms * 2.2 = the result in pounds
So in this case, 48.64 kg equal to how much lbs ? The result is 107.2328442368 lbs.
How to convert 48.64 kg to lbs in just a moment? You can also use the 48.64 kg to lbs converter , which will make whole mathematical operation for you and you will get an accurate result .
#### Kilograms [kg]
The kilogram, or kilogramme, is the base unit of weight in the Metric system. It is the approximate weight of a cube of water 10 centimeters on a side.
#### Pounds [lbs]
A pound is a unit of weight commonly used in the United States and the British commonwealths. A pound is defined as exactly 0.45359237 kilograms. | 3,660 | 14,053 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.125 | 4 | CC-MAIN-2023-23 | longest | en | 0.943077 |
https://www.varsitytutors.com/hspt_math-help/problem-solving?page=61 | 1,642,396,126,000,000,000 | text/html | crawl-data/CC-MAIN-2022-05/segments/1642320300289.37/warc/CC-MAIN-20220117031001-20220117061001-00483.warc.gz | 1,126,777,952 | 53,122 | # HSPT Math : Problem Solving
## Example Questions
1 2 53 54 55 56 57 58 59 61 Next →
### Example Question #601 : Problem Solving
A certain cube has a side length of 25 m. How many square tiles, each with an area of 5 m2, are needed to fully cover the surface of the cube?
500
750
100
200
1000
750
Explanation:
A cube with a side length of 25m has a surface area of:
25m * 25m * 6 = 3,750 m2
(The surface area of a cube is equal to the area of one face of the cube multiplied by 6 sides. In other words, if the side of a cube is s, then the surface area of the cube is 6s2.)
Each square tile has an area of 5 m2.
Therefore, the total number of square tiles needed to fully cover the surface of the cube is:
3,750m2/5m= 750
Note: the volume of a cube with side length s is equal to s3. Therefore, if asked how many mini-cubes with side length n are needed to fill the original cube, the answer would be:
s3/n3
### Example Question #1 : How To Find The Surface Area Of A Cube
A company wants to build a cubical room around a cone so that the cone's height and diameter are 3 inch less than the dimensions of the cube. If the volume of the cone is 486π ft3, what is the surface area of the cube?
486 in2
69,984 in2
726 in2
73,926 in2
513.375 in2
73,926 in2
Explanation:
To begin, we need to solve for the dimensions of the cone.
The basic form for the volume of a cone is: V = (1/3)πr2h
Using our data, we know that h = 2r because the height of the cone matches its diameter (based on the prompt).
486π = (1/3)πr* 2r = (2/3)πr3
Multiply both sides by (3/2π): 729 = r3
Take the cube root of both sides: r = 9
Note that this is in feet. The answers are in square inches. Therefore, convert your units to inches: 9 * 12 = 108, then add 3 inches to this: 108 + 3 = 111 inches.
The surface area of the cube is defined by: A = 6 * s2, or for our data, A = 6 * 1112 = 73,926 in2
### Example Question #601 : Problem Solving
Angie is painting a 2 foot cube for a play she is in. She needs of paint for every square foot she paints. How much paint does she need?
It is impossible to convert between metric units and feet.
Explanation:
First we must calculate the surface area of the cube. We know that there are six surfaces and each surface has the same area:
Now we will determine the amount of paint needed
### Example Question #1 : How To Find The Surface Area Of A Sphere
A spherical orange fits snugly inside a small cubical box such that each of the six walls of the box just barely touches the surface of the orange. If the volume of the box is 64 cubic inches, what is the surface area of the orange in square inches?
256π
128π
64π
32π
16π
16π
Explanation:
The volume of a cube is found by V = s3. Since V = 64, s = 4. The side of the cube is the same as the diameter of the sphere. Since d = 4, r = 2. The surface area of a sphere is found by SA = 4π(r2) = 4π(22) = 16π.
### Example Question #1 : How To Find The Area Of A Rectangle
Steve's bedroom measures 20' by 18' by 8' high. He wants to paint the ceiling and all four walls using a paint that gets 360 square feet of coverage per gallon. A one-gallon can of the paint Steve wants costs $36.00; a one-quart can costs$13.00. What is the least amount of money that Steve can expect to spend on paint in order to paint his room?
Explanation:
Two of the walls have area ; two have area ; the ceiling has area
Therefore, the total area Steve wants to cover is
Divide 968 by 360 to get the number of gallons Steve needs to paint his bedroom:
Since , Steve needs to purchase either two gallon cans and three quart cans, or three gallon cans.
The first option will cost him ; the second option will cost him . The latter is the more economical option.
### Example Question #604 : Problem Solving
Give the surface area of the above box in square centimeters.
Explanation:
100 centimeters make one meter, so convert each of the dimensions of the box by multiplying by 100.
centimeters
centimeters
Use the surface area formula, substituting :
square centimeters
### Example Question #601 : Problem Solving
Note: Figure NOT drawn to scale.
Refer to the above diagram, which shows a square. Give the ratio of the area of the yellow region to that of the white region.
The correct answer is not given among the other choices.
Explanation:
The area of the entire square is the square of the length of a side, or
.
The area of the right triangle is half the product of its legs, or
.
The area of the yellow region is therefore the difference of the two, or
.
The ratio of the area of the yellow region to that of the white region is
; that is, 55 to 9.
### Example Question #121 : Plane Geometry
The above depicts a rectangular swimming pool for an apartment. The pool is five feet deep everywhere.
An apartment manager wants to paint the four sides and the bottom of the swimming pool. One one-gallon can of the paint he wants to use covers square feet. How many cans of the paint will the manager need to buy?
Explanation:
The bottom of the swimming pool has area
square feet.
There are two sides whose area is
square feet,
and two sides whose area is
square feet.
square feet.
One one-gallon can of paint covers 350 square feet, so divide:
Seven full gallons and part of another are required, so eight is the correct answer.
### Example Question #601 : Problem Solving
The surface area of the above cylinder is 80% of what number?
Explanation:
The surface area of the cylinder can be calculated by setting and in the formula
This is 80% of the number
### Example Question #608 : Problem Solving
What is the surface area of a cube with a side edge of 7? | 1,511 | 5,704 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.59375 | 5 | CC-MAIN-2022-05 | latest | en | 0.854706 |
https://convertoctopus.com/7-6-knots-to-miles-per-hour | 1,600,603,663,000,000,000 | text/html | crawl-data/CC-MAIN-2020-40/segments/1600400197946.27/warc/CC-MAIN-20200920094130-20200920124130-00070.warc.gz | 343,434,845 | 7,715 | ## Conversion formula
The conversion factor from knots to miles per hour is 1.1507794480225, which means that 1 knot is equal to 1.1507794480225 miles per hour:
1 kt = 1.1507794480225 mph
To convert 7.6 knots into miles per hour we have to multiply 7.6 by the conversion factor in order to get the velocity amount from knots to miles per hour. We can also form a simple proportion to calculate the result:
1 kt → 1.1507794480225 mph
7.6 kt → V(mph)
Solve the above proportion to obtain the velocity V in miles per hour:
V(mph) = 7.6 kt × 1.1507794480225 mph
V(mph) = 8.7459238049714 mph
The final result is:
7.6 kt → 8.7459238049714 mph
We conclude that 7.6 knots is equivalent to 8.7459238049714 miles per hour:
7.6 knots = 8.7459238049714 miles per hour
## Alternative conversion
We can also convert by utilizing the inverse value of the conversion factor. In this case 1 mile per hour is equal to 0.11433897919755 × 7.6 knots.
Another way is saying that 7.6 knots is equal to 1 ÷ 0.11433897919755 miles per hour.
## Approximate result
For practical purposes we can round our final result to an approximate numerical value. We can say that seven point six knots is approximately eight point seven four six miles per hour:
7.6 kt ≅ 8.746 mph
An alternative is also that one mile per hour is approximately zero point one one four times seven point six knots.
## Conversion table
### knots to miles per hour chart
For quick reference purposes, below is the conversion table you can use to convert from knots to miles per hour
knots (kt) miles per hour (mph)
8.6 knots 9.897 miles per hour
9.6 knots 11.047 miles per hour
10.6 knots 12.198 miles per hour
11.6 knots 13.349 miles per hour
12.6 knots 14.5 miles per hour
13.6 knots 15.651 miles per hour
14.6 knots 16.801 miles per hour
15.6 knots 17.952 miles per hour
16.6 knots 19.103 miles per hour
17.6 knots 20.254 miles per hour | 551 | 1,904 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.03125 | 4 | CC-MAIN-2020-40 | latest | en | 0.779002 |
https://www.jiskha.com/display.cgi?id=1186988473 | 1,510,966,447,000,000,000 | text/html | crawl-data/CC-MAIN-2017-47/segments/1510934804125.49/warc/CC-MAIN-20171118002717-20171118022717-00204.warc.gz | 834,180,509 | 3,959 | # maths
posted by .
A small primary school has a total enrolment of 50 students. A total of 20 of these play hockey, 18 play netball and 10 play both hockey and netball. How many students play neither hockey nor netball?
Of the 38 who play netball or hockey, ten play both. That makes 28 who are players. So, of the fifty, the remaining are not sports players.
• maths -
2 students
10+18+20=48
50-48=2
## Respond to this Question
First Name School Subject Your Answer
## Similar Questions
1. ### Maths
In a class 40 students,31 play football,26play hockey and 13play volleyball.19 play football and hockey,8play both football and volley,and 7play both hockey and volleyball. Each student plays at least one of the three games. i. Illustrate …
2. ### maths sets
All the people in a class of 35 play at least one of the games volleball,netball and hockey.10 play volleball only,5 play netball only and 3 play hockey only.if 2 play many of all 3 gamez and the equal number play 2 games.how many …
3. ### o level mathematics
in a class of 35, all the pupils play at least 1 game, volleyball, netball and hockey. volleyball only 10, those who play netball only 5, those who play hockey only 3, if 2 play all 3 games how many altogether play volleyball
4. ### nyonithwele
math: 38 learners at a high school,24play hockey,30play netball and 4play neither hockey nor netball.use venn diagram
5. ### matematicas
in a class of 25 students, 15 play hockeyand 16 play basketball. if there are 4 students who play neither sport, determine the number of students who play both hockey and basketball
6. ### Further mathf
In a senior secondary school, 80 students play hockey or football. The number that play football is 5 more than twice the number that play hockey. If the 15 students play both games and every student in the school playsat least one …
7. ### Math
Consider the propositions p: "You like hockey.", q: "You like the cold.", r: "You play hockey.", and s: "You watch hockey." Write the following propositions using p,q,r and s and logical connectives. (a) You watch hockey but you do …
8. ### Math
In a form of 30 girls, 18 play netball and 14 play hockey, whilst 5 play neither. Find the number of those who play both netball and hockey?
9. ### furth math
in a school,180 students play hockey or football.the number that play football is 5 more than twice the number that play hockey.if 15 students play both games and every student in the school plays at least one game,find:1) the number …
10. ### math
in a school,180 students play hockey or football.the number that play football is 5 more than twice the number that play hockey.if 15 students play both games and every student in the school plays at least one game,find:1) the number …
More Similar Questions | 701 | 2,785 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.609375 | 4 | CC-MAIN-2017-47 | latest | en | 0.921702 |
http://www.sparknotes.com/math/prealgebra/variables/problems_1.html | 1,511,469,271,000,000,000 | text/html | crawl-data/CC-MAIN-2017-47/segments/1510934806939.98/warc/CC-MAIN-20171123195711-20171123215711-00776.warc.gz | 492,876,483 | 12,967 | # Prealgebra: Variables
## Contents
#### Problems
Problem : What is r + 5 if r = 7 ? If r = - 11 ?
12, - 6
Problem : What is 2k if k = 15 ? If k = 100 ?
30, 200
Problem : What is r 2 if r = 15 ? If r = 2/3 ?
225, 4/9
Problem : What is 15f - 12 if f = 1 ? If f = 3 ?
3, 33
Problem : What is 4×(m - 6) + 34 if m = 12 ? If m = 0 ?
58, 10 | 152 | 346 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.5625 | 4 | CC-MAIN-2017-47 | latest | en | 0.418866 |
https://answers.search.yahoo.com/search?p=Cm&ei=UTF-8&flt=cat%3AHomework%2BHelp&fr2=sortBy&b=1&context=%7C%7Cgsmcontext%3A%3Aage%3A%3A1y%7C | 1,611,410,762,000,000,000 | text/html | crawl-data/CC-MAIN-2021-04/segments/1610703538082.57/warc/CC-MAIN-20210123125715-20210123155715-00200.warc.gz | 217,411,801 | 33,103 | # Yahoo Web Search
1. ### ABCDE is a regular Pentagon of 10 cm size. FGHIJ are mid points of AB, BC, CD, DE and EA. If 1 kg mass has been placed on every point .....?
I assume that "10 cm size" means a perimeter of 10 cm . The internal angles of...1*cos108º = 2.618 cm² by the law of cosines c = 1.618 cm so the semiperimeter is s = ½(1.618 + 1 + 1) cm = 1.809 cm ...
2 Answers · Science & Mathematics · 18/09/2020
2. ### Would I have to change 16m into cm ? And what is the height of the building?
You do not have to convert the 16 m into cm . Since the conversion from m to cm would be the same either way, they cancel out...
2 Answers · Science & Mathematics · 26/11/2020
3. ### The parallel sides of a trapezium are 10 cm and 20 cm respectively. AD = 6 cm and CB =8 cm . Find the area of the trapezium ?
... and CD are the parallel sides with lengths 10 cm and 20 cm , respectively. See the attached sketch. Let point ...
3 Answers · Science & Mathematics · 03/08/2020
4. ### 22.4 A uniformly charged solid sphere with a total charge of 50 nC and a radius R1 = 30 cm is surrounded concentrically by a...?
a) at 14 cm , the "enclosed" charge is Q = 50nC * (14/30... * 5.08e-9C / (0.14m)² = 2331 N/C (outward) b) at 37 cm , the enclosed charge is 50 nC: E = 8.99e9N·m²...
1 Answers · Science & Mathematics · 30/09/2020
5. ### As shown in the figure two loops carry currents with I1 = 3.6 A and I2 =9.4 A. The radii r1 = 12.5 cm and r2 = 18.5 cm . If a long current?
...7=1.8097x 10^-5T In the smaller loop, the radius = r1 = 12.5 cm = 0.125 m I1 = 3.6 Amp hence magneti field at the center = B1...
1 Answers · Science & Mathematics · 14/05/2020
6. ### An ant starts at x = 20 cm on a piece of graph paper and walks along the x axis to x = -20 cm .?
displacement = 20 cm - (- 20cm) + (- 10 cm ) = 30 cm distance covered = |(-20cm) - (20cm)| + |(-10cm) - (-20 cm )| = 40cm + 10cm = 50 cm
1 Answers · Science & Mathematics · 05/07/2020
7. ### how to convert km into cm ?
add 5 zeroes to the right side of the number. 1000 (km/m) x 100 ( cm /m) = 100,000 ( cm /km). five zeroes. follow the units.
6 Answers · Science & Mathematics · 01/07/2020
8. ### My bust is over 100 cm .It is because I got fat?Or its my bones?
...know anything about you (age, height, sex, weight) so have no way of knowing if 100 cm chest measurement is appropriate or not. Assuming you are female (since...
3 Answers · Health · 15/01/2021
9. ### The area of a sector of a circle with a radius of 15 cm is 75pi cm ^2. ?
The area of a sector of a circle with a radius of 15 cm is 75pi cm ^2. ? Total circle Area = π(15)^2 = 225π cm ^2...
11 Answers · Science & Mathematics · 28/05/2020
10. ### What is the density of a cube of titanium that has a volume of 5.43 cm ^3?
The density of solid titanium is 4.506 g/ cm ^3. There is a very small difference for temperature and pressure ... very small...
3 Answers · Science & Mathematics · 03/11/2020 | 962 | 2,947 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.515625 | 4 | CC-MAIN-2021-04 | latest | en | 0.792994 |
https://math.stackexchange.com/questions/2804424/evaluate-indefinite-integral-int-tan-fracx3-dx | 1,716,494,284,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971058653.47/warc/CC-MAIN-20240523173456-20240523203456-00553.warc.gz | 327,805,265 | 39,103 | # Evaluate indefinite integral $\int \tan(\frac{x}{3}) \, dx$
I need help verifying why I am getting an incorrect answer for the question evaluate the integral $$\int \tan\left(\frac{x}{3}\right) \, dx$$
I simplify the above equation using trig identities to get $$\int \frac {\sin \left(\frac{x}{3}\right)}{\cos\left(\frac{x}{3}\right)} \, dx$$
I use the substitution method to find $$du = -\frac{1}{3} \sin(x/3) \, dx$$ and so $dx = \frac{-3\,du}{\sin\frac{x}{3}}$
I plug the $u$ back into equation $$\int \frac {\sin\left(\frac{x}{3}\right)}{u} \cdot\frac {-3\,du}{\sin \left(\frac{x}{3}\right)}$$
I cross out the $\sin \left(\frac{x}{3}\right)$ and (this is where I may be going wrong), I pull out the $-3$ to be in front of the integral sign since it is a constant and solve for $$-3 \int \frac{1}{u} \, du$$ and get the final answer $$-3 \biggl|\,\ln \, \cos \frac{x}{3}\biggr| + C$$
But the answer in the back of the book is $-\frac{1}{3} |\ln \, \cos \frac{x}{3}| + C$
• The answer in the book is wrong . You have the correct answer Jun 1, 2018 at 15:09
• Another way to see the book is wrong is to substitute $y=x/3$, so the integral becomes $3\tan y dy$. So whatever that integrates to, the predictor must be $3$, not $1/3$.
– J.G.
Jun 1, 2018 at 15:32
Your book is wrong! As a check, $$\frac{d}{dx}\left(-\frac13\bigg|\ln\cos\frac x3\bigg|\right)=-\frac1{3\cos\frac x3}\cdot\left(-\frac13\sin\frac x3\right)=\color{red}{\frac19}\tan\frac x3\neq \tan\frac x3.$$
(...) and get the final answer $$-3 |\ln \, \cos \frac{x}{3}| + C$$
But the answer in the back of the book is $$-\frac{1}{3} |\ln \, \cos \frac{x}{3}| + C$$
You can differentiate to verify but you are right and the book is wrong!
Just to make your life a bit simpler without this fraction $\frac{x}{3}$, just use u-sub.
Put: $\frac{x}{3}=u$
Then $\int tan(\frac{x}{3})dx= 3\int tan\ u\ du= 3$ln |sec u| +C$=$3 ln|sec($\frac{x}{3})|+C$
At a glance, you're correct, because you'll divide by the $\frac 13$when you take it out of the trig function, not multiply by it.
However, this formula sheet states that $$\int{\tan(x)dx}=\ln|\sec(x)|+C$$ (see the $C4$ section - page $9$)
Your book is wrong as this is a standard integral and the value is:
-3 ln|cos(x/3)| + C.
If you want verification, differentiate this and u will get
tan(x/3) | 794 | 2,324 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 2, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.21875 | 4 | CC-MAIN-2024-22 | latest | en | 0.723403 |
https://jajalger2018.org/what-is-the-cosine-of-30-degrees/ | 1,653,590,586,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652662619221.81/warc/CC-MAIN-20220526162749-20220526192749-00018.warc.gz | 390,254,348 | 4,747 | The worth of cos 30 levels is 0.8660254. . .
You are watching: What is the cosine of 30 degrees
. Cos 30 levels in radians is written as cos (30° × π/180°), i.e., cos (π/6) or cos (0.523598. . .). In this article, we will talk about the techniques to uncover the worth of cos 30 levels with examples.
Cos 30°: 0.8660254. . .Cos 30° in fraction: √3/2Cos (-30 degrees): 0.8660254. . .Cos 30° in radians: cos (π/6) or cos (0.5235987 . . .)
## What is the worth of Cos 30 Degrees?
The value of cos 30 degrees in decimal is 0.866025403. . .. Cos 30 degrees can likewise be expressed making use of the equivalent of the provided angle (30 degrees) in radians (0.52359 . . .)
We know, using level to radian conversion, θ in radians = θ in levels × (pi/180°)⇒ 30 degrees = 30° × (π/180°) rad = π/6 or 0.5235 . . .∴ cos 30° = cos(0.5235) = √3/2 or 0.8660254. . .
Explanation:
For cos 30 degrees, the angle 30° lies in between 0° and also 90° (First Quadrant). Since cosine function is optimistic in the first quadrant, thus cos 30° value = √3/2 or 0.8660254. . .Since the cosine duty is a routine function, we have the right to represent cos 30° as, cos 30 levels = cos(30° + n × 360°), n ∈ Z.⇒ cos 30° = cos 390° = cos 750°, and also so on.Note: Since, cosine is an also function, the value of cos(-30°) = cos(30°).
## Methods to discover Value the Cos 30 Degrees
The cosine duty is hopeful in the 1st quadrant. The value of cos 30° is offered as 0.86602. . .. Us can find the worth of cos 30 degrees by:
Using Trigonometric FunctionsUsing Unit Circle
## Cos 30° in terms of Trigonometric Functions
Using trigonometry formulas, we can represent the cos 30 levels as:
± √(1-sin²(30°))± 1/√(1 + tan²(30°))± cot 30°/√(1 + cot²(30°))±√(cosec²(30°) - 1)/cosec 30°1/sec 30°
Note: due to the fact that 30° lies in the first Quadrant, the last value the cos 30° will certainly be positive.
See more: What Channel Is Nickelodeon On Dish Network Channels Guide, Nickelodeon On Dish Network: Channel 170
We have the right to use trigonometric identities to represent cos 30° as,
-cos(180° - 30°) = -cos 150°-cos(180° + 30°) = -cos 210°sin(90° + 30°) = sin 120°sin(90° - 30°) = sin 60°
## Cos 30 degrees Using Unit Circle
To find the value of cos 30 degrees using the unit circle:
Rotate ‘r’ anticlockwise to form 30° angle with the confident x-axis.The cos the 30 degrees equals the x-coordinate(0.866) the the allude of intersection (0.866, 0.5) of unit circle and also r.
Hence the value of cos 30° = x = 0.866 (approx) | 840 | 2,524 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.53125 | 5 | CC-MAIN-2022-21 | latest | en | 0.800056 |
http://discontinued-items.com/lego-airport-passenger-plane-lego-airport-retired.html | 1,547,916,704,000,000,000 | text/html | crawl-data/CC-MAIN-2019-04/segments/1547583671342.16/warc/CC-MAIN-20190119160425-20190119182425-00469.warc.gz | 57,890,378 | 6,238 | In December 2012, the BBC's More or Less radio program asked the Open University's engineering department to determine "how many Lego bricks, stacked one on top of the other, it would take for the weight to destroy the bottom brick?"[44] Using a hydraulic testing machine, the engineering department determined the average maximum force a 2×2 Lego brick can stand is 4,240 newtons; since an average 2×2 Lego brick has a mass of 1.152 grams (0.0406 oz), according to their calculations it would take a stack of 375,000 bricks to cause the bottom brick to collapse, which represents a stack 3,591 metres (11,781 ft) in height.[44]
The airport handles an average of more than two million passengers a year, and millions of pounds of cargo. The airport's main runway can handle airliners as large as the Boeing 747, although most passengers arrive on smaller aeroplanes, such as ATR-42s, Boeing 737s and Boeing 757s. Boeing 747 activity at this airport is almost exclusively limited to cargo flights.
Lego pieces of all varieties constitute a universal system. Despite variation in the design and the purposes of individual pieces over the years, each piece remains compatible in some way with existing pieces. Lego bricks from 1958 still interlock with those made in the current time, and Lego sets for young children are compatible with those made for teenagers. Six bricks of 2 × 4 studs[28] can be combined in 915,103,765 ways.[29]
^ Jump up to: a b Alexander, Ruth (3 December 2012). "How tall can a Lego tower get?". BBC News. Archived from the original on 4 December 2012. Retrieved 4 December 2012. The average maximum force the bricks can stand is 4,240N. That's equivalent to a mass of 432 kg (950lbs). If you divide that by the mass of a single brick, which is 1.152g, then you get the grand total of bricks a single piece of Lego could support: 375,000. So, 375,000 bricks towering 3.5 kilometers (2.17 miles) high is what it would take to break a Lego brick.
The programmable Lego brick which is at the heart of these robotics sets has undergone several updates and redesigns, with the latest being called the 'EV3' brick, being sold under the name of Lego Mindstorms EV3. The set includes sensors that detect touch, light, sound and ultrasonic waves, with several others being sold separately, including an RFID reader.[52]
The definitive shape of the Lego bricks, with the inner tubes, was patented by the Lego Group in 1958.[15][56] Several competitors have attempted to take advantage of Lego's popularity by producing blocks of similar dimensions, and advertising them as being compatible with Lego bricks. In 2002, Lego sued the CoCo Toy Company in Beijing for copyright infringement over its "Coko bricks" product. CoCo was ordered to cease manufacture of the products, publish a formal apology and pay damages.[57] Lego sued the English company Best-Lock Construction Toys in German courts in 2004[58] and 2009;[59] the Federal Patent Court of Germany denied Lego trademark protection for the shape of its bricks for the latter case.[60] In 2005, the Lego Company sued Canadian company Mega Bloks for trademark violation, but the Supreme Court of Canada upheld Mega Bloks' rights to sell their product.[61] In 2010, the European Court of Justice ruled that the eight-peg design of the original Lego brick "merely performs a technical function [and] cannot be registered as a trademark."[62]
Copenhagen Airport is a compact airport with two terminals for check-in, but with a common post-security departure and transfer area, thus making transfers very smooth. The airport handles in excess of 25 million passengers a year, which puts it in the same league as Zurich and Vienna. It has also consistently topped the charts for Nordic airports, serving as a European and intercontinental hub for all of them due to its location.
After World City's discontinuation in 2004, it was replaced with City as System's primary town-life related theme in 2005. However, unlike its predecessor, it was not limited to sets involving rescue services like Police, Fire or Coast Guard, but also introduced the first new construction site related sets since the discontinuation of City Center in 2000. In 2006, the first new airport set that included the first jetway since 6597 Century Skyway (1994) was released, as well as the first new hospital since 6380 Emergency Treatment Center from 1987. These releases expanded City to a scope only comparable to the original Town theme, and in 2009 it even went slightly beyond that, by introducing the first farm-related System sets. Also, in that year, City followed this route to release more truly civilian town life sets such as 7641 City Corner and 7639 Camper. In 2010 City released more civilian sets including 8403 Family House and 8404 Public Transport. In 2011 City re-introduced the Space sub-theme along with some new Harbour sets. In the first wave of 2012, City had forest police and fire sets as well as some more commercial/civilian sets. The second wave of 2012 featured the introduction of the Mining theme and respective sets, as well as a hospital. In the winter of 2013, Police and Fire sets were once again focused on but, contrary to those released in the previous year, were set in the city. In the summer of 2013, the Coast Guard and Cargo subthemes were brought back.
My six year old grandson loves Lego's. His daddy is a pilot so we got him this Lego City Airport for his birthday. Wow! He couldn't wait to put it together! It was organized in bags (which I didn't know) and this prevents overwhelming a little person. At least that's what my son said! He did indeed help our grandson put it together, but that was expected as a father/son project. It is so realistic! I love the tower especially. It has moving parts and all kinds of nifty little things - like luggage! The plane has a bathroom too! This is a great addition to anyone who loves Lego's!
@DrDave : My thoughts were not that the hangar should be big enough to hold *all* aircraft simultaneously - which would, indeed, make it huge! - but that it should at least be big enough to hold any one of them. Just a few studs wider should cover the wingspan of either jet (judging from the photos). Add a few centimetres to the height - by whatever construction means - and that'll accommodate the oversized tail on the Viggen-styled plane. If/when I pick up this set, that's the mods I'll be making to it.
Merlin Entertainments operates seven Legoland amusement parks, the original in Billund, Denmark, the second in Windsor, England, the third in Günzburg, Germany, the fourth in Carlsbad, California, the fifth in Winter Haven, Florida, the sixth in Nusajaya, Malaysia[66] and the seventh in Dubai, United Arab Emirates.[67] and the eighth in Shanghai, Peoples of Republic of China.[68] On 13 July 2005, the control of 70% of the Legoland parks was sold for \$460 million to the Blackstone Group of New York while the remaining 30% is still held by Lego Group.[69] There are also eight Legoland Discovery Centres, two in Germany, four in the United States, one in Japan and one in the United Kingdom. Two Legoland Discovery Centres opened in 2013: one at the Westchester Ridge Hill shopping complex in Yonkers, NY and one at the Vaughan Mills in Vaughan, Ontario, Canada. Another has opened at the Meadowlands complex in East Rutherford, New Jersey in 2014.[70]
Since 1963, Lego pieces have been manufactured from a strong, resilient plastic known as acrylonitrile butadiene styrene (ABS).[12][30] As of September 2008, Lego engineers use the NX CAD/CAM/CAE PLM software suite to model the elements. The software allows the parts to be optimised by way of mould flow and stress analysis. Prototype moulds are sometimes built before the design is committed to mass production. The ABS plastic is heated to 232 °C (450 °F) until it reaches a dough-like consistency. It is then injected into the moulds at pressures between 25 and 150 tonnes, and takes approximately 15 seconds to cool. The moulds are permitted a tolerance of up to twenty micrometres, to ensure the bricks remain connected.[33] Human inspectors check the output of the moulds, to eliminate significant variations in colour or thickness. According to the Lego Group, about eighteen bricks out of every million fail to meet the standard required.[37] Lego factories recycle all but about 1 percent of their plastic waste from the manufacturing process. If the plastic cannot be re-used in Lego bricks, it is processed and sold on to industries that can make use of it.[38][39] Lego has a self-imposed 2030 deadline to find a more eco-friendly alternative to the ABS plastic it currently uses in its bricks.[40] | 1,981 | 8,707 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.734375 | 4 | CC-MAIN-2019-04 | latest | en | 0.938734 |
http://techcodebit.com/sparse-matrix-and-its-representations-set-1-using-arrays-and-linked-lists/ | 1,527,093,332,000,000,000 | text/html | crawl-data/CC-MAIN-2018-22/segments/1526794865691.44/warc/CC-MAIN-20180523161206-20180523181206-00505.warc.gz | 293,119,726 | 14,989 | # Sparse Matrix and its representations | Set 1 (Using Arrays and Linked Lists)
A matrix is a two-dimensional data object made of m rows and n columns, therefore having total m x n values. If most of the elements of the matrix have 0 value, then it is called a sparse matrix.
Why to use Sparse Matrix instead of simple matrix ?
• Storage: There are lesser non-zero elements than zeros and thus lesser memory can be used to store only those elements.
• Computing time: Computing time can be saved by logically designing a data structure traversing only non-zero elements..
Example:
```0 0 3 0 4
0 0 5 7 0
0 0 0 0 0
0 2 6 0 0
```
Representing a sparse matrix by a 2D array leads to wastage of lots of memory as zeroes in the matrix are of no use in most of the cases. So, instead of storing zeroes with non-zero elements, we only store non-zero elements. This means storing non-zero elements with triples- (Row, Column, value).
Sparse Matrix Representations can be done in many ways following are two common representations:
1. Array representation
Method 1: Using Arrays
2D array is used to represent a sparse matrix in which there are three rows named as
• Row: Index of row, where non-zero element is located
• Column: Index of column, where non-zero element is located
• Value: Value of the non zero element located at index – (row,column)
`// C program for Sparse Matrix Representation`
`// using Linked Lists`
`#include<stdio.h>`
`int` `main()`
`{`
` ``// Assume 4x5 sparse matrix`
` ``int` `sparseMatrix[4][5] =`
` ``{`
` ``{0 , 0 , 3 , 0 , 4 },`
` ``{0 , 0 , 5 , 7 , 0 },`
` ``{0 , 0 , 0 , 0 , 0 },`
` ``{0 , 2 , 6 , 0 , 0 }`
` ``};`
` ``int` `size = 0;`
` ``for` `(``int` `i = 0; i < 4; i++)`
` ``for` `(``int` `j = 0; j < 5; j++)`
` ``if` `(sparseMatrix[i][j] != 0)`
` ``size++;`
` ``// number of columns in compactMatrix (size) must be`
` ``// equal to number of non - zero elements in`
` ``// sparseMatrix`
` ``int` `compactMatrix[3][size];`
` ``// Making of new matrix`
` ``int` `k`
` ``for` `(i = 0; i < 4; i++)`
` ``for` `(j = 0; j < 5; j++)`
` ``if` `(sparseMatrix[i][j] != 0)`
` ``{`
` ``compactMatrix[0][k] = i;`
` ``compactMatrix[1][k] = j;`
` ``compactMatrix[2][k] = sparseMatrix[i][j];`
` ``k++;`
` ``}`
` ``for` `(i=0; i<3; i++)`
` ``{`
` ``for` `(j=0; j<size; j++)`
` ``printf``(``"%d "``, compactMatrix[i][j]);`
` ``printf``(``"\n"``);`
` ``}`
` ``return` `0;`
`}`
Output:
```0 0 1 1 3 3
2 4 2 3 1 2
3 4 5 7 2 6
```
In linked list, each node has four fields. These four fields are defined as:
• Row: Index of row, where non-zero element is located
• Column: Index of column, where non-zero element is located
• Value: Value of the non zero element located at index – (row,column)
• Next node: Address of the next node
`// C program for Sparse Matrix Representation`
`// using Linked Lists`
`#include<stdio.h>`
`#include<stdlib.h>`
`// Node to represent sparse matrix`
`struct` `Node`
`{`
` ``int` `value;`
` ``int` `row_position;`
` ``int` `column_postion;`
` ``struct` `Node *next;`
`};`
`// Function to create new node`
`void` `create_new_node(``struct` `Node** start, ``int` `non_zero_element,`
` ``int` `row_index, ``int` `column_index )`
`{`
` ``struct` `Node *temp, *r;`
` ``temp = *start;`
` ``if` `(temp == NULL)`
` ``{`
` ``// Create new node dynamically`
` ``temp = (``struct` `Node *) ``malloc` `(``sizeof``(``struct` `Node));`
` ``temp->value = non_zero_element;`
` ``temp->row_position = row_index;`
` ``temp->column_postion = column_index;`
` ``temp->next = NULL;`
` ``*start = temp;`
` ``}`
` ``else`
` ``{`
` ``while` `(temp->next != NULL)`
` ``temp = temp->next;`
` ``// Create new node dynamically`
` ``r = (``struct` `Node *) ``malloc` `(``sizeof``(``struct` `Node));`
` ``r->value = non_zero_element;`
` ``r->row_position = row_index;`
` ``r->column_postion = column_index;`
` ``r->next = NULL;`
` ``temp->next = r;`
` ``}`
`}`
`// This function prints contents of linked list`
`// starting from start`
`void` `PrintList(``struct` `Node* start)`
`{`
` ``struct` `Node *temp, *r, *s;`
` ``temp = r = s = start;`
` ``printf``(``"row_position: "``);`
` ``while``(temp != NULL)`
` ``{`
` ``printf``(``"%d "``, temp->row_position);`
` ``temp = temp->next;`
` ``}`
` ``printf``(``"\n"``);`
` ``printf``(``"column_postion: "``);`
` ``while``(r != NULL)`
` ``{`
` ``printf``(``"%d "``, r->column_postion);`
` ``r = r->next;`
` ``}`
` ``printf``(``"\n"``);`
` ``printf``(``"Value: "``);`
` ``while``(s != NULL)`
` ``{`
` ``printf``(``"%d "``, s->value);`
` ``s = s->next;`
` ``}`
` ``printf``(``"\n"``);`
`}`
`// Driver of the program`
`int` `main()`
`{`
` ``// Assume 4x5 sparse matrix`
` ``int` `sparseMatric[4][5] =`
` ``{`
` ``{0 , 0 , 3 , 0 , 4 },`
` ``{0 , 0 , 5 , 7 , 0 },`
` ``{0 , 0 , 0 , 0 , 0 },`
` ``{0 , 2 , 6 , 0 , 0 }`
` ``};`
` ``/* Start with the empty list */`
` ``struct` `Node* start = NULL;`
` ``for` `(``int` `i = 0; i < 4; i++)`
` ``for` `(``int` `j = 0; j < 5; j++)`
` ``// Pass only those values which are non - zero`
` ``if` `(sparseMatric[i][j] != 0)`
` ``create_new_node(&start, sparseMatric[i][j], i, j);`
` ``PrintList(start);`
` ``return` `0;`
`}`
Output:
```row_position: 0 0 1 1 3 3
column_postion: 2 4 2 3 1 2
Value: 3 4 5 7 2 6
Disclaimer: This content belongs to geeksforgeeks, source: http://geeksforgeeks.org``` | 2,084 | 5,902 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.5 | 4 | CC-MAIN-2018-22 | latest | en | 0.746495 |
https://www.assignmentexpert.com/homework-answers/mathematics/statistics-and-probability/question-15911 | 1,582,264,786,000,000,000 | text/html | crawl-data/CC-MAIN-2020-10/segments/1581875145443.63/warc/CC-MAIN-20200221045555-20200221075555-00438.warc.gz | 620,266,563 | 13,428 | 83 173
Assignments Done
99,1%
Successfully Done
In February 2020
# Answer to Question #15911 in Statistics and Probability for lim
Question #15911
Find the indicated probability Weekly salaries of teachers in one state are normally distrubuted with a mean of \$490 and a standard deviation of \$45. What is the probblity that a randomly selected teacher earns more than \$525 a week?
1
2012-10-04T10:30:49-0400
For any normal random variable X with mean μ and standard deviation σ , X ~
Normal( μ , σ ), (note that in most textbooks and literature the notation is
with the variance, i.e., X ~ Normal( μ , σ² ). Most software denotes the normal
with just the standard deviation.)
You can translate into standard normal
units by:
Z = ( X - μ ) / σ
Moving from the standard normal back to
the original distribuiton using:
X = μ + Z * σ
Where Z ~ Normal( μ =
0, σ = 1). You can then use the standard normal cdf tables to get
probabilities.
If you are looking at the mean of a sample, then remember
that for any sample with a large enough sample size the mean will be normally
distributed. This is called the Central Limit Theorem.
If a sample of
size is is drawn from a population with mean μ and standard deviation σ then the
sample average xBar is normally distributed
with mean μ and standard
deviation σ /√(n)
In this question we have
X ~ Normal( μx = 490 , σx²
= 2025 )
X ~ Normal( μx = 490 , σx = 45 )
Find P( X > 525 )
P(
( X - μ ) / σ > ( 525 - 490 ) / 45 )
= P( Z > 0.7777778 )
= P( Z
< -0.7777778 )
= 0.21835
Need a fast expert's response?
Submit order
and get a quick answer at the best price
for any assignment or question with DETAILED EXPLANATIONS! | 464 | 1,672 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.734375 | 4 | CC-MAIN-2020-10 | latest | en | 0.851369 |
https://www.jiskha.com/questions/1808962/deshaun-is-going-to-rent-a-truck-for-one-day-there-are-two-companies-he-can-choose-from | 1,657,004,093,000,000,000 | text/html | crawl-data/CC-MAIN-2022-27/segments/1656104514861.81/warc/CC-MAIN-20220705053147-20220705083147-00285.warc.gz | 880,664,627 | 4,896 | # Algebra
Deshaun is going to rent a truck for one day. There are two companies he can choose from, and they have the following prices.
Company A charges \$118 and allows unlimited mileage.
Company B has an initial fee of \$55 and charges an additional \$0.90 for every mile driven.
For what mileages will Company A charge less than Company B?
Use for the number of miles driven, and solve your inequality for .
1. 👍
2. 👎
3. 👁
4. ℹ️
5. 🚩
1. A = 118
B = 55+.9x
A <= B
118 <= 55 + .9x
.9x >= 63
x >= 70
1. 👍
2. 👎
3. ℹ️
4. 🚩
👤
Reiny
## Similar Questions
Still need help? You can ask a new question. | 203 | 603 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.5625 | 4 | CC-MAIN-2022-27 | latest | en | 0.911346 |
https://homework.cpm.org/category/CCI_CT/textbook/cc4/chapter/8/lesson/8.1.4/problem/8-61 | 1,579,995,335,000,000,000 | text/html | crawl-data/CC-MAIN-2020-05/segments/1579251681625.83/warc/CC-MAIN-20200125222506-20200126012506-00342.warc.gz | 484,395,272 | 15,317 | ### Home > CC4 > Chapter 8 > Lesson 8.1.4 > Problem8-61
8-61.
. Evaluate each expression below. Homework Help ✎
1. $\sqrt [ 3 ] { - 64 }$
What value of $x$ times itself three times is equal to $−64$,
$(x)(x)(x) = −64$?
$x = −4$
1. $\sqrt [ 5 ] { 32 }$
What value of $x$ times itself $5$ times equals $32$,
$(x)(x)(x)(x)(x) = 32$?
$x = 2$
1. $\sqrt [ 3 ] { 27 }$
1. $\sqrt [ 4 ] { 10000 }$ | 166 | 398 | {"found_math": true, "script_math_tex": 13, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.703125 | 4 | CC-MAIN-2020-05 | latest | en | 0.539151 |
https://www.mathvids.com/lesson_series/85/lessons/1157-square-roots-and-radicals-17 | 1,695,373,112,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233506339.10/warc/CC-MAIN-20230922070214-20230922100214-00745.warc.gz | 995,521,883 | 9,148 | # Square Roots and Radicals 17
Taught by YourMathGal
• Currently 4.0/5 Stars.
6465 views | 1 rating
Part of video series
Meets NCTM Standards:
Lesson Summary:
In this lesson on Square Roots and Radicals, we learn how to divide with higher order roots using the quotient rule. The instructor explains how to rationalize the denominator for monomials with higher order roots in the denominator. By factoring and multiplying the numerator and denominator by the appropriate factors, we can simplify expressions involving cube roots and 4th roots. The lesson includes examples with variables and emphasizes the importance of having the correct number of factors underneath a root in order to simplify the expression.
Lesson Description:
Square Roots and Radicals Part 17 covers dividing with higher order roots using the quotient rule. Explains rationalizing the denominator for monomials with higher order roots in the denominator.
More free YouTube videos by Julie Harland are organized at http://yourmathgal.com
• How do you divide radical expressions and rationalize the denominator of fractions with higher order roots?
• How do you simplify the cube root of 8/27?
• How do you simplify the cube root of -8/27?
• How do you simplify and rationalize the denominator of the cube root of 1/4?
• How do you simplify and rationalize the denominator of 3/(the cube root of 6)?
• How do you rationalize the denominator with cube roots and variables with exponents?
• How do you simplify and rationalize the denominator of the fourth root of 3/2?
• How do you rationalize the denominator with fourth roots?
• How do you simplify and rationalize the denominator of the cube root of 5/9?
• How do you simplify and rationalize the denominator of the cube root of 1/3y^10?
• #### Staff Review
• Currently 4.0/5 Stars.
This lesson explains the same topics as the last few lessons have described, but this time, higher order roots are used. Instead of just square roots, we will now be working with cube roots, fourth roots, and other roots higher than 2. Similar methods are used for rationalizing the denominator and simplifying, but the techniques are a bit different. Many good example problems are done in this lesson and all steps are explained very well. | 490 | 2,255 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.65625 | 5 | CC-MAIN-2023-40 | latest | en | 0.894658 |
https://www.mathway.com/examples/finite-math/functions/determining-if-a-function-is-proper-or-improper?id=737 | 1,656,130,905,000,000,000 | text/html | crawl-data/CC-MAIN-2022-27/segments/1656103034170.1/warc/CC-MAIN-20220625034751-20220625064751-00113.warc.gz | 942,654,490 | 27,905 | # Finite Math Examples
Determine if Proper or Improper
Step 1
A rational function is any function which can be written as the ratio of two polynomial functions where the denominator is not .
is a rational function
Step 2
A rational function is proper when the degree of the numerator is less than the degree of the denominator, otherwise it is improper.
Degree of numerator is less than the degree of denominator implies a proper function
Degree of numerator is greater than the degree of denominator implies an improper function
Degree of numerator is equal to the degree of denominator implies an improper function
Step 3
Find the degree of the numerator.
Remove parentheses.
Identify the exponents on the variables in each term, and add them together to find the degree of each term.
The largest exponent is the degree of the polynomial.
Step 4
Find the degree of the denominator.
Remove parentheses.
Identify the exponents on the variables in each term, and add them together to find the degree of each term.
The largest exponent is the degree of the polynomial.
Step 5
The degree of the numerator is less than the degree of the denominator .
Step 6
The degree of the numerator is less than the degree of the denominator, which means that is a proper function.
Proper | 262 | 1,272 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.8125 | 4 | CC-MAIN-2022-27 | longest | en | 0.873833 |
https://socratic.org/questions/588ec0fb7c01496070e5f24d | 1,586,522,590,000,000,000 | text/html | crawl-data/CC-MAIN-2020-16/segments/1585371896913.98/warc/CC-MAIN-20200410110538-20200410141038-00205.warc.gz | 689,547,950 | 5,922 | # What is the range of y = 2x-2?
Aug 24, 2017
Range of $y = \left(- \infty , + \infty\right)$
#### Explanation:
$y = 2 x - 2$
$y$ is a straight line defined $\forall x \in \mathbb{R}$
$y$ has no upper or lower bounds.
Hence, the range of $y = \left(- \infty , + \infty\right)$
This can be inferred from the graph of $y$ below.
graph{2x-2 [-32.47, 32.48, -16.24, 16.23]} | 149 | 378 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 7, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.1875 | 4 | CC-MAIN-2020-16 | longest | en | 0.71793 |
http://www.cps.brockport.edu/~little/CLASSES/F2018/ESC251/NOTES/sumfun.html | 1,558,637,705,000,000,000 | text/html | crawl-data/CC-MAIN-2019-22/segments/1558232257361.12/warc/CC-MAIN-20190523184048-20190523210048-00040.warc.gz | 256,555,491 | 4,791 | 0in
The Sum Function
Leigh J. Little SUNY Brockport Sep 14, 2018
Do the following:
• Configure Windows to show file name extensions by clicking on View at the top of any operating system window and checking the box that says 'Show Filetypes'.
• Change MATLAB into your usual working directory and copy these two files into that directory.
• Make sure your browser did not change the file extensions (Firefox sometimes does this). Change the filenames to the correct ones if necessary.
• Run the sc1 script.
You should recognize the output. In this case, MATLAB computed and plotted the best-fit line through a set of data points. If you look at the sc1.m script, you will notice that some commands are familiar, some commands we have not looked at, but their purpose seems clear (max function). Some operations are a little less clear. The meat of the whole calculation is in the polyfit function, which we will come back to later in the semester.
If you have taken a statistics course, you know that there is more to the best-fit line than just what you are looking at now. Equally as important as the graphical part is assessing the quallity of the line itself. Is the line reasonably approximating the data and can we use this line to reliably make other calculations?
This is a good example of a real problem that illustrates the need to be able to augment MATLAB with our own calculations. MATLAB provides a large number of general purpose functions for calculations, but it can't predict everything we will need. The basic quantities we need to assess the quality of this best-fit line are all easily computed, but we are going to have to do these calcullations ourselves.
Let x be the vector of x-coordinates and y be the vector of y-coordinates. We would like to compute the following:
• The Pearson correlation coefficient. This is defined as r = [cov(x,y)/(σx σy)] where σx is the standard deviation in x, σy is the standard deviation in y and cov(x,y) is the covariance between the vectors x and y. This covariance is defined as
cov(x,y) = n∑ i=1 (xi − xmean) (yi − ymean)
n−1
where xmean and ymean are the means of x and y respectively.
• The R2 value. This is defined as R2 = r2.
There are other things we would typically compute, but these are the most common values needed.
As it turns out, all of these quantities can be quickly computed using the MATLAB operations we have seen, provided you understand what the summation notation means. Recall that for a vector x = x1,x2,…,xn of length n,
n∑ i=1 xi = x1 + x2 + …+ xn.
This basic sum simply adds up the elements of the vector x. From this sum, we can quickly determine the mean value of x.
xmean = 1 n n∑ i=1 xi.
This formula says add up the elements of x and divide the sum by the number of elements.
Fortunately, we have a MATLAB command for computing sums of vectors.
``` >> x = [1 3 4 2 6]
x =
1 3 4 2 6
>> sum(x)
ans =
16
```
As can be seen, the sum function will add up the elements of a vector.
Note that if the input is a matrix, the sum function does something slightly different
``` >> A = [1 2 3; 4 5 6; 7 8 9]
A =
1 2 3
4 5 6
7 8 9
>> sum(A)
ans =
12 15 18
>> sum(sum(A))
ans =
45
```
If the input is a matrix, sum returns a vector consisting of the sums down the columns of A. If we want to add up all of the elements of A, we need to do a sum of the sum. This last example illustrates an important concept and this is the idea of nested function calls. It is possible to use the output of one function directly as the input to another function, for example
``` >> sum(sqrt(x))
ans =
8.5958
```
To begin building our computation, we need the means of x and y. This is easy since we have the sum function and another function, the length function
``` >> length(x)
ans =
5
```
This function will give you the number of elements in a vector. Thus, we can compute the mean of x by doing
``` >> meanx = sum(x)/length(x)
ans =
3.2000
```
The next quantity we need is the standard deviation. This is defined as
σx = ⎛
1 n−1 n∑ i=1 (xi − xmean)2
.
This formula looks bad, but it turns out that by combining the sum function and the basic operations, it can be done almost as easily as the calculation of meanx. The worst-looking part of this formula is the the sum
n∑ i=1 (xi − xmean)2 = (x1 − xmean)2 + (x2 − xmean)2 + …+ (xn − xmean)2
Consider the calculation
``` >> x = [1 3 4 2 6]
x =
1 3 4 2 6
>> x - 4
ans =
-3 -1 0 -2 2
```
If you subtract a constant from a vector, that constant gets subtracted from each element of the vector. Moreover, we can easily square each element of this result
``` >> (x-4)^2
Error using ^
Incorrect dimensions for raising a matrix to a power. Check that the matrix
is square and the power is a scalar. To perform elementwise matrix powers,
use '.^'.
>> (x-4).^2
ans =
9 1 0 4 4
```
We need to remember to use the .^ operator since we want to square each element of x−4 and not multiply x−4 by itself (which can't be done). We can then do
``` >> sum( (x-4).^2 )
ans =
18
```
This calculation subtracts 4 from each element of x, squares each element of the resulting vector and sums the elements of the vector. Essentially, this is the ugly part of the standard deviation formula. Putting it all together, we have
``` >> stdx = sqrt( sum( (x-meanx).^2 )/(length(x)-1))
stdx =
1.9235
```
The last part of our calculation that we need is the nasty looking sum in the formula for covariance
cov(x,y) =
n∑ i=1 (xi − xmean) (yi − ymean)
n−1
However, this is almost as easy as the standard deviation. We need to take x subtract its mean value, then do the same for y. We then need to multiply these two vectors element by element, then sum the result. This gives
```>> cov = sum( (x-meanx).*(y-meany) )/(length(x)-1)
```
assuming that we had defined a vector y and computed its mean.
Finally, we compute the Pearson correlation coefficient using r = [cov(x,y)/(σx σy)] The sequence of steps in a script file would look like
``` n = length(x);
meanx = sum(x)/n;
meany = sum(y)/n;
stdx = sqrt( sum( (x-meanx).^2 )/(n-1));
stdy = sqrt( sum( (y-meany).^2 )/(n-1));
cov = sum( (x-meanx).*(y-meany) )/(n-1);
r = cov/(stdx*stdy);
rsquared = r^2;
```
This represents a generic process for computing r and R2. You would need to replace x and y with the variable names that you are actually using in your script file. For example, the script we started with initially used xval and yval instead of x and y.
File translated from TEX by TTH, version 4.12.
On 14 Sep 2018, 10:11. | 1,889 | 6,632 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.90625 | 4 | CC-MAIN-2019-22 | latest | en | 0.951956 |
https://www.coursehero.com/file/5951367/Problem-Set3/ | 1,488,261,840,000,000,000 | text/html | crawl-data/CC-MAIN-2017-09/segments/1487501174135.70/warc/CC-MAIN-20170219104614-00370-ip-10-171-10-108.ec2.internal.warc.gz | 791,463,143 | 89,372 | Problem Set3
# Problem Set3 - DGP: Y = X 1 &amp;amp; 1 + X 2...
This preview shows pages 1–2. Sign up to view the full content.
Problem Set 3 Problem 1 Suppose that Y = + e where e s N (0 T 1 ; and 6 = ± 2 I T . (1) Show that the OLS estimator is unbiased. (2) Can we still use the t-statistics by simply assuming homoscedasticity? Why or why not? (3) Derive the GLS estimator when is known. Is the GLS estimator BLUE? Why or why not? (4) Show that the OLS estimator is not BLUE. (5) If the GLS estimator is not feasible, what would be an alternative for doing t-test? Problem 2 Consider the following regression equation Y = X 1 1 + X 2 2 + X 3 3 + e; e s N (0 T 1 ; ± 2 I T ) Suppose that X 2 = 3 X 1 2 X 3 . Problem 3 Consider the following regression equation DGP: Y = X 1 1 + e; e s N (0 T 1 ; ± 2 I T ) Reg. equation: Y = X 1 1 + X 2 2 + e; e s N (0 T 1 ; ± 2 I T ) Is ^ 1 unbiased? How does adding irrelevant variable X 2 a/ect the distribution of ^ 1 in comparison with the case that X 2 is not included? Prove your answer. Problem 4 Consider the following regression equation
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up to access the rest of the document.
Unformatted text preview: DGP: Y = X 1 & 1 + X 2 & 2 + e; e s N (0 T & 1 ; 2 I T ) Reg. equation: Y = X 1 & 1 + e; e s N (0 T & 1 ; 2 I T ) Is ^ & 1 unbiased? Prove your answer. Problem 5 Sang-Hee wants to forecast the dependent variable at time T + 1 , y T +1 by estimating the following regression equation Y = X& + e; e s N (0 T & 1 ; 2 I T ) (1) What is the forecasts of y T +1 ? (2) What is the nature of prediction error? Explain your answer by deriving the variance of prediction error. 1 Problem 6 Brie&y explain the following notions: (1) Unit root process (2) Unit root test (3) Cointegration (4) Stationary process (5) Spurious regression Problem 7 Consider the following model y i = & + e i , e i s i : i : d :N (0 ; 1) (1) Construct the log likelihood function. (2) Derive the maximum likelihood estimator of & . (3) Derive Var( ^ & ). 2...
View Full Document
## This note was uploaded on 09/12/2010 for the course GERAS 099876f taught by Professor Gtewewa during the Spring '09 term at Aberystwyth University.
### Page1 / 2
Problem Set3 - DGP: Y = X 1 &amp;amp; 1 + X 2...
This preview shows document pages 1 - 2. Sign up to view the full document.
View Full Document
Ask a homework question - tutors are online | 806 | 2,555 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.609375 | 4 | CC-MAIN-2017-09 | longest | en | 0.841074 |
https://www.studytadka.com/general-science/material/ | 1,555,783,818,000,000,000 | text/html | crawl-data/CC-MAIN-2019-18/segments/1555578529962.12/warc/CC-MAIN-20190420180854-20190420202854-00279.warc.gz | 832,449,285 | 33,341 | # General Science - Material Questions Answers
1. A student follows the rule 3x 3+ 3x 3= 12, 4x 4+4x 4=16, 5 x 5 + 5 x 5 = 20, then find the value of the expression 12 x 12 + '12 x 12 according to this rule.
• Option : B
• Explanation : Given pattemis 3 x 3 + 3 x 3 == 12,4 x 4 + 4 x 4 == 16 and 5 x 5 + 5 x 5 == 20 Here, the numbers of every expression are added
Page 1 of 11 | 153 | 378 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.703125 | 4 | CC-MAIN-2019-18 | longest | en | 0.703138 |
https://www.nathaniel.ai/modular-multiplication/ | 1,719,318,534,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198865972.21/warc/CC-MAIN-20240625104040-20240625134040-00776.warc.gz | 785,449,812 | 3,294 | # Modular multiplication
inspired by the work of mathematician Simon Plouffe
In mathematics, a multiplication table or times table defines a multipli-cation operation for an algebraic system. In the States, many educators teach elementary decimal multiplication with times tables up to 9 × 9.
In modular arithmetic, for a positive integer n, two numbers a and b are said to be congruent modulo n, if their difference ab is an integer multiple of n (that is, there is an integer k such that ab = nk). This congruence relation is typically denoted ab (mod n).
A familiar example of modular arithmetic is the 12-hour clock where numbers "wrap around" upon reaching the modulus (12). We can determine congruence between a certain hour on the 24-hour clock with an hour on the 12-hour clock using modular arithmetic: 15:00 is congruent to 3:00 because 15 - 3 is 12, an integer multiple 12.
By relating points on a unit circle, modular multiplication can be visualized another way. In the figure below, b and n remain constant at b = 2 and n = 12, while a is incremented from zero. For each a, we take the product of ab (mod n) and trace a line from a on the unit circle to the product ab (mod n).
As a is incremented, the same relations are repeated. Because of this, all values of a when b = 2 and n = 12 can be displayed simultaneously. This is akin to displaying a single row or column in a times table.
Staying within this single row/column, we can still vary the modulus n. For b = 2 and all a, increasing the modulus has the effect of revealing better and better approximations of a cardioid.
The resulting cardioid occurs in many places: in the epicycloid of two circles with equal radii where (R + r) / r = 2, in elliptical catacaustics, and as the main continent in the Mandelbrot set where fc(z) = z2 + c
Moving to the next column for b = 3 and for all a, increasing the modulus has the effect of revealing a nephroid.
The resulting nephroid occurs in many similar places: in the epicycloid of two circles where (R + r) / r = 3, in circular catacaustics, and as the main continent in the Multibrot set where fc(z) = z3 + c.
Predictably, the relations hold for the next column as well where b = 4.
This epicycloid appears when (R + r) / r = 4. It also appears as the main continent in the Multibrot set where fc(z) = z4 + c.
The most obvious method for constructing the full multiplication table in this visual style is to concatenate each row/column, creating a figure very much like the tables above:
But we can also pan across columns or rows in the multiplication table by continuously varying b for constant a and constant n. | 640 | 2,648 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.65625 | 5 | CC-MAIN-2024-26 | latest | en | 0.904978 |
https://assignmentchef.com/product/solved-cmpt280-assignment8-union-find-adt/ | 1,723,110,498,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640726723.42/warc/CC-MAIN-20240808093647-20240808123647-00602.warc.gz | 92,762,176 | 46,459 | # [Solved] CMPT280 Assignment8 –Union-find ADT
30 \$
SKU: [Solved] CMPT280 Assignment8 –Union-find ADT Category: Tag:
In this section we present material required for Question 1.
A union-find ADT (also called a disjoint-set ADT) keeps track of a set of elements which are partitioned into disjoint subsets. It is useful for establishing equivalencies of groups of items in a set about which nothing is known initially. For example, suppose we have an initial set of cities:
Vancouver, Edmonton, Regina, Saskatoon, Winnipeg, Toronto, Montreal, Calgary
Let’s then suppose that we decide that Vancouver and Edmonton are “equivalent” (this can be defined in any number of ways), that Regina, Saskatoon, and Winnipeg are equivalent, and that Montreal and Calgary are equivalent. Now we would have four subsets of equivalent elements of our overall set:
{Vancouver, Edmonton}, {Regina, Saskatoon, Winnipeg}, {Toronto}, {Montreal, Calgary}
Note that since Toronto was not deemed equivalent to anything, it is in its own subset by itself. Now, let’s suppose we want to find out which set a particular city is in. This is done by choosing from each subset a representative (also called an equivalence-class label) which acts as the identifier for that set. Suppose for the sake of simplicity, that we choose the first item in each set as its representative (shown in bold):
{Vancouver, Edmonton}, {Regina, Saskatoon, Winnipeg}, {Toronto}, {Montreal, Calgary}
If we were to now ask which subset Winnipeg belongs to, the answer would be Regina. Asking which subset an element belongs to is called the find operation. The find operation applied to an element returns the representative of the set to which it belongs, for example, find(Winnipeg) = Regina, or find(Calgary) = Montreal, or find(Vancouver) = Vancouver. The find operation is one of the two main operations supported by the Union-Find ADT.
The Union-Find ADT unsurprisingly supports a second operation called union. The union operation takes two elements as arguments, and establishes them as being “equivalent”, meaning, they should be in the same set. So union(Edmonton, Calgary) would place Calgary and Edmonton in the same subset. But if Edmonton and Calgary are equivalent, then by transitivity, everything in the subsets to which Edmonton and Calgary belong must also be equivalent, so the union operation actually merges two subsets into one — so this is just familiar set union operation!. Thus, union(Edmonton, Calgary) would alter our group of subsets so they look like this:
{Vancouver, Edmonton, Montreal, Calgary}, {Regina, Saskatoon, Winnipeg}, {Toronto}
So now Find(Calgary) would result in an answer of Vancouver. You may be wondering why we chose Vancouver as the representative element of the merged subset instead of Montreal. This is an implementationlevel decision. In principle, either one could be chosen.
In summary, the Union-Find data structure keeps track of a set of disjoint subsets of a set of elements. It supports the operations find(X) (look up the name of the subset to which element X belongs) and union(X,Y) (merge the subsets containing X and Y). In this assignment we will implement the union-find ADT using a directed, unweighted graph.
## 1.2 Minimum Spanning Tree
Given a connected, weighted, undirected graph, its minimum spanning tree consists of the subset of the graph’s edges of smallest total weight such that the graph remains connected. Such a set of edges always forms a tree because if it weren’t a tree there would be a cycle, which implies that it wouldn’t be the minimum cost set of edges that keeps the graph connected because you could remove one edge from the cycle and the graph would still be connected. Here is a weighed, undirected graph, and its minimum spanning tree (denoted by thicker, red edges):
No other set of edges that keeps the above graph connected has a smaller sun of weights.
The minimum spanning tree has many applications since many optimization problems can be reduced to a minimum spanning tree algorithm. Suppose you have identified several sites at which to build network routers and you know what it would cost to connect each pair of network routers by a physical wire. You would like to know what is the cheapest possible way to connect all your routers. This is an instance of the minimum spanning tree problem.
Finding the minimum spanning tree isn’t as straightforward as it might seem. There are various algorithms for finding the minimum spanning tree. We will be using Kruskal’s algorithm which, conveniently, can be implemented efficiently with a union-find ADT.
Question 1 :
For this problem you will implement Kruskal’s algorithm for finding the minimum spanning tree of an undirected weighted graph. Kruskal’s algorithm uses a union-find data structure to keep track of subsets of vertices of the input graph G. Initially, every vertex of G is in a subset by itself. The intuition for Kruskal’s algorithm is that the edges of the input graph G are sorted in ascending order of weight (smallest weights first), then each such edge (a, b) is examined in order, and if a and b are currently in different subsets we merge the two sets containing a and b and add (a, b) to the graph of the minimum spanning tree. This works because vertices in the same subset in the union-find structure are all connected. Once all of the vertices are in the same subset, we know that they are all connected. Since we always add the next smallest edge possible to the minimum spanning tree, the result is the smallest-cost set of edges that cause the graph to be completely connected, i.e. the minimum spanning tree! Here’s Kruskal’s algorithm, in pseudocode:
Algoirthm minimumSpanningTreeKruskal(G) G – A weighted, undirected graph.minST = an undirected, weighted graph with the same node set as G, but no edges.UF = a union-find data structure containing the node set of G in which each node is initially in its own subset.Sort the edges of G in order from smallest to largest weight.for each edge e=(a,b) in sorted order if UF.find(a) != UF.find(b) minST.addEdge(a,b)set the weight of (a,b) in minST to the weight of (a,b) in GUF.union(a,b)return minST
In order to implement Kruskal’s algorithm you will first need to implement a union-find ADT. We can implement union-find with a directed (unweighted) graph F. Initially the graph has a node for each item in the set, and no edges. This makes the union operation very easy. The operation union(a,b) can be completed simply by adding the edge (find(a), find(b)) to F, that is, we add an edge that connects the representative elements of the subsets containing a and b. The find(a) operation then works by checking node a to see if it has an outgoing edge, if it does, we follow it and check the node we get to to see if it has an outgoing edge. We continue going in this fashion until we find a node that does not have an outgoing edge. That node is the representative element of the subset that contains a, and we would return that node. Here’s an example of a directed graph that represents a set of subsets of the elements 1 through 8:
If we were to call find(7) on this graph, we would see that 7 has an edge to 3, which has an edge to 2, but 2 has no outgoing edge, so find(7) = 2. Similarly if we called find(4), we would follow the edge to node 6, then its outgoing edge to node 5, and find that 5 has no outgoing edge, so find(4) = 5. Overall, this graph represents that 1, 2, 3, and 7 are in the same subset, which has 2 as its representative element; that 4, 5, and 6 are in the same subset with representative element 5, and 8 is in a subset by itself. Now, suppose we do union(6, 1). This causes an edge to be added from find(6)=5 to find(1)=2, that is an edge from 5 to 2:
This causes the subsets containing 6 and 1 to be merged, and the new merged subset has representative element 2. Convince yourself that if you call find() on any element except 8, you will get a result of 2 – follow the arrows from the starting node and you’ll always end up at 2.
Here are the algorithms for the union and find operations using a graph as the underlying data structure:
Algorithm union(a, b) a, b – elements whose subsets are to be merged// If a and b are already in the same set, do nothing.if find(a) == find(b) return// Otherwise, merge the sets add the edge (find(a), find(b)) to the union-find graph.Algorithm find(a) a – element for which we want to determine set membership// Follow the chain of directed edges starting from a x = awhile x has an outgoing edge (x,y) in the union-find graph x = y// Since at this point x has no outgoing edge, it must be the // representative element of the set to which a belongs, so…return x
These are the simplest possible algorithms for union() and find(), and they don’t result in the most efficient implementations. There are improvements that we could make, but to keep things simple, we won’t bother with them. Eventually, I’ll provide solutions that use these algorithms, as well as an improved, more efficient solution for those who are interested.
Well, that was a lot of stuff. Now we can finally get to what you actually have to do:
1. Import the project Kruskal-Template (provided) module into IntelliJ workspace. You may need to add the lib280-asn8 project (also provided) as a module depdnency of the Kruskal-Template module (this process is covered in the self-guided tutorials on Moodle).
2. In the UnionFind280 class in the Kruskal-Template project, complete the implementation of the methods union() and find(). Do not modify anything else. You may add a main method to the UnionFind class for testing purposes.
3. In Kruskal.java complete the implementation of the minSpanningTree method. Do not modify anything else.
4. Run the main program in Kruskal.java. The pre-programmed input graph is the same as the one shown in Section 2.2. The input graph and the minimum spanning tree as computed by the minSpanningTree() method are displayed as output. Check the output to see if the minimum spanning tree that is output matches the one in Section 2.2.
## Implementation Hints
When implementing Kruskal’s algorithm, you should be able to avoid having to write your own sorting algorithm, or putting the edges into an array to sort the edges by their weights. You can take advantage of ADTs already in lib280-asn8a. All you need is to put the edges in a dispenser which, when you remove an item, will always give you the edge with the smallest weight (hint: look in the lib280.tree package for ArrayedMinHeap280). Conveniently, WeightedEdge280 objects are Comparable based on their weight.
Question 2 :
For this question you will implement Dijkstra’s algorithm. The implementation will be done within the NonNegativeWeightedGraphAdjListRep280 class which you can find in the lib280-asn8.graph package. This class is an extension of WeightedGraphAdjListRep280 which restricts the graph edges to have nonnegative weights. This works well for us since Dijkstra’s algorithm can only be used on graphs with nonnegative weights.
1. Implement the shortestPathDijkstra method in NonNegativeWeightedGraphAdjListRep280. The method’s javadoc comment explains the inputs and outputs of the method.
2. Implement the extractPath method in NonNegativeWeightedGraphAdjListRep280. The method’s javadoc comment explains the inputs and outputs of the method.
The pseudocode for Dijkstra’s algorithm is reproduced below.
Algoirthm dijkstra(G, s)G is a weighted graph with non-negative weights.s is the start vertex.Postcondition: v.tentativeDistance is the length of the shortest path from s to v.v.predecessorNode is the node that appears before v on the shortest path from s to v.Let V be the set of vertices in G.For each v in Vv.tentativeDistance = infinityv.visited = falsev.predecessorNode = nulls.tentativeDistance = 0while there is an unvisited vertex cur = the unvisited vertex with the smallest tentative distance. cur.visited = true// update tentative distances for adjacent vertices if needed // note that w(i,j) is the cost of the edge from i to j.For each z adjacent to cur if (z is unvisited and z.tentativeDistance > cur.tentativeDistance + w(cur,z) )z.tentativeDistance = cur.tentativeDistance + w(cur,z)z.predecessorNode = cur
## Implementation Hints
Even though the pseudocode implies that tentativeDistance, visited and predecessorNode are properties of vertices and perhaps should be stored in vertex objects, it is easiest to just use a set of parallel arrays in the implementation of Dijstra’s algorithm, much like the way we represented these as arrays during the in-class examples. E.g. an array boolean visited[] such that if visisted[i] is true, it means that vertex i has been visited. This is quite easy to use since vertices are always numbered 1 through n.
## Sample Output
If you done things right, then you should get the following outputs for start vertices 1 and 9 respectively.
Enter the number of the start vertex:
1
The length of the shortest path from vertex 1 to vertex 1 is: 0.0 Not reachable.
The length of the shortest path from vertex 1 to vertex 2 is: 1.0
The path to 2 is: 1, 2
The length of the shortest path from vertex 1 to vertex 3 is: 3.0
The path to 3 is: 1, 3
The length of the shortest path from vertex 1 to vertex 4 is: 23.0
The path to 4 is: 1, 3, 5, 6, 4
The length of the shortest path from vertex 1 to vertex 5 is: 7.0
The path to 5 is: 1, 3, 5
The length of the shortest path from vertex 1 to vertex 6 is: 16.0
The path to 6 is: 1, 3, 5, 6
The length of the shortest path from vertex 1 to vertex 7 is: 42.0 The path to 7 is: 1, 3, 5, 6, 4, 8, 9, 7
The length of the shortest path from vertex 1 to vertex 8 is: 31.0
The path to 8 is: 1, 3, 5, 6, 4, 8
The length of the shortest path from vertex 1 to vertex 9 is: 36.0
The path to 9 is: 1, 3, 5, 6, 4, 8, 9
Enter the number of the start vertex:
9
The length of the shortest path from vertex 9 to vertex 1 is: 36.0 The path to 1 is: 9, 8, 4, 6, 5, 3, 1
The length of the shortest path from vertex 9 to vertex 2 is: 35.0 The path to 2 is: 9, 8, 4, 6, 5, 3, 2
The length of the shortest path from vertex 9 to vertex 3 is: 33.0
The path to 3 is: 9, 8, 4, 6, 5, 3
The length of the shortest path from vertex 9 to vertex 4 is: 13.0
The path to 4 is: 9, 8, 4
The length of the shortest path from vertex 9 to vertex 5 is: 29.0
The path to 5 is: 9, 8, 4, 6, 5
The length of the shortest path from vertex 9 to vertex 6 is: 20.0
The path to 6 is: 9, 8, 4, 6
The length of the shortest path from vertex 9 to vertex 7 is: 6.0
The path to 7 is: 9, 7
The length of the shortest path from vertex 9 to vertex 8 is: 5.0
The path to 8 is: 9, 8
The length of the shortest path from vertex 9 to vertex 9 is: 0.0 Not reachable.
# 3 Files Provided
lib280-asn8: A copy of lib280 which includes:
• solutions to assignment 7;
• graph classes necessary for questions 1 and 2.
Kruskal-template An IntelliJ module with templates template for question 1. | 3,710 | 15,033 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.546875 | 4 | CC-MAIN-2024-33 | latest | en | 0.950821 |
http://nrich.maths.org/public/leg.php?code=47&cl=3&cldcmpid=4739 | 1,502,970,862,000,000,000 | text/html | crawl-data/CC-MAIN-2017-34/segments/1502886103270.12/warc/CC-MAIN-20170817111816-20170817131816-00454.warc.gz | 323,856,650 | 10,287 | # Search by Topic
#### Resources tagged with Creating expressions/formulae similar to Flower Show:
Filter by: Content type:
Stage:
Challenge level:
### There are 85 results
Broad Topics > Algebra > Creating expressions/formulae
### Gutter
##### Stage: 4 Challenge Level:
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
### The Pillar of Chios
##### Stage: 3 Challenge Level:
Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .
### Pick's Theorem
##### Stage: 3 Challenge Level:
Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.
### Hallway Borders
##### Stage: 3 Challenge Level:
A hallway floor is tiled and each tile is one foot square. Given that the number of tiles around the perimeter is EXACTLY half the total number of tiles, find the possible dimensions of the hallway.
### Semi-square
##### Stage: 4 Challenge Level:
What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
### Snookered
##### Stage: 4 and 5 Challenge Level:
In a snooker game the brown ball was on the lip of the pocket but it could not be hit directly as the black ball was in the way. How could it be potted by playing the white ball off a cushion?
### Reasonable Algebra
##### Stage: 4 Challenge Level:
Use algebra to reason why 16 and 32 are impossible to create as the sum of consecutive numbers.
### Three Four Five
##### Stage: 4 Challenge Level:
Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.
### How Big?
##### Stage: 3 Challenge Level:
If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?
### Square Pizza
##### Stage: 4 Challenge Level:
Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?
### Always the Same
##### Stage: 3 Challenge Level:
Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?
### Areas of Parallelograms
##### Stage: 4 Challenge Level:
Can you find the area of a parallelogram defined by two vectors?
### Balance Point
##### Stage: 4 Challenge Level:
Attach weights of 1, 2, 4, and 8 units to the four attachment points on the bar. Move the bar from side to side until you find a balance point. Is it possible to predict that position?
### Pythagoras Proofs
##### Stage: 4 Challenge Level:
Can you make sense of these three proofs of Pythagoras' Theorem?
### Hike and Hitch
##### Stage: 4 Challenge Level:
Fifteen students had to travel 60 miles. They could use a car, which could only carry 5 students. As the car left with the first 5 (at 40 miles per hour), the remaining 10 commenced hiking along the. . . .
### How Do You React?
##### Stage: 4 Challenge Level:
To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...
### Generating Triples
##### Stage: 4 Challenge Level:
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
### Pareq Calc
##### Stage: 4 Challenge Level:
Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .
### Terminology
##### Stage: 4 Challenge Level:
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
### Number Pyramids
##### Stage: 3 Challenge Level:
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
### More Number Pyramids
##### Stage: 3 Challenge Level:
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
### Chocolate Maths
##### Stage: 3 Challenge Level:
Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .
### Hand Swap
##### Stage: 4 Challenge Level:
My train left London between 6 a.m. and 7 a.m. and arrived in Paris between 9 a.m. and 10 a.m. At the start and end of the journey the hands on my watch were in exactly the same positions but the. . . .
### Perfectly Square
##### Stage: 4 Challenge Level:
The sums of the squares of three related numbers is also a perfect square - can you explain why?
### One and Three
##### Stage: 4 Challenge Level:
Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .
### Lower Bound
##### Stage: 3 Challenge Level:
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
### Unit Interval
##### Stage: 4 and 5 Challenge Level:
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
### Special Sums and Products
##### Stage: 3 Challenge Level:
Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.
### Pair Products
##### Stage: 4 Challenge Level:
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
### Triangles Within Pentagons
##### Stage: 4 Challenge Level:
Show that all pentagonal numbers are one third of a triangular number.
### Christmas Chocolates
##### Stage: 3 Challenge Level:
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
### How Much Can We Spend?
##### Stage: 3 Challenge Level:
A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?
### Crossed Ends
##### Stage: 3 Challenge Level:
Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?
### Partly Painted Cube
##### Stage: 4 Challenge Level:
Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?
### Steel Cables
##### Stage: 4 Challenge Level:
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
### Marbles in a Box
##### Stage: 3 Challenge Level:
How many winning lines can you make in a three-dimensional version of noughts and crosses?
### Always a Multiple?
##### Stage: 3 Challenge Level:
Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...
### Seven Squares
##### Stage: 3 Challenge Level:
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
### Painted Cube
##### Stage: 3 Challenge Level:
Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?
### Magic Sums and Products
##### Stage: 3 and 4
How to build your own magic squares.
### Triangles Within Triangles
##### Stage: 4 Challenge Level:
Can you find a rule which connects consecutive triangular numbers?
### ' Tis Whole
##### Stage: 4 and 5 Challenge Level:
Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed?
##### Stage: 3 Challenge Level:
Think of a number and follow my instructions. Tell me your answer, and I'll tell you what you started with! Can you explain how I know?
### AMGM
##### Stage: 4 Challenge Level:
Can you use the diagram to prove the AM-GM inequality?
### Screen Shot
##### Stage: 4 Challenge Level:
A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .
### Interactive Number Patterns
##### Stage: 4 Challenge Level:
How good are you at finding the formula for a number pattern ?
### Cubes Within Cubes Revisited
##### Stage: 3 Challenge Level:
Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?
##### Stage: 3 Challenge Level:
Think of a number... follow the machine's instructions. I know what your number is! Can you explain how I know?
### Attractive Tablecloths
##### Stage: 4 Challenge Level:
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
### Quick Times
##### Stage: 3 Challenge Level:
32 x 38 = 30 x 40 + 2 x 8; 34 x 36 = 30 x 40 + 4 x 6; 56 x 54 = 50 x 60 + 6 x 4; 73 x 77 = 70 x 80 + 3 x 7 Verify and generalise if possible. | 2,343 | 9,869 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.96875 | 4 | CC-MAIN-2017-34 | latest | en | 0.890499 |
http://mathhelpforum.com/algebra/59305-polynomial-function-3-zeros.html | 1,480,965,711,000,000,000 | text/html | crawl-data/CC-MAIN-2016-50/segments/1480698541783.81/warc/CC-MAIN-20161202170901-00505-ip-10-31-129-80.ec2.internal.warc.gz | 172,399,193 | 10,099 | # Thread: polynomial function with 3 zeros
1. ## polynomial function with 3 zeros
i have a problem that states to find and simplify the polynomial function that has zeros of -3, 0, and 2i.
my current work:
f(x)=x(x+3)(x-2i)
f(x)=(x^2 + 3x)(x-2i)
f(x)=x^3 - 2ix^2 +3x^2 - 6xi
Just wanting to see if I am doing this problem in the correct way.
i have a problem that states to find and simplify the polynomial function that has zeros of -3, 0, and 2i.
my current work:
f(x)=x(x+3)(x-2i)
f(x)=(x^2 + 3x)(x-2i)
f(x)=x^3 - 2ix^2 +3x^2 - 6xi
Just wanting to see if I am doing this problem in the correct way.
$= x^3 + (3 - 2i) x^2 - 6i x$.
3. Does your problem forbid your function from having additional zeros? If it does no then you could try the function:
f(x)=x(x+3)(x-2i)(x+2i)
So
$f(x)=(x^3 - 2ix^2 +3x^2 - 6xi)(x+2i)$
$f(x)=(x^4 - 2ix^3 +3x^3 - 6ix^2+2ix^3 + 4x^2 +6ix^2 + 12x)$
$f(x)=(x^4 +3x^3+ 4x^2 + 12x)$
The advantage of this expression being that the coefficients are all real and it has all of the required zeros. | 401 | 1,035 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.1875 | 4 | CC-MAIN-2016-50 | longest | en | 0.795352 |
https://homework.cpm.org/cpm-homework/homework/category/CON_FOUND/textbook/CA/chapter/Ch2/lesson/2.2.3/problem/2-114 | 1,568,732,512,000,000,000 | text/html | crawl-data/CC-MAIN-2019-39/segments/1568514573080.8/warc/CC-MAIN-20190917141045-20190917163045-00225.warc.gz | 517,712,372 | 15,595 | ### Home > CA > Chapter Ch2 > Lesson 2.2.3 > Problem2-114
2-114.
1. Solve the equations below for x and check your solutions. Homework Help ✎
1. −3 + x = −2x + 6
2. 5 − x = 3x + 1
3. −4x = 2x + 9
4. −(x − 3) = −4x
Isolate the variable on one side of the equation and solve for x.
x = 1 Show your work.
Use the same strategy as in part (a).
Use the same strategy as in part (a).
x = −1.5 Show your work.
Start by distributing the negative sign, then use the same strategy as in part (a). | 171 | 497 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.734375 | 4 | CC-MAIN-2019-39 | latest | en | 0.870907 |
https://www.doorsteptutor.com/Exams/ISS/Paper-1/Questions/Topic-Probability-0/Subtopic-Standard-Probability-Distributions-9/Part-4.html | 1,503,164,064,000,000,000 | text/html | crawl-data/CC-MAIN-2017-34/segments/1502886105700.94/warc/CC-MAIN-20170819162833-20170819182833-00403.warc.gz | 912,355,701 | 28,879 | # Probability-Standard Probability Distributions (ISS Statistics Paper I (Old Subjective Pattern)): Questions 18 - 22 of 22
Get 1 year subscription: Access detailed explanations (illustrated with images and videos) to 165 questions. Access all new questions we will add tracking exam-pattern and syllabus changes. View Sample Explanation or View Features.
Rs. 550.00 or
## Question number: 18
» Probability » Standard Probability Distributions » Poisson
Appeared in Year: 2015
Essay Question▾
### Describe in Detail
Prove that for r = 1, 2, …, n
### Explanation
We known that the L. H. S. is an incomplete gamma function and R. H. S. is a cumulative density function of Poisson distribution.
The incomplete gamma function is
Then
provided that r is an integer. Thus recall that Γ (r) = (r… (20 more words) …
## Question number: 19
» Probability » Standard Probability Distributions » Uniform
Appeared in Year: 2011
Essay Question▾
### Describe in Detail
Let X have the continuous c. d. f. F (x). Define U = F (x). Show that both - log U arid -log (1 - U) are exponential random variables:
### Explanation
Let U = F (x), then the distribution function of G of U is given by
Since F is non-increasing and its continuous.
G (u) =F (F -1 (u) ) implies G (u) =u
Then the p. d. f is
Since F is a distribution function takes value… (54 more words) …
## Question number: 20
» Probability » Standard Probability Distributions » Poisson
Appeared in Year: 2011
Essay Question▾
### Describe in Detail
Show that the sum of two independent Poisson random variables with parameters λ and µ respectively is a Poisson random variable with parameter λ+µ.
### Explanation
Let X and Y are independent Poisson random variables with parameters λ and µ respectively. We proof this by moment generating function. The moment generating function of Poisson distribution is
So, the sum of X and Y moment generating function is
because X and Y are independent
## Question number: 21
» Probability » Standard Probability Distributions » Normal
Appeared in Year: 2011
Essay Question▾
### Describe in Detail
Let the joint p. d. f. of (X, Y) be f (x, y) = e -y, 0 < x < y < ∞.
Obtain the probability P (X + Y ≤ 1).
### Explanation
The Joint p. d. f. is
f (x, y) = e -y, 0 < x < y < ∞.
Let assume X + Y =U and Y = V, then X = U-V
Using Jacobian technique
The range is 0 < ∞, u ≤ v < ∞
The… (23 more words) …
## Question number: 22
» Probability » Standard Probability Distributions » Exponential
Appeared in Year: 2009
Essay Question▾
### Describe in Detail
Explain “Memoryless property” of a distribution. Show that the exponential distribution has memoryless property.
### Explanation
The property of memory less is that these distributions of “time from now to the next period” are exactly the same. The property is most easily explained in terms of “waiting times.
Suppose X is a discrete random variable whose values is a non-negative. In probability theory, a distribution is… (84 more words) …
f Page | 777 | 3,033 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.734375 | 4 | CC-MAIN-2017-34 | longest | en | 0.766602 |
https://tbc-python.fossee.in/convert-notebook/Electrical_Machines_by_M._V._Despande/Chapter_11_1.ipynb | 1,701,197,681,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679099942.90/warc/CC-MAIN-20231128183116-20231128213116-00029.warc.gz | 646,070,628 | 40,217 | # CHAPTER 11 - THREE PHASE INDUCTION MOTORS:PRINCIPLES AND CHARACTERISTICS¶
## Example E1 - Pg 233¶
In [1]:
#Caption: Find (a)Number of poles and (b)% slip
#Exa:11.1
import math
f=50.#Frequency(in hertz)
n=960.#Speed of induction motor on full load(in r.p.m)
n_s=1000.#Synchronous speed(in r.p.m)
p=(f*120.)/(n_s)
print '%s %.f' %('(a)Number of poles is=',p)
s=n_s-n
S=(s/n_s)*100.
print '%s %.f' %('(b)Slip is(in%)=',S)
(a)Number of poles is= 6
(b)Slip is(in%)= 4
## Example E2 - Pg 233¶
In [2]:
#Caption: Find (a)Speed of motor (b)%Slip
#Exa:11.2
p=6.#Number of poles
f_s=50.#Stator frequency(in c/s)
f_r=2.#Rotor frequency(in c/s)
n_s=(120.*f_s)/p
n=(f_r*120.)/p
s=n_s-n
print '%s %.f' %('Speed of motor(in r.p.m)=',s)
S=(n/n_s)*100
print '%s %.f' %('Slip(in %)=',S)
Speed of motor(in r.p.m)= 960
Slip(in %)= 4
## Example E3 - Pg 233¶
In [3]:
#Caption: Calculate (a)Number of poles (b)Slip (c)Slip for full load torque if total resistance in rotor circuit is doubled
#Exa:11.3
n=970.#Speed of induction motor(in r.p.m)
f=50.#Frequency(in hertz)
n_s=1000.#Synchronous speed(in r.p.m)
p=(f*120.)/n_s
print '%s %.f' %('(a)Number of poles=',p)
s=((n_s-n)/n_s)*100
print '%s %.f' %('(b)Slip(in%)=',s)
S=((s/100)*2)*100
print '%s %.f' %('(c)Required slip(in%)=',S)
(a)Number of poles= 6
(b)Slip(in%)= 3
(c)Required slip(in%)= 6
## Example E4 - Pg 234¶
In [4]:
#Caption: Calculate (a)Mechanical power output (b)Torque (c)Maximum Torque (d)Speed at maximum torque (e)Power output when torque is maximum
#Exa:11.4
import math
p=4.#Number of poles
V=440.#Voltage of motor(in volts)
f=50.#Frequency(in hertz)
n_s=1500.#Synchronous speed(r.p.m)
sp=1440.#Speed of motor at load(in r.p.m)
t=1.8#Stator to rotor turns ratio
R_r=0.1#Resistance of rotor per phase(in ohms)
X_r=0.8#Reactance of rotor per phase at standstill(in ohms)
r_r=R_r*(t**2.)#Rotor resistance referred to stator(in ohms)
x_r=X_r*(t**2.)#Reactance of rotor at stanstill referred to stator(in ohms)
E=V/(math.sqrt(3.))
P=((s/100.)*(E**2.)*r_r)/((r_r**2.)+((s/100.)**2.)*(x_r**2.))
T=(3.*P)/(2.*(math.pi)*(n_s/60.))
P_M=(3.*P*sp)/n_s
print '%s %.f' %('(a)Mechanical power output(in watt)=',P_M)
print '%s %.f' %('(b)Torque(in N-m)=',T)
s_m=R_r/X_r
N=n_s*(1-s_m)
P_1=((s_m)*(E**2)*(r_r))/((r_r**2.)+((s_m**2)*(x_r**2)))
T_m=(3.*P_1)/(2.*(math.pi)*(n_s/60.))
print '%s %.1f' %('(c)Speed at maximum torque(in r.p.m)=',N)
print '%s %.f' %('(d)Maximum torque(in N-m)=',T_m)
P_o=(3*P_1*N)/n_s
print '%s %.f' %('(e)Output power at maximum torque(in watt)=',P_o)
(a)Mechanical power output(in watt)= 20814
(b)Torque(in N-m)= 138
(c)Speed at maximum torque(in r.p.m)= 1312.5
(d)Maximum torque(in N-m)= 238
(e)Output power at maximum torque(in watt)= 32677
## Example E5 - Pg 235¶
In [5]:
#Caption:Find (a)Speed of the motor (b)Speed at which torque will be maximum (c)Ratio of maximum to full load torque
#Exa:11.5
import math
V=3300.#Voltage supplied to induction motor(in volts)
p=10.#Number of poles
f=50.#frequency(in hertz)
R_r=0.015#Rotor resistance per phase(in ohms)
X_r=0.25#Standstill reactance per phase(in ohms)
s=2.5#Slip(in %)
n_s=(f*120.)/p
n=n_s*(1.-(s/100.))
print '%s %.f' %('(a)Speed of the motor(in r.p.m)=',n)
S=R_r/X_r
N=n_s*(1-S)
print '%s %.f' %('(b)Speed at which torque will be maximum(in r.p.m)=',N)
T_f=(s/100.)*R_r/((R_r**2.)+(((s/100.)**2.)*(X_r**2.)))
T_m=S*R_r/((R_r**2.)+((S**2.)*(X_r**2.)))
#R=T_m/T_f
R=2.316
print '%s %.3f' %('(c)Ratio of maximum to full load torque=',R)
(a)Speed of the motor(in r.p.m)= 585
(b)Speed at which torque will be maximum(in r.p.m)= 564
(c)Ratio of maximum to full load torque= 2.316
## Example E6 - Pg 236¶
In [6]:
#Caption: Calculate (a)Speed at which mechanical power from rotor will be maximum (b)Maximum power
#Exa:11.6
import math
p=4.#Number of poles
f=50.#Frequency(in hertz)
V=440.#Supplied voltage to induction motor(in volts)
R_r=0.1#Rotor resistance per phase(in ohm)
X_r=0.8#Rotor reactance at standstill per phase(in ohm)
t=1.3#Ratio of stator turns per phase to rotor turns per phase
a=R_r/X_r
s=(-(a**2.))+math.sqrt(1.+(a**2.))
n_s=120.*f/p
#N=n_s*(1.-s)
N=1334.5
print '%s %.1f' %('(a)Required speed(in r.p.m)=',N)
r=R_r*t
x=X_r*t
E=V/math.sqrt(3.)
#P_m=(3.*s*(E**2.)*r*(1.-s))/((r**2.)+((s**2.)+(x**2.)))
P_m=62.72
print '%s %.2f' %('(b)Maximum power(in kwatts)=',P_m)
(a)Required speed(in r.p.m)= 1334.5
(b)Maximum power(in kwatts)= 62.72
## Example E7 - Pg 236¶
In [7]:
#Caption: Find Current per phase in the rotor (a)when rotor is at standstill and star connected impedance of 4.1+%i2 per phase is connected in series with rotor (b)when rotor runs at 3% slip with short circuit at the slip rings
#Exa:11.7
import math,cmath
V=69.28#Induced e.m.f(in volts)
r=0.9#Resistance of rotor per phase(in ohm)
x=6.#Standstill rectance of rotor per phase(in ohm)
z=4.1+(1j*2)
s=3#Slip(in%)
V_r=V/math.sqrt(3.)
R_r=r+z.real
X_r=(1j*2.)+(1j*x)
Z=R_r+X_r
I_r=V_r/Z
print ('(a)Current when rotor is at standstill=',I_r)
E=(s/100.)*V_r
Imp=r+(1j*(s/100.)*x)
i_r=E/Imp
print ('(b)Current when rotor runs at 3% slip=',i_r)
('(a)Current when rotor is at standstill=', (2.2471250926661392-3.595400148265823j))
('(b)Current when rotor runs at 3% slip=', (1.2820136746620923-0.25640273493241844j))
## Example E8 - Pg 237¶
In [8]:
#Caption:Find (a)% reduction in stator voltage and (b)the power factor of the rotor circuit
#Exa:11.8
import math
R_r=0.02#Rotor resistance per phase(in ohm)
X_r=0.1#Rotor reactance per phase(in ohm)
s=4#Slip(in%)
S=100.-s
T_f=((s/100.)*R_r)/((R_r**2.)+(((s/100.)**2.)*(X_r**2.)))
S_r=1-(.5*(S/100.))
T=(S_r*R_r)/((R_r**2.)+((S_r**2.)*(X_r**2.)))
Re=(1-math.sqrt(T_f/T))*100.
print '%s %.2f' %('(a)% reduction in stator voltage(in %)=',Re)
pf=R_r/(math.sqrt((R_r**2.)+((S_r**2.)*(X_r**2.))))
print '%s %.2f' %('(b)Power factor=',pf)
(a)% reduction in stator voltage(in %)= 24.24
(b)Power factor= 0.36
## Example E9 - Pg 242¶
In [9]:
#Caption:Find (a)the rotor copper loss per phase if motor is running at slip of 4% (b)Mechanical power developed
#Exa:11.9
P_i=100000.#Input power(in watt)
P_sc=2000.#Stator copper loss(in watt)
s=4.#slip(in %)
P_r=P_i-P_sc
P_rc=((s/100.)*P_r)/3.
print '%s %.f' %('(a)Rotor copper lossper phase(in watt)=',P_rc)
P_m=P_r-(P_rc*3)
print '%s %.f' %('(b)Mechanical power developed(in watt)=',P_m)
(a)Rotor copper lossper phase(in watt)= 1307
(b)Mechanical power developed(in watt)= 94080
## Example E10 - Pg 242¶
In [10]:
#Caption: Calculate (a)% slip (b)Rotor copper loss (c)Output from the rotor (d)Efficiency
#Exa:11.10
import math
V=440.#Supplied voltage(in volts)
f=50.#frequency(in hertz)
p=6.#Number of poles
n=960.#Speed of motor(in r.p.m)
P_i=50000.#Input power(in watt)
P_wf=1800.#Winding and friction losses(in watt)
P_s=1200.#Stator losses(in watt)
n_s=(120.*f)/p
S=((n_s-n)/n_s)*100.
print '%s %.f' %('(a)% slip=',S)
P_r=P_i-P_s
P_rc=(S/100.)*P_r
print '%s %.f' %('(b)Rotor copper loss(in watt)=',P_rc)
P_o=P_r-P_rc-P_wf
print '%s %.f' %('(c)Output of rotor(in watt)=',P_o)
eff=(P_o/P_i)*100.
print '%s %.f' %('(d)Efficiency(in%)=',eff)
(a)% slip= 4
(b)Rotor copper loss(in watt)= 1952
(c)Output of rotor(in watt)= 45048
(d)Efficiency(in%)= 90
## Example E11 - Pg 243¶
In [11]:
#Caption:Find (a)Equivalent rotor current per phase (b)Stator current per phase (c)Power factor (d)Rotor input (e)Rotor copper losses (f)Torque (g)Mechanical power output from rotor (h)Stator input (i)Efficiency
#Exa:11.11
import math,cmath
from math import cos,atan,sqrt
V=440.#Voltage supplied(in volts)
p=8.#Number of poles
f=50.#Frequency(in hertz)
r1=0.2#Stator resistance(in ohm)
x1=1.2#Stator reactance(in ohm)
r2=0.3#Equivalent resistance of rotor referred to stator(in ohm)
x2=1.2#Equivalent reactance of rotor referred to stator(in ohm)
r_m=150.#Magnetising resistance(in ohms)
x_m=18.#Magnetising reactance(in ohms)
P_wf=750.#Winding and friction losses(in watt)
s=0.04#Slip
n_s=(f*120.)/(p*60.)
y1=1./r_m
y2=1./(1j*x_m)
y3=1./((r2/s)+(1j*x2))
Y=y1+y2+y3
Z=1./Y
Z_t=Z+(r1+(1j*x1))
E=V*Z/(Z_t)
z3=1./y3
i2=E/z3
print '(a)Rotor current per phase(in A)=',i2
i1=V/Z_t
print '(b)Stator current per phase(in A)=',i1
#pf=cos(atan(-(Z_t.imag)/(Z_t.real))*57.3)*57.3
pf=0.793
print '(c)Power factor=',pf
#P_r=(i2*(i2.conjugate()))*(r2/s)
P_r=20.569
print '(d)Rotor input(in kwatt)=',P_r
#P_rc=(i2*(i2.conjugate()))*r2
P_rc=0.822
print '(e)Rotor copper loss(in kwatt)=',P_rc
#T=3*P_r/(2*math.pi*n_s)
T=785.67
print '(f)Torque(in N-m)=',T
#P_me=P_r-P_rc-(P_wf/3)
P_me=19497
print '(g)Mechanical output from rotor(in watts per phase)=',P_me
#P_st=V*((i1*(i1.conjugate()))**.5)*pf
P_st=21713
print '(h)Stator input(watts per phase)=',P_st
#eff=(P_me/P_st)*100
eff=89.77
print '(i)Efficiency(in %)=',eff#Caption:Find (a)Equivalent rotor current per phase (b)Stator current per phase (c)Power factor (d)Rotor input (e)Rotor copper losses (f)Torque (g)Mechanical power output from rotor (h)Stator input (i)Efficiency
(a)Rotor current per phase(in A)= (49.1649692861-14.7144544702j)
(b)Stator current per phase(in A)= (48.8875753093-36.52322494j)
(c)Power factor= 0.793
(d)Rotor input(in kwatt)= 20.569
(e)Rotor copper loss(in kwatt)= 0.822
(f)Torque(in N-m)= 785.67
(g)Mechanical output from rotor(in watts per phase)= 19497
(h)Stator input(watts per phase)= 21713
(i)Efficiency(in %)= 89.77
## Example E12 - Pg 245¶
In [12]:
#Caption:Find (a)Equivalent rotor current per phase (b)Stator current per phase (c)Power factor (d)Rotor input (e)Rotor copper losses (f)Torque (g)Mechanical power output from rotor (h)Stator input (i)Efficiency.Solve it by APPROXIMATE equivalent circuit method
#Exa:11.12
import math,cmath
from math import sqrt,cos,atan
V=440.#Voltage supplied(in volts)
p=8.#Number of poles
f=50.#Frequency(in hertz)
r1=0.2#Stator resistance(in ohm)
x1=1.2#Stator reactance(in ohm)
r2=0.3#Equivalent resistance of rotor referred to stator(in ohm)
x2=1.2#Equivalent reactance of rotor referred to stator(in ohm)
r_m=150.#Magnetising resistance(in ohms)
x_m=18.#Magnetising reactance(in ohms)
P_wf=750.#Winding and friction losses(in watt)
s=0.04#Slip
I2=V/((r1+(r2/s))+(1j*x1)+(1j*x2))
print '(a)Equivalent rotor current per phase(in A)=',I2
y1=1./r_m
y2=1./(1j*x_m)
I_o=V*(y1+y2)
I_1=I2+I_o
print '(b)Stator current per phase(in A)=',I_1
#pf=cos(atan(I_1.imag/I_1.real)*57.3)*57.3
pf=0.804
print '(c)Power factor=',pf
#P_r=(I2*I2.conjugate())*(r2/s)
P_r=22.317
print '(d)Rotor input(in kwatt)=',P_r
#P_rc=(I2*I2.conjugate())*r2
P_rc=892
print '(e)Rotor copper losses(in watts)=',P_rc
#T=P_r/(2.*math.pi*((f*120.)/(p*60.)))
T=852.42
print '(f)Torque(in N-m)=',T
#P_me=P_r-P_rc-(P_wf/3.)
P_me=21175
print '(g)Mechanical power output from rotor(in watts per phase)=',P_me
#P_si=V*pf*((I_1*I_1.conjugate()))*5
P_si=24100
print '(h)Stator input(in watts per phase)=',P_si
#e=(P_me/P_si)*100.
e=87.86
print '(i)Efficiency (in %)=',e
(a)Equivalent rotor current per phase(in A)= (52.0830130669-16.2336664105j)
(b)Stator current per phase(in A)= (55.0163464002-40.6781108549j)
(c)Power factor= 0.804
(d)Rotor input(in kwatt)= 22.317
(e)Rotor copper losses(in watts)= 892
(f)Torque(in N-m)= 852.42
(g)Mechanical power output from rotor(in watts per phase)= 21175
(h)Stator input(in watts per phase)= 24100
(i)Efficiency (in %)= 87.86
## Example E13 - Pg 246¶
In [13]:
#Caption:Find (a)Equivalent rotor current (b)Stator current (c)Power factor (d)Stator input (e)Rotor input (f)Efficiency
#Exa:11.13
import math,cmath
from math import sqrt,cos,atan
V=440#Voltage supplied(in volts)
f=50#frequency(in hertz)
Z_s=1.5+(1j*3)#Stator impedance per phase(in ohms)
Z_r=1.6+(1j*1)#Rotor impedance per phase(in ohms)
Z_m=3+(1j*40)#Magnetising impedance per phase(in ohms)
P_wf=300#Friction and winding loss(in watt)
s=0.04#Slip
Z=40+(1j*1)
z=Z*Z_m/(Z+Z_m)
Zt=z+Z_s
I1=(V/sqrt(3))/Zt
E=(V/sqrt(3))-(I1*Z_s)
I2=E/Z
print '(a)Equivalent Rotor current(in A)=',I2
print '(b)Stator current(in A)=',I1
#pf=cos(atan(Zt.imag/Zt.real))
pf=0.7
print '(c)Power factor=',pf
#P_s=sqrt(3)*V*(I1*I1.conjugate())*pf
P_s=4486.5
print '(d)Stator input(in watt)=',P_s
#P_r=3*(I2*I2.conjugate())*(Z_r.real/s)
P_r=3709.6
print '(e)Rotor input(in watt)=',P_r
P_ro=P_r*(1-s)
P_me=P_ro-P_wf
#e=(P_me/P_s)*100
e=72.68
print '(f)Efficiency(in%)=',e
(a)Equivalent Rotor current(in A)= (5.66994104723-0.356954905389j)
(b)Stator current(in A)= (5.87947093141-6.02010508394j)
(c)Power factor= 0.7
(d)Stator input(in watt)= 4486.5
(e)Rotor input(in watt)= 3709.6
(f)Efficiency(in%)= 72.68 | 4,926 | 12,466 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.125 | 4 | CC-MAIN-2023-50 | latest | en | 0.384774 |
https://socratic.org/questions/how-do-you-write-y-9-3-x-2-in-standard-form-2 | 1,653,779,535,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652663021405.92/warc/CC-MAIN-20220528220030-20220529010030-00113.warc.gz | 593,497,055 | 6,031 | How do you write y + 9 = -3(x - 2) in standard form?
Nov 19, 2017
$3 x + y = - 3$
Explanation:
$\text{the equation of a line in "color(blue)"standard form }$ is.
$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{A x + B y = C} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
$\text{where A is a positive integer and B, C are integers}$
$\text{rearrange "y+9=-3(x-2)" into this form}$
$\Rightarrow y + 9 = - 3 x + 6$
$\text{add 3x to both sides}$
$3 x + y + 9 = \cancel{- 3 x} \cancel{+ 3 x} + 6$
$\Rightarrow 3 x + y + 9 = 6$
$\text{subtract 9 from both sides}$
$3 x + y \cancel{+ 9} \cancel{- 9} = 6 - 9$
$\Rightarrow 3 x + y = - 3 \leftarrow \textcolor{red}{\text{in standard form}}$ | 301 | 740 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 12, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.5625 | 5 | CC-MAIN-2022-21 | latest | en | 0.472857 |
https://www.convertunits.com/from/foot+of+air/to/decipascal | 1,631,971,250,000,000,000 | text/html | crawl-data/CC-MAIN-2021-39/segments/1631780056476.66/warc/CC-MAIN-20210918123546-20210918153546-00623.warc.gz | 754,219,200 | 23,359 | ## ››Convert foot of air [0 °C] to decipascal
foot of air decipascal
Did you mean to convert foot of air [0 °C] foot of air [15 °C] to decipascal
How many foot of air in 1 decipascal? The answer is 0.025879322442072.
We assume you are converting between foot of air [0 °C] and decipascal.
You can view more details on each measurement unit:
foot of air or decipascal
The SI derived unit for pressure is the pascal.
1 pascal is equal to 0.25879322442072 foot of air, or 10 decipascal.
Note that rounding errors may occur, so always check the results.
Use this page to learn how to convert between feet of air and decipascals.
Type in your own numbers in the form to convert the units!
## ››Quick conversion chart of foot of air to decipascal
1 foot of air to decipascal = 38.64089 decipascal
2 foot of air to decipascal = 77.28178 decipascal
3 foot of air to decipascal = 115.92266 decipascal
4 foot of air to decipascal = 154.56355 decipascal
5 foot of air to decipascal = 193.20444 decipascal
6 foot of air to decipascal = 231.84533 decipascal
7 foot of air to decipascal = 270.48622 decipascal
8 foot of air to decipascal = 309.1271 decipascal
9 foot of air to decipascal = 347.76799 decipascal
10 foot of air to decipascal = 386.40888 decipascal
## ››Want other units?
You can do the reverse unit conversion from decipascal to foot of air, or enter any two units below:
## Enter two units to convert
From: To:
## ››Definition: Decipascal
The SI prefix "deci" represents a factor of 10-1, or in exponential notation, 1E-1.
So 1 decipascal = 10-1 pascals.
The definition of a pascal is as follows:
The pascal (symbol Pa) is the SI unit of pressure.It is equivalent to one newton per square metre. The unit is named after Blaise Pascal, the eminent French mathematician, physicist and philosopher.
## ››Metric conversions and more
ConvertUnits.com provides an online conversion calculator for all types of measurement units. You can find metric conversion tables for SI units, as well as English units, currency, and other data. Type in unit symbols, abbreviations, or full names for units of length, area, mass, pressure, and other types. Examples include mm, inch, 100 kg, US fluid ounce, 6'3", 10 stone 4, cubic cm, metres squared, grams, moles, feet per second, and many more! | 647 | 2,308 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.53125 | 4 | CC-MAIN-2021-39 | latest | en | 0.879672 |
https://www.hackmath.net/en/example/785 | 1,563,920,605,000,000,000 | text/html | crawl-data/CC-MAIN-2019-30/segments/1563195529737.79/warc/CC-MAIN-20190723215340-20190724001340-00265.warc.gz | 718,660,934 | 6,393 | # Circle
What is the radius of the circle whose perimeter is 6 cm?
Result
r = 1 cm
#### Solution:
Leave us a comment of example and its solution (i.e. if it is still somewhat unclear...):
Be the first to comment!
## Next similar examples:
1. Bicycle wheel
After driving 157 m bicycle wheel rotates 100 times. What is the radius of the wheel in cm?
2. Velocipede
The front wheel of velocipede from year 1880 had a diameter 1.8 m. If the front wheel turned again one then rear wheel 6 times. What was the diameter of the rear wheel?
3. Circumference - a simple
What is the ratio of the circumference of any circle and its diameter? Write the result as a real number rounded to 2 decimal places.
4. Circle r,D
Calculate the diameter and radius of the circle if it has length 52.45 cm.
5. Circle - easy 2
The circle has a radius 6 cm. Calculate:
6. Circle from string
Martin has a long 628 mm string . He makes circle from it. Calculate the radius of the circle.
7. Athlete
How long length run athlete when the track is circular shape of radius 120 meters and an athlete runs five times in the circuit?
8. Clock hands
The second hand has a length of 1.5 cm. How long does the endpoint of this hand travel in one day?
9. Flowerbed
In the park there is a large circular flowerbed with a diameter of 12 m. Jakub circulated him ten times and the smaller Vojtoseven times. How many meters each went by and how many meters did Jakub run more than Vojta?
10. Circle - simple
The circumference of a circle is 930 mm. How long in mm is its diameter?
11. Mine
Wheel in traction tower has a diameter 5 m. How many meters will perform an elevator cabin if wheel rotates in the same direction 49 times?
12. Well
Rope with a bucket is fixed on the shaft with the wheel. The shaft has a diameter 50 cm. How many meters will drop bucket when the wheels turn 15 times?
13. Bicycle wheel
Bicycle wheel diameter is 62 cm. How many times turns the bicycle on the road 1 km long?
14. Tree trunk
Calculate the average tree trunk with a circumference of 149 cm.
15. Circle - simple
Calculate the area of a circle in dm2, if its circumference is 31.4 cm.
16. Clock
How long is trajectory of second hand of hours for day, if is 15 mm long?
17. Cable
Cable consists of 8 strands, each strand consists of 12 wires with diameter d = 0.5 mm. Calculate the cross-section of the cable. | 598 | 2,359 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.828125 | 4 | CC-MAIN-2019-30 | latest | en | 0.914724 |
https://www.thefinancials.net/future-value-factor/ | 1,558,630,067,000,000,000 | text/html | crawl-data/CC-MAIN-2019-22/segments/1558232257316.10/warc/CC-MAIN-20190523164007-20190523190007-00518.warc.gz | 938,892,310 | 11,613 | Future Value Factor = (1 + r)n
r = rate per period
n = number of periods
Future value factor formula is utilized by financial analysts or investors to calculate the future value of an amount by using its present value. Future value factor can be found on a table which helps to make calculations simple for large amounts.
Basically, time value of money is the underlying concept behind the idea for future value factor formula. The time value of money means that a specific amount received today has more worth than the same amount received at a future date. Furthermore, any amount you have in your hand at present may have an opportunity to invest in a potential project to generate more profits, contrary while on the other side future receivables will not generate any additional earnings.
For instance, an amount of \$150 to be received 6 months after present date seems more profitable rather to get \$135 today. In such a case, a person can earn 10% extra if he opts to receive the amount in future.
## Effect of rate per period
As with other formulas the rate may need some modification to calculate the precise future value factor. The rate must match with the period you are using while on the other side you must also take care about how frequent the amount is compounded. For instance, if annual nominal interest rate is 12% and investment is compounded on monthly basis. Therefore, we will convert annual rate to month which will be 1% and would be used in the formula as you can see in example below.
### Future value factor formula example
We will continue with the above example, the formula for future value factor will be;
Future value factor = (1 + 0.01)12
= 1.1268
In the above equation, 0.01 is rate which is 1% per month in the example above. By solving this equation, the future value factor for 12 months at the rate of 1% will be 1.1268.
Although there is a table available for future value factor which helps in quick calculation but one can also use above formula. By using above example, we add another thing in it where the person gets \$500 after one year with same terms. The future value factor table shows a value of 1.1268 for 12 years at 1% rate. That factor will then be multiplied with \$500 to find the future value of \$500 and is;
= \$563.40
Share. | 502 | 2,302 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.34375 | 4 | CC-MAIN-2019-22 | longest | en | 0.933957 |
https://learn.careers360.com/school/question-for-what-value-of-a-the-quadratic-ax2-2a-1x-a-4-is-always-positive-38265/ | 1,713,173,227,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296816954.20/warc/CC-MAIN-20240415080257-20240415110257-00345.warc.gz | 315,387,095 | 34,257 | #### For what value of a the quadratic ax^2 +(2a-1)x +a+4 is always positive
Solution: We have , $ax^{2}+(2a-1)x+(a+4)$
$\\ \\ \Rightarrow \hspace{1cm}D> 0\hspace{0.5cm} and \hspace{0.5cm}a> 0\\ \\ \Rightarrow \hspace{1cm}D=(2a-1)^{2}-4a(a+4)< 0\\ \\ \Rightarrow \hspace{1cm}-8a+1< 0\Rightarrow a> \frac{1}{8}\\ \\ \Rightarrow \hspace{1cm}a\in (\frac{1}{8},\infty)$ | 173 | 370 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.0625 | 4 | CC-MAIN-2024-18 | latest | en | 0.20518 |
https://www.physicsforums.com/threads/voltage-in-the-middle-of-an-electric-dipole.167578/ | 1,531,850,429,000,000,000 | text/html | crawl-data/CC-MAIN-2018-30/segments/1531676589757.30/warc/CC-MAIN-20180717164437-20180717184437-00621.warc.gz | 937,011,826 | 13,790 | Homework Help: Voltage in the Middle of an Electric Dipole
1. Apr 25, 2007
kaotak
Problem Diagram (Ignore the tildes, they're just placeholders):
Below: An electric dipole
~~~~~~~y-axis~~~~~~~~~~~~~~~~~~
~~~~~~~|~~~~~~~~~~~~~~~~~~~~~
~~~~~~~|~~~~~~~~~~~~~~~~~~~~~
~~~~~~~|~~~~~~~~~~~~~~~~~~~~~
~~<---a---> <---a--->~~~~~~~~~~~~~~~
+Q --------- X --------- -Q~~~-------------- x-axis
Problem Statement: Find the voltage at X.
My answer is that V_x = 0, since the potentials from each side of the dipole sum to zero. And I'm pretty sure this is right. But my question is... how does this fit in with the following definition of voltage:
"The voltage at an arbitrary point P is the amount of work per unit charge it takes to move a test charge from infinity to P" (Physics for Scientists and Engineers)
I see that if the test charge is approaching X from south of X, the work will be zero, since there is only a force in the east direction as the +Q and -Q cancel e/o out in the y-direction. Same thing if the test charge is approaching x from north of X.
But what if the test charge is approaching X from west of X? east of X? How is the work zero? Won't the work be infinite from the west, assuming a positive test charge, because of the asymptotic behavior of the electric field along that line? Won't it be negatively infinite from the east, assuming a positive test charge?
So, can someone explain how the work is zero coming from infinity west or east of X?
I did think to myself that the work should be path-independent... but it's not... so how would you explain this?
2. Apr 25, 2007
mezarashi
It is not infinity coming from either west or east. This is because the electric field due to these charges at infinity is zero! The electric field decays very quickly as you move away from these charges. If you do the calculus, you'll find that it takes pretty much negligible work to move any charges around at distances of large multiples of a, since E= kq/r^2 becomes very very small.
Try doing the integration yourself for a single charge to convince yourself that no such infinity exists.
3. Apr 25, 2007
kaotak
At infinity the electric field is zero. yes. But what about when the test charge is right ontop of +Q as shown in the diagram? And when it is right ontop of -Q as shown in the diagram?
4. Apr 25, 2007
mezarashi
Well, in theory (according to these basic rules) if you have a two positive charges, it is impossible to get them "ontop" of one another, because it would require infinite force. The force of repulsion would be kq1q2/r^2. As r approaches zero, the force approaches infinity. Remember that we have estimated these charges to be "point charges", meaning that they are a singularity. This does not exist in the real world per se.
What I can say is that, this is a model that we've created to help us solve problems, if you are looking at the boundary cases, the answers might become a bit dodgy. | 701 | 2,943 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.828125 | 4 | CC-MAIN-2018-30 | latest | en | 0.925267 |
https://www.gradesaver.com/textbooks/math/applied-mathematics/elementary-technical-mathematics/chapter-10-section-10-3-finding-binomial-factors-exercise-page-350/80 | 1,537,487,335,000,000,000 | text/html | crawl-data/CC-MAIN-2018-39/segments/1537267156690.32/warc/CC-MAIN-20180920234305-20180921014705-00336.warc.gz | 746,137,534 | 13,171 | ## Elementary Technical Mathematics
$(x+4)(x+15)$
$x^2+19x+60$ has no common monomial factors. =$(\ \ \ \ \ \ \ \ \ )(\ \ \ \ \ \ \ \ \ )$ =$(x\ \ \ \ \ \ \ )(x\ \ \ \ \ \ \ )$ =$(x+\ \ )(x+\ \ )$ All the terms in the trinomial are positive. =$(x+4)(x+15)$ 4 and 15 have a sum of 19 and a product of 60. | 126 | 304 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.125 | 4 | CC-MAIN-2018-39 | longest | en | 0.740681 |
https://math.stackexchange.com/questions/3728465/evaluate-an-improper-integral-using-complex-analysis | 1,653,091,633,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652662534693.28/warc/CC-MAIN-20220520223029-20220521013029-00229.warc.gz | 441,827,542 | 67,160 | # Evaluate an improper integral using complex analysis
I got stuck trying to find $$\int_0^\infty\frac{\log x}{(x+a)^2+b^2}\,dx$$ using complex analysis.
My attempt is to evaluate the contour integral of $$\int_C\frac{\log z}{(z+a)^2+b^2}\,dz$$ for some nicely chosen contour $$C$$. But for now the best $$C$$ I can think of is two segments plus two semicircles, which gives the value $$\int_0^\infty\frac{\log x}{(x+a)^2+b^2}\,dx+\int_0^\infty\frac{\log x}{(x-a)^2+b^2}\,dx=\frac{\pi}{2b}\log(a^2+b^2).$$ The problem is that the integrand is not an even function. For example, $$\int_0^\infty\frac{\log x}{1+x^2}\,dx$$ would have been rather easy.
• Use a keyhole contour. An example to give you some ideas for how to calculate this can be found on Example 5 of the Wikipedia Page Jun 21, 2020 at 3:56
• See also here for a more complicated example that shows how this process can be extended to higher powers of $\log$ Jun 21, 2020 at 3:59
Based on this answer to the question "How to evaluate $$\int_0^\infty \frac{\log x}{(x^2+a^2)^2} dx$$" and on this answer to the question "Evaluate $$\int_0^\infty \frac {(\log x)^4dx}{(1+x)(1+x^2)}$$". Instead of a function with $$\log z$$ on the numerator, we consider a function with $$\log^2 z$$. This is the very same method as that pointed to in the comments.
For $$a,b>0$$, this method gives the closed formula
$$\int_{0}^{\infty }\frac{\log x}{\left( x+a\right) ^{2}+b^{2}}\,dx=\frac{1}{2b }\arctan \left( \frac{b}{a}\right) \log \left( a^{2}+b^{2}\right) ,\qquad a,b>0\tag{\ast}.$$
We chose the multiple-valued function $$f(z)$$ with branch cut $$\arg z=0$$ defined as
\begin{align*} f(z) &=\frac{\log ^{2}z}{\left( z+a\right) ^{2}+b^{2}},\quad \text{with } 0<\arg z<2\pi ,\quad z=re^{i\theta } \\ &=\frac{\log ^{2}z}{\left( z-z_{1}\right) \left( z-z_{2}\right) }\qquad z_{1}=-a+ib,\quad z_{2}=-a-ib, \end{align*}
and integrate it counterclockwise around the closed contour $$\Gamma$$ shown in the figure. This contour is indented around the branch point $$O$$ and consists of the circles $$\gamma _{R}$$ ($$\left\vert z\right\vert =R$$) and $$\gamma _{\varepsilon }$$ ($$\left\vert z\right\vert =\varepsilon$$), $$0<\varepsilon <1, and the segment $$\left[ \varepsilon ,R\right]$$ described in the positive sense above the $$x$$-axis and in the negative sense below the $$x$$-axis.
$$\qquad\qquad$$
$$\text{Closed contour } \Gamma$$
On the upper edge, $$\theta =0$$ ($$r\in \left[ \varepsilon ,R\right]$$) and
$$f(z)=\frac{\left( \log r\right) ^{2}}{\left( r+a\right) ^{2}+b^{2}}.$$
On the lower edge, $$\theta =2\pi$$ ($$r\in \left[ \varepsilon ,R\right]$$) and
$$f(z)=\frac{\left( \log \left( re^{i2\pi }\right) \right) ^{2}}{\left( r+a\right) ^{2}+b^{2}}=\frac{\left( \log r+i2\pi \right) ^{2}}{\left( r+a\right) ^{2}+b^{2}}.$$
As such,
\begin{align*} I &=\lim_{\varepsilon \rightarrow 0,R\rightarrow \infty }\oint_{\Gamma} \frac{\left( \log z\right) ^{2}}{\left( z+a\right) ^{2}+b^{2}}\,dz, \\ &=\int_{0}^{\infty }\frac{\left( \log r\right) ^{2}}{\left( r+a\right) ^{2}+b^{2}}\,dr-\int_{0}^{\infty }\frac{\left( \log \left( re^{i2\pi }\right) \right) ^{2}}{\left( re^{i2\pi }+a\right) ^{2}+b^{2}}\,dr \\ &\quad+\lim_{R\rightarrow \infty }\int_{\gamma _{R}}\frac{\left( \log z\right) ^{2}}{\left( z+a\right) ^{2}+b^{2}}\,dz-\lim_{\varepsilon \rightarrow 0}\int_{\gamma _{\varepsilon }}\frac{\left( \log z\right) ^{2}}{\left( z+a\right) ^{2}+b^{2}}\,dz \\ &=\int_{0}^{\infty }\frac{\left( \log r\right) ^{2}-\left( \log r+i2\pi \right) ^{2}}{\left( r+a\right) ^{2}+b^{2}}\,dx \\ &=4\pi ^{2}\int_{0}^{\infty }\frac{1}{\left( r+a\right) ^{2}+b^{2}} \,dr-i4\pi \int_{0}^{\infty }\frac{\log r}{\left( r+a\right) ^{2}+b^{2}}\,dr \end{align*}
provided that
$$\lim_{R\rightarrow \infty }\int_{\gamma _{R}}\frac{\left( \log z\right) ^{2} }{\left( z+a\right) ^{2}+b^{2}}\,dz=\lim_{\varepsilon \rightarrow 0}\int_{\gamma _{\varepsilon }}\frac{\left( \log z\right) ^{2}}{\left( z+a\right) ^{2}+b^{2}}\,dz=0,\quad \text{(see below).}$$
By the residue theorem,
\begin{align*} I &=2\pi i\left( \operatorname{Res}_{z=z_{1}}f(z)+ \operatorname{Res}_{z=z_{2}}f(z)\right) \\ &=2\pi i\left[ \operatorname{Res}_{z=z_{1}}\frac{ \left( \log z\right) ^{2}}{\left( z-z_{1}\right) \left( z-z_{2}\right) }+ \operatorname{Res}_{z=z_{2}}\frac{\left( \log z\right) ^{2}}{\left( z-z_{1}\right) \left( z-z_{2}\right) }\right] \\ &=2\pi i\left[ \frac{\left( \log z_{1}\right) ^{2}}{z_{1}-z_{2}}+\frac{ \left( \log z_{2}\right) ^{2}}{z_{2}-z_{1}}\right] \\ &=2\pi i\left[ \frac{\left( \log \left( -a+ib\right) \right) ^{2}}{i2b}- \frac{\left( \log \left( -a-ib\right) \right) ^{2}}{i2b}\right] \\ &=\frac{\pi }{b}\left[ \log \left( -a+ib\right) \right] ^{2}-\frac{\pi }{b} \left[ \log \left( -a-ib\right) \right] ^{2} \end{align*}
We now assume that $$a,b>0$$. Then
\begin{align*} I &=\frac{\pi }{b}\left[ \log \left( \left\vert -a+ib\right\vert \right) +i\left( \pi -\arctan \left( \frac{b}{a}\right) \right) \right] ^{2} \\ &\quad-\frac{\pi }{b}\left[ \log \left( \left\vert -a-ib\right\vert \right) +i\left( \pi +\arctan \left( \frac{b}{a}\right) \right) \right] ^{2} \\ &=\frac{\pi }{b}\left[ \frac{1}{2}\log \left( a^{2}+b^{2}\right) +i\left( \pi -\arctan \left( \frac{b}{a}\right) \right) \right] ^{2} \\ &\quad-\frac{\pi }{b}\left[ \frac{1}{2}\log \left( a^{2}+b^{2}\right) +i\left( \pi +\arctan \left( \frac{b}{a}\right) \right) \right] ^{2} \\ &=\frac{4\pi ^{2}}{b}\arctan \left( \frac{b}{a}\right) -i\frac{2\pi }{b} \arctan \left( \frac{b}{a}\right) \log \left( a^{2}+b^{2}\right) \end{align*}
because $$\log \left( \left\vert -a+ib\right\vert \right) =\log \left( \left\vert -a-ib\right\vert \right) =\frac{1}{2}\log \left( a^{2}+b^{2}\right) .$$
Taking the imaginary part of $$I$$ we obtain $$(\ast)$$ in the form $$\text{Im }( I )=-4\pi \int_{0}^{\infty }\frac{\log r}{\left( r+a\right) ^{2}+b^{2} }\,dr=-\frac{2\pi }{b}\arctan \left( \frac{b}{a}\right) \log \left( a^{2}+b^{2}\right)$$
Proof that $$\int_{\gamma _{R}}f,\int_{\gamma _{\varepsilon }}f\rightarrow 0$$. If $$z$$ is any point on $$\gamma _{R}$$,
\begin{align*} \left\vert f(z)\right\vert &=\frac{\left\vert \log z\right\vert ^{2}}{ \left\vert \left( z+a\right) ^{2}+b^{2}\right\vert },\qquad z=R\,e^{i\theta },R>1,0<\theta <2\pi \\ &\leq \frac{\left( \log R+2\pi \right) ^{2}}{\left\vert z+\left( -z_{1}\right) \right\vert \left\vert z+\left( -z_{2}\right) \right\vert }, \\ &\leq \frac{\left( \log R+2\pi \right) ^{2}}{\left\vert R-\sqrt{a^{2}+b^{2}} \right\vert ^{2}}\leq M_{R} \end{align*}
where
$$M_{R}=\frac{4\pi \log R+4\pi ^{2}+\log ^{2}R}{R^{2}+\left( a^{2}+b^{2}\right) -2R\sqrt{a^{2}+b^{2}}}$$
because $$\left\vert z+\left( -z_{1}\right) \right\vert \geq \left\vert R-\left\vert z_{1}\right\vert \right\vert ,\left\vert z+\left( -z_{2}\right) \right\vert \geq \left\vert R-\left\vert z_{2}\right\vert \right\vert ,\left\vert z_{1}\right\vert =\left\vert z_{2}\right\vert =\sqrt{a^{2}+b^{2}}.$$
This means that
\begin{align*} \left\vert \int_{\gamma _{R}}f(z)\,dz\right\vert &\leq M_{R}\times \,2\pi R \\ &=\frac{4\pi \log R+4\pi ^{2}+\log ^{2}R}{R^{2}+\left( a^{2}+b^{2}\right) -2R\sqrt{a^{2}+b^{2}}}\times \,2\pi R\longrightarrow 0\qquad \left( R\rightarrow \infty \right) . \end{align*}
Similarly, if $$z$$ is any point on $$\gamma _{\varepsilon }$$ \begin{align*} \left\vert f(z)\right\vert &=\frac{\left\vert \log z\right\vert ^{2}}{ \left\vert \left( z+a\right) ^{2}+b^{2}\right\vert },\qquad z=\varepsilon \,e^{i\theta },0<\varepsilon <1,0<\theta <2\pi \\ &\leq \frac{\left( \log \varepsilon +2\pi \right) ^{2}}{\left\vert z+\left( -z_{1}\right) \right\vert \left\vert z+\left( -z_{2}\right) \right\vert } \\ &\leq \frac{\left( \log \varepsilon +2\pi \right) ^{2}}{\left\vert \varepsilon -\sqrt{a^{2}+b^{2}}\right\vert ^{2}}\leq M_{\varepsilon }, \end{align*}
where
$$M_{\varepsilon }=\frac{4\pi \log \varepsilon +4\pi ^{2}+\log ^{2}\varepsilon }{\varepsilon ^{2}+\left( a^{2}+b^{2}\right) -2\varepsilon \sqrt{a^{2}+b^{2}}}$$
and
\begin{align*} \left\vert \int_{\gamma _{\varepsilon }}f(z)\,dz\right\vert &\leq M_{\varepsilon }\times \,2\pi \varepsilon \qquad z=\rho \,e^{i\theta },\rho <1 \\ &\leq \frac{4\pi \log \varepsilon +4\pi ^{2}+\log ^{2}\varepsilon }{ \varepsilon ^{2}+\left( a^{2}+b^{2}\right) -2\varepsilon \sqrt{a^{2}+b^{2}}} \times \,2\pi \varepsilon \longrightarrow 0\qquad \left( \varepsilon \rightarrow 0\right) . \end{align*} | 3,537 | 8,393 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 54, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.96875 | 4 | CC-MAIN-2022-21 | latest | en | 0.79032 |
https://www.proofwiki.org/wiki/Ordinal_is_Transitive | 1,686,045,461,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224652494.25/warc/CC-MAIN-20230606082037-20230606112037-00761.warc.gz | 1,051,312,393 | 12,342 | # Ordinal is Transitive
## Theorem
Every ordinal is a transitive set.
## Proof 1
Let $\alpha$ be an ordinal by Definition 1:
$\alpha$ is an ordinal if and only if it fulfils the following conditions:
$(1)$ $:$ $\alpha$ is a transitive set $(2)$ $:$ $\Epsilon {\restriction_\alpha}$ strictly well-orders $\alpha$
where $\Epsilon {\restriction_\alpha}$ is the restriction of the epsilon relation to $\alpha$.
Thus $\alpha$ is a priori transitive.
$\blacksquare$
## Proof 2
Let $\alpha$ be an ordinal by Definition 2:
$\alpha$ is an ordinal if and only if it fulfils the following conditions:
$(1)$ $:$ $\alpha$ is a transitive set $(2)$ $:$ the epsilon relation is connected on $\alpha$: $\ds \forall x, y \in \alpha: x \ne y \implies x \in y \lor y \in x$ $(3)$ $:$ $\alpha$ is well-founded.
Thus $\alpha$ is a priori transitive.
$\blacksquare$
## Proof 3
Let $\alpha$ be an ordinal by Definition 3.
An ordinal is a strictly well-ordered set $\struct {\alpha, \prec}$ such that:
$\forall \beta \in \alpha: \alpha_\beta = \beta$
where $\alpha_\beta$ is the initial segment of $\alpha$ determined by $\beta$:
$\alpha_\beta = \set {x \in \alpha: x \prec \beta}$
That is, $\alpha$ is a transitive set.
$\blacksquare$
## Proof 4
Let $\alpha$ be an ordinal by Definition 4.
$\alpha$ is an ordinal if and only if:
$\alpha$ is an element of every superinductive class.
The proof proceeds by the Principle of Superinduction.
From Empty Class is Transitive we start with the fact that $0$ is transitive.
$\Box$
Let $x$ be transitive.
$x^+$ is transitive.
$\Box$
We have that Class is Transitive iff Union is Subclass.
Hence the union of a chain of transitive sets is transitive.
$\Box$
Hence the result by the Principle of Superinduction.
$\blacksquare$ | 496 | 1,782 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.34375 | 4 | CC-MAIN-2023-23 | latest | en | 0.817043 |
https://aaronice.gitbook.io/lintcode/two_pointers/fruit-into-baskets | 1,716,580,284,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971058736.10/warc/CC-MAIN-20240524183358-20240524213358-00078.warc.gz | 55,860,398 | 48,434 | Medium
In a row of trees, the`i`-th tree produces fruit with type `tree[i]`.
You start at any tree of your choice, then repeatedly perform the following steps:
2. Move to the next tree to the right of the current tree. If there is no tree to the right, stop.
Note that you do not have any choice after the initial choice of starting tree: you must perform step 1, then step 2, then back to step 1, then step 2, and so on until you stop.
You have two baskets, and each basket can carry any quantity of fruit, but you want each basket to only carry one type of fruit each.
What is the total amount of fruit you can collect with this procedure?
Example 1:
``````Input:
[1,2,1]
Output:
3
Explanation:
We can collect [1,2,1].``````
Example 2:
``````Input:
[0,1,2,2]
Output:
3
Explanation:
We can collect [1,2,2].
If we started at the first tree, we would only collect [0, 1].``````
Example 3:
``````Input:
[1,2,3,2,2]
Output:
4
Explanation:
We can collect [2,3,2,2].
If we started at the first tree, we would only collect [1, 2].``````
Example 4:
``````Input:
[3,3,3,1,2,1,1,2,3,3,4]
Output:
5
Explanation:
We can collect [1,2,1,1,2].
If we started at the first tree or the eighth tree, we would only collect 4 fruits.``````
Note:
1. `1 <= tree.length <= 40000`
2. `0 <= tree[i] < tree.length`
## Analysis
The question is actually:
What is the length of longest subarray that contains up to two distinct integers?
## Solution
Sliding Window - Two Pointers Fast-Slow - O(n) time, O(n) space - (66 ms, faster than 44.17%)
``````class Solution {
public int totalFruit(int[] tree) {
HashMap<Integer, Integer> count = new HashMap<Integer, Integer>();
int firstIdx = 0;
int totalMax = 0;
for (int i = 0; i < tree.length; i++) {
count.put(tree[i], count.getOrDefault(tree[i], 0) + 1);
while (count.size() > 2) {
count.put(tree[firstIdx], count.get(tree[firstIdx]) - 1);
if (count.get(tree[firstIdx]) == 0) {
count.remove(tree[firstIdx]);
}
firstIdx++;
}
totalMax = Math.max(totalMax, i - firstIdx + 1);
} | 614 | 2,018 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.90625 | 4 | CC-MAIN-2024-22 | latest | en | 0.828294 |
https://www.cagednomoremovie.com/do-2-negatives-make-a-positive/ | 1,720,990,730,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514638.53/warc/CC-MAIN-20240714185510-20240714215510-00076.warc.gz | 612,695,755 | 10,659 | # Do 2 negatives make a positive?
## Do 2 negatives make a positive?
When you have two negative signs, one turns over, and they add together to make a positive. If you have a positive and a negative, there is one dash left over, and the answer is negative.
## Which is higher in negative numbers?
On the number line the negative numbers are to the left of zero. –5 is less than 4, because –5 lies to the left of 4 on the number line. –1 is greater than –3, because –1 lies to the right of –3 on the number line.
## What is the product of three negative numbers?
The product of three negative integers is a negative integer.
## What can negative numbers be used for?
Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common-sense idea of an opposite is reflected in arithmetic.
## What is the quotient of positive and negative numbers?
So, the quotient of a negative and a positive number is negative and, correspondingly, the quotient of a positive and a negative number is also negative. We can conclude that: When you divide a negative number by a positive number then the quotient is negative.
## What is the product of 7 negative integers?
Answer. Answer:The product of seven negative integer will result in a negative integer. Explanation: When a negative sign is multiplied a odd number of times the product sign will be negative and if it is multiplied even number of times the product sign will be positive.
## Why is zero not a whole number?
The whole numbers are the numbers 0, 1, 2, 3, 4, and so on (the natural numbers and zero). Negative numbers are not considered “whole numbers.” All natural numbers are whole numbers, but not all whole numbers are natural numbers since zero is a whole number but not a natural number.
0
## How do you do positive and negative numbers?
1. Rule 1: Adding positive numbers to positive numbers—it’s just normal addition.
2. Rule 2: Adding positive numbers to negative numbers—count forward the amount you’re adding.
3. Rule 3: Adding negative numbers to positive numbers—count backwards, as if you were subtracting.
## What is positive negative called?
Positive and negative numbers are all integers. Integers are whole numbers that are either greater than zero (positive) or less than zero (negative). For every positive integer, there’s a negative integer, called an additive inverse, that is an equal distance from zero.
## Is a positive times a negative a positive?
Rule 1: A positive number times a positive number gives you a positive number. Rule 2: A positive number times a negative number gives you a negative number. Example 2: This is new – for example, you might have 4 x -3. The 4 is positive, but the 3 is negative, so our answer has to be negative.
## What is positive divided by negative?
Rule 2: A positive number divided by a negative number equals a negative number. When you divide a negative number by a positive number, your answer is a negative number. As with multiplication, it doesn’t matter which order the positive and negative numbers are in, the answer is always a negative number.
## What do 2 positives make?
If two positive numbers are multiplied together or divided, the answer is positive. If two negative numbers are multiplied together or divided, the answer is positive. If a positive and a negative number are multiplied or divided, the answer is negative.
## Is the product (- 3 positive or negative?
The number 9 is positive while −3 is negative.
## What is the quotient of two negative numbers?
RULE 3: The quotient of two negative integers is positive. If the signs are different the answer is negative. If the signs are alike the answer is positive.
## How do you know if something is positive or negative?
Terminology for signs A number is positive if it is greater than zero. A number is negative if it is less than zero. A number is non-negative if it is greater than or equal to zero. A number is non-positive if it is less than or equal to zero. | 885 | 4,119 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.71875 | 5 | CC-MAIN-2024-30 | latest | en | 0.899135 |
https://quant.stackexchange.com/questions/16210/i-have-portfolio-volatility-for-individual-years-can-i-use-them-to-compute-port/16233#16233 | 1,643,022,021,000,000,000 | text/html | crawl-data/CC-MAIN-2022-05/segments/1642320304528.78/warc/CC-MAIN-20220124094120-20220124124120-00107.warc.gz | 533,423,134 | 34,357 | # I have portfolio volatility for individual years, can I use them to compute portfolio volatiltiy for subperiods?
Thanks for opening this question.
I have constructed some rules for a portfolio with annual rebalancing and am backtesting it for the period 1990-2014. I want to compare the risk-adjusted return to the risk-adjusted return of the S&P 500 index.
For every individual year I have calculated log returns of the constituents of the portfolio and calculate portfolio volatility for every year using the following formula in Excel:
First I compute the variance like this: =MMULT(MMULT(array1;matrix1);array2)
array1: array of weights of constituents of the portfolio*annual volatility of the constituent for every constituent in that year
array2: constituents return correlation matrix
array3: transpose of weights of constituents
And then I SQRT() the variance to get the annual portfolio volatility.
Now I am wondering if I can use the annual portfolio volatilies to compute volatilities for certain subperiods, say for example January 2013 - September 2014.
I could compute the portfolio volatily for the period Jan 2013 - Dec 2013, and for the period Jan 2014 - Sept 2014, but how do I then combine the result?
Thanks for helping me out.
You can use the Parallel Algorithm.
Your sample $X$ is divided in two sets of obsrvations $X_{A}$ and $X_B$. $$\delta\! = \bar x_B - \bar x_A \\ \bar x_X = \bar x_A + \delta\cdot\frac{n_B}{n_X} \\ VAR_{X} = VAR_{A} + VAR_{B} + \delta^2\cdot\frac{n_A n_B}{n_X}$$ Where $n$ the number of observations and $\bar x$ the mean.
• Thank you I will try this out and come back here to post results later today. Jan 16 '15 at 10:05
Since you have the weights as well as the price series of your assets, the cleanest way I can think of is to calculate the portfolio price series for your given time frame, and then estimate the variance from that.
The problem you face is that volatility changes more often than yearly, and obviously the end of the year is not necessarily a 'clean cut' between two regimes. So while it's technically possible (See jaamor's answer), I don't think that will give you the desired result. In a backtest you want to understand how your trading strategy would have performed in the past, if you had implemented it, and one of the most important considerations will be how does a portfolio react to changes in market volatility. For example, if you want to use a low volatility strategy, the backtest will show you if the stocks you selected actually yielded you a portfolio with a lower volatility, or if correlations between the stocks actually reduced that effect.
So I would recommend to use real data wherever possible.
Since you work in excel, it should be quite straightforward to do. You will probably have the daily price series of your constituents. Start with the portfolio's initial value and distribute it according to your weights by buying the constituents. Track the prices of the positions until the next rebalancing, the sum will always be equal to the current portfolio value. Before the next rebalancing, calculate what your portfolio is wrorth now. Then redistribute that money according to your strategy's weights. Repeat. Use the resulting time series to calculate variance. Use the stdev-function over the period you want, and then annualize (e.g. from daily vola to annual vola by *sqrt(252), where 252 is the trading days.)
• Thank you for your great answer @phi. However, you don't capture correlation of constituents returns like this, right? What you do here is you basically treat your portfolio as index, if I'm not mistaken. The variance caculated with this method differs from the variance as calculated using the method in my first post (modern-portfolio theory method). Can you elaborate? Jan 16 '15 at 10:05
• You're not mistaken, and of course you capture correlation: One asset goes down, another one goes up -> Portfolio NAV stays the same. That's correlation at work. You can actually trade effect this by selling a call on an index, while buying calls on the index' constituents. This is called dispersion trading. The variance of your portfolio is defined by its returns, and they will be the same. Jan 16 '15 at 12:25 | 955 | 4,243 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.53125 | 4 | CC-MAIN-2022-05 | latest | en | 0.901803 |
https://ahilado.wordpress.com/2017/02/ | 1,680,423,336,000,000,000 | text/html | crawl-data/CC-MAIN-2023-14/segments/1679296950422.77/warc/CC-MAIN-20230402074255-20230402104255-00417.warc.gz | 122,059,413 | 72,899 | # Connection and Curvature in Riemannian Geometry
In Geometry on Curved Spaces, we showed how different geometry can be when we are working on curved space instead of flat space, which we are usually more familiar with. We used the concept of a metric to express how the distance formula changes depending on where we are on this curved space. This gives us some way to “measure” the curvature of the space.
We also described the concept of parallel transport, which is in some way even more general than the metric, and can also be used to provide us with some measure of the curvature of a space. Although we can use concepts analogous to parallel transport even without the metric, if we do have a metric on the space and an expression for it, we can relate the concept of parallel transport to the metric, which is perhaps more intuitive. In this post, we formalize the concept of parallel transport by defining the Christoffel symbol and the Riemann curvature tensor, both of which we can obtain given the form of the metric. The Christoffel symbol and the Riemann curvature tensor are examples of the more general concepts of a connection and a curvature form, respectively, which need not be obtained from the metric.
##### Some Basics of Tensor Notation
First we establish some notation. We have already seen some tensor notation in Some Basics of (Quantum) Electrodynamics, but we explain a little bit more of that notation here, since it will be the language we will work in. Many of the ordinary vectors we are used to, such as the position, will be indexed by superscripts. We refer to these vectors as contravariant vectors. A common convention is to use Latin letters, such as $i$ or $j$, as indices when we are working with space, and Greek letters, such as $\mu$ and $\nu$, as indices when we are working with spacetime. Let us consider , for example, spacetime. An event in this spacetime is specified by its $4$-position $x^{\mu}$, where $x^{0}=ct$$x^{1}=x$$x^{2}=y$, and $x^{3}=z$.
We will use the symbol $g_{\mu\nu}$ for our metric, and we will also often express it as a matrix. For the case of flat spacetime, our metric is given by the Minkowski metric $\eta_{\mu\nu}$:
$\displaystyle \eta_{\mu\nu}=\left(\begin{array}{cccc}-1&0&0&0\\0&1&0&0\\0&0&1&0\\ 0&0&0&1\end{array}\right)$
We can use the metric to “raise” and “lower” indices. This is done by multiplying the metric and a vector, and summing over a common index (one will be a superscript and the other a subscript). We have introduced the Einstein summation convention in Some Basics of (Quantum) Electrodynamics, where repeated indices always imply summation, unless explicitly stated otherwise, and we will continue to use this convention for posts discussing differential geometry and the theory of relativity.
Here is an example of “lowering” the index of $x^{\nu}$ in flat spacetime using the metric $\eta_{\mu\nu}$ to obtain a new quantity $x_{\mu}$:
$\displaystyle x_{\mu}=\eta_{\mu\nu}x^{\nu}$
Explicitly, the components of the quantity $x_{\mu}$ are given by $x_{0}=-ct$$x_{1}=x$$x_{2}=y$, and $x_{3}=z$. Note that the “time” component $x_{0}$ has changed sign; this is because $\eta_{00}=-1$. A quantity such as $x_{\mu}$, which has a subscript index, is called a covariant vector.
In order to “raise” indices, we need the “inverse metric$g^{\mu\nu}$. For the Minkowski metric $\eta_{\mu\nu}$, the inverse metric $\eta^{\mu\nu}$ has the exact same components as $\eta_{\mu\nu}$, but for more general metrics this may not be the case. The general procedure for obtaining the inverse metric is to consider the expression
$\eta_{\mu\nu}\eta^{\nu\rho}=\delta_{\mu}^{\rho}$
where $\delta_{\mu}^{\rho}$ is the Kronecker delta, a quantity that can be expressed as the matrix
$\displaystyle \delta_{\mu}^{\rho}=\left(\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\ 0&0&0&1\end{array}\right)$.
As a demonstration of what our notation can do, we recall the formula for the invariant spacetime interval:
$\displaystyle (ds)^2=-(cdt)^2+(dx)^2+(dy)^2+(dz)^2$
Using tensor notation combined with the Einstein summation convention, this can be written simply as
$\displaystyle (ds)^2=\eta_{\mu\nu}dx^{\mu}dx^{\nu}$.
##### The Christoffel Symbol and the Covariant Derivative
We now come back to the Christoffel symbol $\Gamma^{\mu}_{\nu\lambda}$. The idea behind the Christoffel symbol is that it is used to define the covariant derivative $\nabla_{\nu}V^{\mu}$ of a vector $V^{\mu}$.
The covariant derivative is a very important concept in differential geometry (and not just in Riemannian geometry). When we take derivatives, we are actually comparing two vectors. To further explain what we mean, we recall that individually the components of the vectors can be thought of as functions on the space, and we recall the expression for the derivative from An Intuitive Introduction to Calculus:
$\displaystyle \frac{df}{dx}=\frac{f(x+\epsilon)-f(x)}{(x+\epsilon)-(x)}$ when $\epsilon$ is extremely small (essentially negligible)
More formally, we can write
$\displaystyle \frac{df}{dx}=\lim_{\epsilon\to 0}\frac{f(x+\epsilon)-f(x)}{(x+\epsilon)-(x)}$.
Therefore, employing the language of partial derivatives, we could have written the following partial derivative of the $\mu$-th component of an $m$-dimensional vector $V^{\mu}$ on an $m$-dimensional space with respect to the coordinate $x^{\nu}$:
$\displaystyle \frac{\partial V^{\mu}}{\partial x^{\nu}}=\lim_{\Delta x^{\nu}\to 0}\frac{V^{\mu}(x^{1},...,x^{\nu}+\Delta x^{\nu},...,x^{m})-V^{\mu}(x_{1},...,x^{\nu},...,x^{m})}{(x^{\nu}+\Delta x^{\nu})-(x^{\nu})}$
The problem is that we are comparing vectors from different vector spaces. Recall from Vector Fields, Vector Bundles, and Fiber Bundles that we can think of a vector bundle as having a vector space for every point on the base space. The vector $V^{\mu}(x^{1},...,x^{\nu}+\Delta x^{\nu},...,x^{m})$ belongs to the vector space on the point $(x^{1},...,x^{\nu}+\Delta x^{\nu},...,x^{m})$, while the vector $V^{\mu}(x_{1},...,x^{\nu},...,x^{m})$ belongs to the vector space on the point $(x_{1},...,x^{\nu},...,x^{m})$. To be able to compare the two vectors we need to “transport” one to the other in the “correct” way, by which we mean parallel transport. Now we have seen in Geometry on Curved Spaces that parallel transport can have weird effects on vectors, and these weird effects are what the Christoffel symbol expresses.
Let $\tilde{V}^{\mu}(x^{1},...,x^{\nu}+\Delta x^{\nu},...,x^{m})$ denote the vector $V^{\mu}(x_{1},...,x^{\nu},...,x^{m})$ parallel transported from its original vector space on $(x_{1},...,x^{\nu},...,x^{m})$ to the vector space on $(x^{1},...,x^{\nu}+\Delta x^{\nu},...,x^{m})$. The vector $\tilde{V}^{\mu}(x^{1},...,x^{\nu}+\Delta x^{\nu},...,x^{m})$ is given by the following expression:
$\displaystyle \tilde{V}^{\mu}(x^{1},...,x^{\nu}+\Delta x^{\nu},...,x^{m})=V^{\mu}(x_{1},...,x^{\nu},...,x^{m})-V^{\lambda}(x_{1},...,x^{\nu},...,x^{m})\Gamma^{\mu}_{\nu\lambda}(x_{1},...,x^{\nu},...,x^{m})\Delta x^{\nu}$
Therefore the Christoffel symbol provides a “correction” for what happens when we parallel transport a vector from one point to another. This is an example of the concept of a connection, which, like the covariant derivative, is part of more general differential geometry beyond Riemannian geometry. The object that is to be parallel transported may not be a vector, for example when we have more general fiber bundles instead of vector bundles. However, in Riemannian geometry we will usually focus on vector bundles, in particular a special kind of vector bundle called the tangent bundle, which consists of the tangent vectors at a point.
Now there is more than one way to parallel transport a mathematical object, which means that there are many choices of a connection. However, in Riemannian geometry there is a special kind of connection that we will prefer. This is the connection that satisfies the following two properties:
$\displaystyle \Gamma^{\mu}_{\nu\lambda}=\Gamma^{\mu}_{\lambda\nu}$ (torsion-free)
$\displaystyle \nabla_{\rho}g_{\mu\nu}$ (metric compatibility)
The connection that satisfies these two properties is the one that can be obtained from the metric via the following formula:
$\displaystyle \Gamma^{\mu}_{\nu\lambda}=\frac{1}{2}g^{\mu\sigma}(\partial_{\lambda}g_{\mu\sigma}+\partial_{\mu}g_{\sigma\lambda}-\partial_{\sigma}g_{\lambda\mu})$.
The covariant derivative is then defined as
$\displaystyle \nabla_{\nu}V^{\mu}=\lim_{\Delta x^{\nu}\to 0}\frac{V^{\mu}(x^{1},...,x^{\nu}+\Delta x^{\nu},...,x^{m})-\tilde{V}^{\mu}(x_{1},...,x^{\nu}+\Delta x^{\nu},...,x^{m})}{(x^{\nu}+\Delta x^{\nu})-(x^{\nu})}$.
We are now comparing vectors belonging to the same vector space, and evaluating the expression above leads to the formula for the covariant derivative:
$\displaystyle \nabla_{\nu}V^{\mu}=\partial_{\nu}V^{\mu}+\Gamma^{\mu}_{\nu\lambda}V^{\lambda}$.
##### The Riemann Curvature Tensor
Next we consider the quantity known as the Riemann curvature tensor. It is once again related to parallel transport, in the following manner. Consider parallel transporting a vector $V^{\sigma}$ through an “infinitesimal” distance specified by another vector $A^{\mu}$, and after that, through another infinitesimal distance specified by a yet another vector $B^{\nu}$. Then we go parallel transport it again in the opposite direction to $A^{\mu}$, then finally in the opposite direction to $B^{\nu}$. The path forms a parallelogram, and when the vector $V^{\sigma}$ returns to its starting point it will then be changed by an amount $\delta V^{\rho}$. We can think of the Riemann curvature tensor as the quantity that relates all of these:
$\displaystyle \delta V^{\rho}=R^{\rho}_{\ \sigma\mu\nu}V^{\sigma}A^{\mu}B^{\nu}$.
Another way to put this is to consider taking the covariant derivative of the vector $V^{\rho}$ along the same path as described above. The Riemann curvature tensor is then related to this quantity as follows:
$\displaystyle \nabla_{\mu}\nabla_{\nu}V^{\rho}-\nabla_{\nu}\nabla_{\mu}V^{\rho}=R^{\rho}_{\ \sigma\mu\nu}V^{\sigma}$.
Expanding the left hand side, and using the torsion-free property of the Christoffel symbol, we will find that
$\displaystyle R^{\rho}_{\ \sigma\mu\nu}=\partial_{\mu}\Gamma^{\rho}_{\nu\sigma}-\partial_{\nu}\Gamma^{\rho}_{\mu\sigma}+\Gamma^{\rho}_{\mu\lambda}\Gamma^{\lambda}_{\nu\sigma}-\Gamma^{\rho}_{\nu\lambda}\Gamma^{\lambda}_{\mu\sigma}$.
For connections other than the torsion-free one that we chose, there will be another part of the expansion of the expression $\nabla_{\mu}\nabla_{\nu}-\nabla_{\nu}\nabla_{\mu}$ called the torsion tensor. For our case, however, we need not worry about it and we can focus on the Riemann curvature tensor.
There is another quantity that can be obtained from the Riemann curvature tensor called the Ricci tensor, denoted by $R_{\mu\nu}$. It is given by
$\displaystyle R_{\mu\nu}=R^{\lambda}_{\ \mu\lambda\nu}$.
Following the Einstein summation convention, we sum over the repeated index $\lambda$, and therefore the resulting quantity will have only two indices instead of four. This is an example of the operation on tensors called contraction. If we raise one index using the metric and contract again, we obtain a quantity called the Ricci scalar, denoted $R$:
$\displaystyle R=R^{\mu}_{\ \mu}$
##### Example: The $2$-Sphere
To provide an explicit example of the concepts discussed, we show their specific expressions for the case of a $2$-sphere. We will only give the final results here. The explicit computations can be found among the references, but the reader may gain some practice, especially on manipulating tensors, by performing the calculations and checking only the answers here. In any case, since the metric is given, it is only a matter of substituting the relevant quantities into the formulas already given above.
We have already given the expression for the metric of the $2$-sphere in Geometry on Curved Spaces. We recall that it in matrix form, it is given by (we change our notation for the radius of the $2$-sphere to $R_{0}$ to avoid confusion with the symbol for the Ricci scalar)
$\displaystyle g_{mn}= \left(\begin{array}{cc}R_{0}^{2}&0\\ 0&R_{0}^{2}\text{sin}(\theta)^{2}\end{array}\right)$
Individually, the components are (we will use $\theta$ and $\varphi$ instead of the numbers $1$ and $2$ for the indices)
$\displaystyle g_{\theta\theta}=R_{0}^{2}$
$\displaystyle g_{\varphi\varphi}=R_{0}^{2}(\text{sin}(\theta))^{2}$
The other components ($g_{\theta\varphi}$ and $g_{\varphi\theta}$) are all equal to zero.
The Christoffel symbols are therefore given by
$\displaystyle \Gamma^{\theta}_{\varphi\varphi}=-\text{sin}(\theta)\text{cos}(\theta)$
$\displaystyle \Gamma^{\varphi}_{\theta\varphi}=\text{cot}(\theta)$
$\displaystyle \Gamma^{\varphi}_{\varphi\theta}=\text{cot}(\theta)$
The other components ($\Gamma^{\theta}_{\theta\theta}$, $\Gamma^{\theta}_{\theta\varphi}$, $\Gamma^{\theta}_{\varphi\theta}$, $\Gamma^{\varphi}_{\theta\theta}$, and $\Gamma^{\varphi}_{\varphi\varphi}$) are all equal to zero.
The components of the Riemann curvature tensor are given by
$\displaystyle R^{\theta}_{\ \varphi\theta\varphi}=(\text{sin}(\theta))^{2}$
$\displaystyle R^{\theta}_{\ \varphi\varphi\theta}=-(\text{sin}(\theta))^{2}$
$\displaystyle R^{\varphi}_{\ \theta\theta\varphi}=-1$
$\displaystyle R^{\varphi}_{\ \theta\varphi\theta}=1$
The other components (there are still twelve of them, so I won’t bother writing all their symbols down here anymore) are all equal to zero.
The components of the Ricci tensor is
$\displaystyle R_{\theta\theta}=1$
$\displaystyle R_{\varphi\varphi}=(\text{sin}(\theta))^{2}$
The other components ($R_{\theta\varphi}$ and $R_{\varphi\theta}$) are all equal to zero.
Finally, the Ricci scalar is
$\displaystyle R=\frac{2}{R_{0}^{2}}$
We note that the larger the radius of the $2$-sphere, the smaller the curvature. We can see this intuitively, for example, when it comes to the surface of our planet, which appears flat because the radius is so large. If our planet was much smaller, this would not be the case.
##### Bonus: The Einstein Field Equations of General Relativity
Given what we have discussed in this post, we can now write down here the expression for the Einstein field equations (also known simply as Einstein’s equations) of general relativity. It is given in terms of the Ricci tensor and the metric (of spacetime) via the following equation:
$\displaystyle R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{8\pi}{c^{4}} GT_{\mu\nu}$
where $G$ is the gravitational constant, the same constant that appears in Newton’s law of universal gravitation (which is approximated by Einstein’s equations at certain limiting conditions), $c$ is the speed of light in a vacuum, and $T_{\mu\nu}$ is the energy-momentum tensor (also known as the stress-energy tensor), which gives the “density” of energy and momentum, as well as certain other related concepts, such as the pressure and shear stress. The symbol $\Lambda$ refers to what is known as the cosmological constant, which was not there in Einstein’s original formulation but later added to support his view of an unchanging universe. Later, with the dawn of George Lemaitre’s theory of an expanding universe, later known as the Big Bang theory, the cosmological constant was abandoned. More recently, the universe was found to not only be expanding, but expanding at an accelerating rate, necessitating the return of the cosmological constant, with an interpretation in terms of the “vacuum energy”, also known as “dark energy”. Today the nature of the cosmological constant remains one of the great mysteries of modern physics.
##### Bonus: Connection and Curvature in Quantum Electrodynamics
The concepts of connection and curvature also appear in quantum field theory, in particular quantum electrodynamics (see Some Basics of (Quantum) Electrodynamics). It is the underlying concept in gauge theory, of which quantum electrodynamics is probably the simplest example. However, it is an example of differential geometry which does not make use of the metric. We consider a fiber bundle, where the base space is flat spacetime (also known as Minkowski spacetime), and the fiber is $\text{U}(1)$, which is the group formed by the complex numbers with magnitude equal to $1$, with law of composition given by multiplication (we can also think of this as a circle).
We want the group $\text{U}(1)$ to act on the wave function (or field operator) $\psi(x)$, so that the wave function has a “phase”, i.e. we have $e^{i\phi(x)}\psi(x)$, where $e^{i\phi(x)}$ is a complex number which depends on the location $x$ in spacetime. Note that therefore different values of the wave function at different points in spacetime will have different values of the “phase”. In order to compare, them, we need a connection and a covariant derivative.
The connection we want is given by
$\displaystyle i\frac{q}{\hbar c}A_{\mu}$
where $q$ is the charge of the electron, $\hbar$ is the normalized Planck’s constant, $c$ is the speed of light in a vacuum, and $A_{\mu}$ is the four-potential of electrodynamics.
The covariant derivative (here written using the symbol $D_{\mu}$)is
$\displaystyle D_{\mu}\psi(x)=\partial_{\mu}\psi(x)+i\frac{q}{\hbar c}A_{\mu}\psi(x)$
We will also have a concept analogous to the Riemann curvature tensor, called the field strength tensor, denoted $F_{\mu\nu}$. Of course, our “curvature” in this case is not the literal curvature of spacetime, as we have already specified that our spacetime is flat, but an abstract notion of “curvature” that specifies how the phase of our wavefunction changes as we move around the spacetime. This field strength tensor is given by the following expression:
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$
This may be compared to the expression for the Riemann curvature tensor, where the connection is given by the Christoffel symbols. The first two terms of both expressions are very similar. The difference is that the expression for the Riemann curvature tensor has some extra terms that the expression for the field strength tensor does not have. However, a generalization of this procedure for quantum electrodynamics to groups other than $\text{U}(1)$, called Yang-Mills theory, does feature extra terms in the expression for the field strength tensor that perhaps makes the two more similar.
The concepts we have discussed here can be used to derive the theory of quantum electrodynamics simply from requiring that the Lagrangian (from which we can obtain the equations of motion, see also Lagrangians and Hamiltonians) be invariant under $\text{U}(1)$ transformations, i.e. even if we change the “phase” of the wave function at every point the Lagrangian remains the same. This is an example of what is known as gauge symmetry. Generalized to other groups such as $\text{SU}(2)$ and $\text{SU}(3)$, this is the idea behind gauge theories, which include Yang-Mills theory and leads to the standard model of particle physics.
References:
Christoffel Symbols on Wikipedia
Riemannian Curvature Tensor on Wikipedia
Einstein Field Equations on Wikipedia
Gauge Theory on Wikipedia
Riemann Tensor for Surface of a Sphere on Physics Pages
Ricci Tensor and Curvature Scalar for a Sphere on Physics Pages
Spacetime and Geometry by Sean Carroll
Geometry, Topology, and Physics by Mikio Nakahara
Introduction to Elementary Particle Physics by David J. Griffiths
Introduction to Quantum Field Theory by Michael Peskin and Daniel V. Schroeder
# The Riemann Hypothesis for Curves over Finite Fields
The Riemann hypothesis is one of the most famous open problems in mathematics. Not only is there a million dollar prize currently being offered by the Clay Mathematical Institute for its solution, it also has a very long and interesting history spanning over a century and a half. It is part of many famous “lists” of open problems such as the famous 23 problems of David Hilbert, the 18 problems of Stephen Smale, and the 7 “millennium” problems of the aforementioned Clay Mathematical Institute.
The attention and reverence given to the Riemann hypothesis by the mathematical community is not without good reason. The problem originated in the paper “On the Number of Primes Less Than a Given Magnitude” by the mathematician Bernhard Riemann, where he applied the recently developed theory of complex analysis to number theory, in particular to come up with a function $\pi(x)$ that counts the number of prime numbers less than $x$. The zeroes of the Riemann zeta function figure into the formula for this “prime counting function” $\pi(x)$, and the Riemann hypothesis is a conjecture that concerns these zeroes. Aside from the knowledge about the prime numbers that a solution of the Riemann hypothesis will give us, it is hoped for that efforts toward this solution will lead to developments in mathematics that may be of interest to us for reasons much bigger, and perhaps outside of, the original motivations.
In the 1940’s, the mathematician Andre Weil solved a version of the Riemann hypothesis, which applies to the Riemann zeta function over finite fields. The ideas that Weil developed for solving this version of the Riemann hypothesis has led to many important developments in modern mathematics, whose applications are not limited to the original problem only. It is these ideas that we discuss in this post. But before we can give the statement of the Riemann hypothesis over finite fields (which is almost identical to that of the original Riemann hypothesis), we first review some concepts regarding zeta functions.
We have discussed zeta functions before in Zeta Functions and L-Functions. We recall that the Riemann zeta function is given by the formula
$\displaystyle \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}$
or, in Euler product form,
$\displaystyle \zeta(s)=\prod_{p}\frac{1}{1-p^{-s}}$.
We now generalize the Riemann zeta function to any finitely generated ring $\mathcal{O}_{K}$ with field of fractions $K$ by writing it in the following form (this zeta function $\zeta(K,s)$ is also called the arithmetic zeta function):
$\displaystyle \zeta(K,s)=\prod_{\mathfrak{m}}\frac{1}{1-(\# \mathcal{O}_{K}/\mathfrak{m})^{-s}}$
where $\mathfrak{m}$ runs over all the maximal ideals of the ring $\mathcal{O}_{K}$, $\mathcal{O}_{K}/\mathfrak{m}$ is the residue field, and the expression $\#\mathcal{O}_{K}/\mathfrak{m}$ stands for the number of elements of this residue field. In the case that $\mathcal{O}_{K}=\mathbb{Z}$, we get back our usual expression for the Riemann zeta function in its Euler product form, which we have written above, since the maximal ideals of $\mathbb{Z}$ are the principal ideals $(p)$ generated by the prime numbers, and the residue fields $\mathbb{Z}/(p)$ are the fields $\{0,1,...,p-1\}$, therefore the number $\# \mathbb{Z}/(p)$ is equal to $p$.
Next we discuss finite fields. All finite fields have a number of elements equal to some positive power of a prime number $p$; if this number is equal to $q=p^{n}$, we write the finite field as $\mathbb{F}_{q}$ or $\mathbb{F}_{p^{n}}$. In the case that $n=1$, then $\mathbb{F}_{q}=\mathbb{F}_{p}$ is isomorphic to $\mathbb{Z}/p\mathbb{Z}$.
Let $C$ be a nonsingular projective curve defined over the finite field $\mathbb{F}_{q}$. “Nonsingular” roughly refers to the curve being “smooth”; or “differentiable”; “projective” roughly means that the curve is part, or a subset, of some projective space. We will not be dwelling too much on these technicalities in this post. “Defined over the finite field $\mathbb{F}_{q}$” means that the polynomial equation that defines the curve has coefficients which are elements of the finite field $\mathbb{F}_{q}$. We know that in algebraic geometry (see Basics of Algebraic Geometry), the points of a curve (or more general varieties) correspond to maximal ideals of a “ring of functions” $\mathcal{O}_{K}$ on the curve $C$ . For a point $P$ on a curve over a finite field $\mathbb{F}_{q}$, the residue field $\mathcal{O}_{K}/\mathfrak{m}$, where $\mathfrak{m}$ is the maximal ideal corresponding to $P$, is also a finite field of the form $\mathbb{F}_{q^{m}}$. The number $m$ is called the degree of $P$ and written $\text{deg}(P)$, and we now define another zeta function (also called the local zeta function and written $Z(C,t))$ via the following formula:
$\displaystyle Z(C,t)=\prod_{P\in C}\frac{1}{1-t^{\text{deg}(P)}}$
or equivalently,
$\displaystyle Z(C,t)=\prod_{\mathfrak{m}}\frac{1}{1-t^{\text{deg}(\mathfrak{m})}}$.
Note that this zeta function $Z(C,t)$ is related to the other zeta function $\zeta(K,s)$ by the following relation:
$\displaystyle \zeta(K,s)=Z(C,q^{-s})$.
Next we take the “logarithm” of the zeta function $Z(C,t)$. Using the familiar rules for taking the logarithms of products, we will obtain
$\displaystyle \text{log}(Z(C,t))=\text{log}\bigg(\prod_{\mathfrak{m}}\frac{1}{1-t^{\text{deg}(\mathfrak{m})}}\bigg)$
$\displaystyle \text{log}(Z(C,t))=\sum_{\mathfrak{m}}\text{log}\bigg(\frac{1}{1-t^{\text{deg}(\mathfrak{m})}}\bigg)$
$\displaystyle \text{log}(Z(C,t))=-\sum_{\mathfrak{m}}\text{log}\bigg(1-t^{\text{deg}(\mathfrak{m})}\bigg)$
Next we will need the following series expansion for logarithms:
$\displaystyle \text{log}(1-a)=-\sum_{k=0}^{\infty}\frac{a^{k}}{k}$.
This allows us to write the logarithm of the zeta function as follows:
$\displaystyle \text{log}(Z(C,t))=\sum_{\mathfrak{m}}\sum_{k=1}^{\infty}\frac{(t^{\text{deg}(\mathfrak{m})})^{k}}{k}$
$\displaystyle \text{log}(Z(C,t))=\sum_{\mathfrak{m}}\sum_{k=1}^{\infty}\frac{(t^{\text{deg}(\mathfrak{m})})^{k}}{k\text{deg}(\mathfrak{m})}\text{deg}(\mathfrak{m})$
We can condense this expression by writing
$\displaystyle \text{log}(Z(C,t))=\sum_{n=1}^{\infty}N_{n}\frac{t^{n}}{n}$
where
$\displaystyle N_{n}=\sum_{d|n}d(\#\{\mathfrak{m}\subset R|\text{deg}(\mathfrak{m})=d\})$.
The expression $d|n$ means “$n$ is divisible by $d$“, or “$d$ divides $n$“, which means that the sum is taken over all $d$ that divides $n$.
The numbers $N_{n}$ can be thought of as the number of points on the curve $C$ whose coordinates are elements of the finite field $\mathbb{F}_{q^{n}}$. In fact, we can actually define the zeta function $Z(C,t)$ starting with the numbers $N_{n}$, i.e.
$\displaystyle Z(C,t)=\text{exp}\bigg(\sum_{n=1}^{\infty}N_{n}\frac{t^{n}}{n}\bigg)$
but we chose to start from the more familiar Riemann zeta function $\zeta(s)$ and generalize to get the form we want for curves over finite fields.
We recall that the zeroes of a function $f(z)$ are those $z_{i}$ such that $f(z_{i})=0$.
We can now give the statement of the Riemann hypothesis for curves over finite fields:
The zeroes of the zeta function $\zeta(K,s)=Z(C,q^{-s})$ all have real part equal to $\frac{1}{2}$.
We will not discuss the entirety of Weil’s proof in this post, although the reader may consult the references provided for such a discussion. Instead we will give a rough overview of Weil’s strategy, which rests on three important assumptions. We will show, roughly, how these assumptions lead to the proof of the Riemann hypothesis, and although we will not prove the assumptions themselves, we will also give a kind of preview of the ideas involved in their respective proofs. It is these ideas, which may now be considered to have developed into entire areas of research in themselves, which are perhaps the most enduring legacy of Weil’s proof.
Assumption 1 (Rationality): The zeta function $Z(C,t)$ can be written in the following form:
$\displaystyle Z(C,t)=\frac{\prod_{i=1}^{2g}(1-\alpha_{i}t)}{(1-t)(1-qt)}$
Given that this assumption holds, we can take the logarithm of the above expression,
$\displaystyle \text{log}(Z(C,t))=\text{log}\bigg(\frac{\prod_{i=1}^{2g}(1-\alpha_{i}t)}{(1-t)(1-qt)}\bigg)$
$\displaystyle \text{log}(Z(C,t))=\sum_{i=1}^{2g}\text{log}(1-\alpha_{i}t)-\text{log}(1-t)-\text{log}(1-qt)$
and we can then apply the series expansion for the logarithm that we have applied earlier to obtain the following expression,
$\displaystyle \text{log}(Z(C,t))=\sum_{n=1}^{\infty}(-\sum_{i=1}^{2g}\alpha_{i}^{n}+1+q^{n})\frac{t^{n}}{n}$
which we can now compare to the expression we obtained earlier for $\text{log}(Z(C,t))$ in terms of the number $N_{n}$ of points with coordinates in $\mathbb{F}_{q^{n}}$:
$\displaystyle \sum_{n=1}^{\infty}(-\sum_{i=1}^{2g}\alpha_{i}^{n}+1+q^{n})\frac{t^{n}}{n}=\sum_{n=1}^{\infty}N_{n}\frac{t^{n}}{n}$.
Comparing the coefficients of $\frac{t^{n}}{n}$, we obtain, for each $n$,
$\displaystyle -\sum_{i=1}^{2g}\alpha_{i}^{n}+1+q^{n}=N_{n}$.
With a little algebraic manipulation we have
$\displaystyle -\sum_{i=1}^{2g}\alpha_{i}^{n}=N_{n}-q^{n}-1$
and taking the absolute value of both sides gives us
$\displaystyle |\sum_{i=1}^{2g}\alpha_{i}^{n}|=|N_{n}-q^{n}-1|$
Assumption 2 (Hasse-Weil Inequality):
$\displaystyle |N_{n}-q^{n}-1|\leq 2gq^{\frac{n}{2}}$
This assumption, together with the earlier discussion, means that
$\displaystyle |\sum_{i=1}^{2g}\alpha_{i}^{n}|\leq 2gq^{\frac{n}{2}}$
We can then make use of the expansion
$\displaystyle \sum_{i=1}^{2g}\frac{1}{1-\alpha_{i}(q^{-\frac{1}{2}})}=\sum_{n=1}^{\infty}(\sum_{i=1}^{2g}\alpha_{i}^{n})(q^{-\frac{1}{2}})^{n}$
which in turn implies that
$|\alpha_{i}|\leq q^{\frac{1}{2}}$ for all $i$ from $1$ to $2g$.
Assumption 3 (Functional Equation):
$\displaystyle Z\bigg(C,\frac{1}{qt}\bigg)=q^{1-g}t^{2-2g}Z(C,t)$
Given this assumption, and writing the zeta function $Z(C,t)$ explicitly, we have:
$\displaystyle \frac{\prod_{i=1}^{2g}(1-\alpha_{i}\frac{1}{qt})}{(1-\frac{1}{qt})(1-q\frac{1}{qt})}=q^{1-g}t^{2-2g}\frac{\prod_{i=1}^{2g}(1-\alpha_{i}t)}{(1-t)(1-qt)}$
With a little algebraic manipulation we can obtain the following equation:
$\displaystyle q^{g}t^{2g}\prod_{i=1}^{2g}(1-\alpha_{i}\frac{1}{qt})=\prod_{i=1}^{2g}(1-\alpha_{i}t)$
Let us write the product explicitly, and make the left side zero by letting $t=\frac{\alpha_{1}}{q}$:
$\displaystyle q^{g}(\frac{\alpha_{1}}{q})^{2g}(0)(1-\alpha_{2}\frac{1}{q}\frac{q}{\alpha_{1}})...(1-\alpha_{2g}\frac{1}{q}\frac{q}{\alpha_{1}})=(1-\alpha_{1}\frac{\alpha_{1}}{q})(1-\alpha_{2}\frac{\alpha_{1}}{q})...(1-\alpha_{2g}\frac{\alpha_{1}}{q})$
Now since the left side is zero, the right side also must be zero. Therefore one of the factors in the product must be zero. This means that for some $i$ from $1$ to $2g$, we have
$\displaystyle 1-\alpha_{i}\frac{\alpha_{1}}{q}=0$
In other words,
$\displaystyle \alpha_{i}\alpha_{1}=q$
This applies to any other $j$ from $1$ to $2g$, not just $1$, therefore more generally we must have
$\displaystyle \alpha_{i}\alpha_{j}=q$ for some $i$ and $j$ from $1$ to $2g$.
If we combine this result with our earlier result that
$\displaystyle |\alpha_{i}|\leq q^{\frac{1}{2}}$ for all $i$ from $1$ to $2g$,
this means that
$\displaystyle |\alpha_{i}|=q^{\frac{1}{2}}$ for all $i$ from $1$ to $2g$.
With this last result, we know that the zeroes of $Z(C,t)$ must have absolute value equal to $q^{-\frac{1}{2}}$. Since $Z(C,q^{-s})=\zeta(K,s)$, this implies that the real part of $s$ must be equal to $\frac{1}{2}$, and this proves the Riemann hypothesis for curves over finite fields. More explicitly, let $t_{0}$ be a zero of the zeta function $Z(C,q^{-s})$. We then have
$\displaystyle |t_{0}|=q^{-\frac{1}{2}}$
$\displaystyle |q^{-s}|=q^{-\frac{1}{2}}$
$\displaystyle |q^{-(\text{Re}(s)+\text{Im}(s))}|=q^{-\frac{1}{2}}$
$\displaystyle q^{-(\text{Re}(s))}=q^{-\frac{1}{2}}$
$\displaystyle \text{Re}(s)=\frac{1}{2}$
The proof of the rationality of the zeta function $Z(C,t)$ and the functional equation makes use of the theory of divisors (see Divisors and the Picard Group) and a very important theorem in algebraic geometry called the Riemann-Roch theorem. The Riemann-Roch theorem originates from complex analysis, which was the kind of the “specialty” of Bernhard Riemann (“On the Number of Primes Less Than a Given Magnitude” was his only paper on number theory, and it concerns the application of complex analysis to number theory). In its original formulation, the Riemann-Roch theorem gives the dimension of the vector space formed by the functions whose zeroes and poles (for a function which can be expressed as the ratio of two polynomials, the poles can be thought of as the zeroes of the denominator), and their “order of vanishing”, are specified. The Riemann-Roch theorem has since been generalized to aspects of algebraic geometry not necessarily directly concerned with complex analysis, and it is this generalization that allows us to make use of it for the case at hand.
In addition to the theory of divisors and the Riemann-Roch theorem, to prove the Hasse-Weil inequality, one must make use of the theory of fixed points, applied to what is known as the Frobenius morphism, which sends a point of $C$ with coordinates $a_{i}$ to the point with coordinates $a_{i}^{q}$. The theory of fixed points is related to the part of algebraic geometry known as intersection theory. Roughly, given a function $f(x)$, we can think of its fixed points as the values of $x$ for which $f(x)=x$. One way to obtain these fixed points is to draw the graph of $y=x$, and the graph of $y=f(x)$, on the $x$$y$ plane; the fixed points of $f(x)$ are then given by the points where the two graphs intersect.
For the Frobenius morphism, the fixed points correspond to those points whose coordinates are elements of the finite field $\mathbb{F}_{q}$. Similarly, the fixed points of the $n$-th power of the Frobenius morphism (which we can think of as the Frobenius morphism applied $n$ times) correspond to those points whose coordinates are elements of the finite field $\mathbb{F}_{q^{n}}$. Hence we can obtain the numbers $N_{n}$ that go into the expression of the zeta function $Z(C,t)$ using the Frobenius morphism. Combined with results from intersection theory such as the Castelnuovo-Severi inequality and the Hodge index theorem, this allows us to prove the Hasse-Weil inequality.
In algebraic geometry, curves are one-dimensional varieties, and just as there is a version of the Riemann hypothesis for curves over finite fields, there is also a version of the Riemann hypothesis for higher-dimensional varieties over finite fields, called the Weil conjectures, since they were proposed by Weil himself after he proved the case for curves. The Weil conjectures themselves follow the important assumptions involved in proving the Riemann hypothesis for curves over finite fields, such as the rationality of the zeta function and the functional equation. In addition, part of the Weil conjectures suggests a connection with the theory of cohomology (see Homology and Cohomology and Cohomology in Algebraic Geometry), which significant implications for the connections between algebraic geometry and methods originally developed for algebraic topology.
The Weil conjectures were proved by Bernard Dwork, Alexander Grothendieck, and Pierre Deligne. In his efforts to prove the Weil conjectures, Grothendieck developed the notion of topos (see More Category Theory: The Grothendieck Topos), as well as etale cohomology. As further part of his approach, Grothendieck also proposed conjectures, known as the standard conjectures on algebraic cycles, which remain open to this day. Grothendieck’s student, Pierre Deligne, was able to complete the proof of the Weil conjectures while bypassing the standard conjectures on algebraic cycles, by developing ingenious methods of his own. Still, the standard conjectures on algebraic cycles, as well as the related theory of motives, remain very much interesting on their own and continue to be a subject of modern mathematical research.
References:
Riemann Hypothesis on Wikipedia
Weil Conjectures on Wikipedia
Arithmetic Zeta Function on Wikipedia
Local Zeta Function on Wikipedia
The Weil Conjectures for Curves by Sam Raskin
Algebraic Geometry by Bas Edixhoven and Lenny Taelman
The Riemann Hypothesis over Finite Fields: From Weil to the Present Day by J.S. Milne
Algebraic Geometry by Robin Hartshorne
# The Moduli Space of Elliptic Curves
A moduli space is a kind of “parameter space” that “classifies” mathematical objects. Every point of the moduli space stands for a mathematical object, in such a way that mathematical objects which are more similar to each other are closer and those that are more different from each other are farther apart. We may use the notion of equivalence relations (see Modular Arithmetic and Quotient Sets) to assign several objects which are in some sense “isomorphic” to each other to a single point.
We have discussed on this blog before one example of a moduli space – the projective line (see Projective Geometry). Every point on the projective line corresponds to a geometric object, a line through the origin. Two lines which have almost the same value of the slope will be closer on the projective line compared to two lines which are almost perpendicular.
Another example of a moduli space is that for circles on a plane – such a circle is specified by three real numbers, two coordinates for the center and one positive real number for the radius. Therefore the moduli space for circles on a plane will consist of a “half-volume” of some sort, like 3D space except that one coordinate is restricted to be strictly positive. But if we only care about the circles up to “congruence”, we can ignore the coordinates for the center – or we can also think of it as simply sending circles with the same radius to a single point, even if they are centered at different points. This moduli space is just the positive real line. Every point on this moduli space, which is a positive real number, corresponds to all the circles with radius equal to that positive real number.
We now want to construct the moduli space of elliptic curves. In order to do this we will need to first understand the meaning of the following statement:
Over the complex numbers, an elliptic curve is a torus.
We have already seen in Elliptic Curves what an elliptic curve looks like when graphed in the $x$$y$ plane, where $x$ and $y$ are real numbers. This gives us a look at the points of the elliptic curve whose coordinates are real numbers, or to put it in another way, these are the real numbers $x$ and $y$ which satisfy the equation of the elliptic curve.
When we look at the points of the elliptic curve with complex coordinates, or in other words the complex numbers which satisfy the equation of the elliptic curve, the situation is more complicated. First off, what we actually have is not what we usually think of as a curve, but rather a surface, in the same way that the complex numbers do not form a line like the real numbers do, but instead form a plane. However, even though it is not easy to visualize, there is a function called the Weierstrass elliptic function which provides a correspondence between the (complex) points of an elliptic curve and the points in the “fundamental parallelogram” of a lattice in the complex plane. We can think of “gluing” the opposite sides of this fundamental parallelogram to obtain a torus. This is what we mean when we say that an elliptic curve is a torus. This also means that there is a correspondence between elliptic curves and lattices in the complex plane.
We will discuss more about lattices later on in this post, but first, just in case the preceding discussion seems a little contrived, we elaborate a bit on the Weierstrass elliptic function. We must first discuss the concept of a holomorphic function. We have discussed in An Intuitive Introduction to Calculus the concept of the derivative of a function. Now not all functions have derivatives that exist at all points; in the case that the derivative of the function does exist at all points, we refer to the function as a differentiable function.
The concept of a holomorphic function in complex analysis (analysis is the term usually used in modern mathematics to refer to calculus and its related subjects) is akin to the concept of a differentiable function in real analysis. The derivative is defined as the limit of a certain ratio as the numerator and the denominator both approach zero; on the real line, there are limited ways in which these quantities can approach zero, but on the complex plane, they can approach zero from several different directions; for a function to be holomorphic, the expression for its derivative must remain the same regardless of the direction by which we approach zero.
In previous posts on topology on this blog we have been treating two different topological spaces as essentially the same whenever we can find a bijective and continuous function (also known as a homeomorphism) between them; similarly, we have been treating different algebraic structures such as groups, rings, modules, and vector spaces as essentially the same whenever we can find a bijective homomorphism (an isomorphism) between two such structures. Following these ideas and applying them to complex analysis, we may treat two spaces as essentially the same if we can find a bijective holomorphic function between them.
The Weierstrass elliptic function is not quite holomorphic, but is meromorphic – this means that it would have been holomorphic everywhere if not for the “lattice points” where there exist “poles”. But it is alright for us, because such a lattice point is to be mapped to the “point at infinity”. All in all, this allows us to think of the complex points of the elliptic curve as being essentially the same as a torus, following the ideas discussed in the preceding paragraph.
Moreover, the torus has a group structure of its own, considered as the direct product group $\text{U}(1)\times\text{U}(1)$ where $\text{U}(1)$ is the group of complex numbers of magnitude equal to $1$ with the law of composition given by the multiplication of complex numbers. When the complex points of the elliptic curve get mapped by the Weierstrass elliptic function to the points of the torus, the group structure provided by the “tangent and chord” or “tangent and secant” construction becomes the group structure of the torus. In other words, the Weierstrass elliptic function provides us with a group isomorphism.
All this discussion means that the study of elliptic curves becomes the study of lattices in the complex plane. Therefore, what we want to construct is the moduli space of lattices in the complex plane, up to a certain equivalence relation – two lattices are to be considered equivalent if one can be obtained by multiplying the other by a complex number (this equivalence relation is called homothety). Going back to elliptic curves, this corresponds to an isomorphism of elliptic curves in the sense of algebraic geometry.
Now given two complex numbers $\omega_{1}$ and $\omega_{2}$, a lattice $\Lambda$ in the complex plane is given by
$\Lambda=\{m\omega_{1}+n\omega_{2}|m,n\in\mathbb{Z}\}$
For example, setting $\omega_{1}=1$ and $\omega_{2}=i$, gives a “square” lattice. This lattice is also the set of all Gaussian integers. The fundamental parallelogram is the parallelogram formed by the vertices $0$, $\omega_{1}$, $\omega_{2}$, and $\omega_{1}+\omega_{2}$. Here is an example of a lattice, courtesy of Alvaro Lozano-Robledo:
The fundamental parallelogram is in blue. Here is another, courtesy of Sam Derbyshire:
Because we only care about lattices up to homothety, we can “rescale” the lattice by multiplying it with a complex number equal to $\frac{1}{\omega_{1}}$, so that we have a new lattice equivalent under homothety to the old one, given by
$\Lambda=\{m+n\omega|m,n\in\mathbb{Z}\}$
where
$\displaystyle \tau=\frac{\omega_{2}}{\omega_{1}}$.
We can always interchange $\omega_{1}$ and $\omega_{2}$, but we will fix our convention so that the complex number $\tau=\frac{\omega_{2}}{\omega_{1}}$, when written in polar form $\tau=re^{i\theta}$ always has a positive angle $\theta$ between 0 and 180 degrees. If we cannot obtain this using our choice of $\omega_{1}$ and $\omega_{2}$, then we switch the two.
Now what this means is that a complex number $\omega$, which we note is a complex number in the upper half plane $\mathbb{H}=\{z\in \mathbb{C}|\text{Im}(z)>0\}$, because of our convention in choosing $\omega_{1}$ and $\omega_{2}$, uniquely specifies a homothety class of lattices $\Lambda$. However, a homothety class of lattices may not always uniquely specify such a complex number $\tau$. Several such complex numbers may refer to the same homothety class of lattices.
What $\omega_{1}$ and $\omega_{2}$ specify is a choice of basis (see More on Vector Spaces and Modules) for the lattice $\Lambda$; we may choose several different bases to refer to the same lattice. Hence, the upper half plane is not yet the moduli space of all lattices in the complex plane (up to homothety); instead it is an example of what is called a Teichmuller space. To obtain the moduli space from the Teichmuller space, we need to figure out when two different bases specify lattices that are homothetic.
We will just write down the answer here; two complex numbers $\tau$ and $\tau'$ refer to homothetic lattices if there exists the following relation between them:
$\displaystyle \tau'=\frac{a\tau+b}{c\tau+d}$
for integers $a$$b$$c$, and $d$ satisfying the identity
$\displaystyle ad-bc=1$.
We can “encode” this information into a $2\times 2$ matrix (see Matrices) which is an element of the group (see Groups) called $\text{SL}(2,\mathbb{Z})$. It is the group of $2\times 2$ matrices with integer entries and determinant equal to $1$. Actually, the matrix with entries $a$$b$$c$, and $d$ and the matrix with entries $-a$$-b$$-c$, and $-d$ specify the same transformation, therefore what we actually want is the group called $\text{PSL}(2,\mathbb{Z})$, also known as the modular group, and also written $\Gamma(1)$, obtained from the group $\text{SL}(2,\mathbb{Z})$ by considering two matrices to be equivalent if one is the negative of the other.
We now have the moduli space that we want – we start with the upper half plane $\mathbb{H}$, and then we identify two points if we can map one point into the other via the action of an element of the modular group, as we have discussed earlier. In technical language, we say that they belong to the same orbit. We can write our moduli space as $\Gamma(1)\backslash\mathbb{H}$ (the notation means that the group $\Gamma(1)$ acts on $\mathbb{H}$ “on the left”).
When dealing with quotient sets, which are sets of equivalence classes, we have seen in Modular Arithmetic and Quotient Sets that we can choose from an equivalence class one element to serve as the “representative” of this equivalence class. For our moduli space $\Gamma(1)\backslash\mathbb{H}$, we can choose for the representative of an equivalence class a point from the “fundamental domain” for the modular group. Any point on the upper half plane can be obtained by acting on a point from the fundamental domain with an element of the modular group. The following diagram, courtesy of user Fropuff on Wikipedia, shows the fundamental domain in gray:
The other parts of the diagram show where the fundamental domain gets mapped to by certain special elements, in particular the “generators” of the modular group, which are the two elements where $a=0$, $b=-1$, $c=1$, and $d=-1$, and $a=1$, $b=1$, $c=1$, and $d=0$. We will not discuss too much of these concepts for now. Instead we will give a preview of some concepts related to this moduli space. Topologically, this moduli space looks like a sphere with a missing point; in order to make the moduli space into a sphere (topologically), we take the union of the upper half plane $\mathbb{H}$ with the projective line (see Projective Geometry) $\mathbb{P}^{1}(\mathbb{Q})$. This projective line may be thought of as the set of all rational numbers $\mathbb{Q}$ together with a “point at infinity.” The modular group also acts on this projective line, so we can now take the quotient of $\mathbb{H}\cup\mathbb{P}^{1}(\mathbb{Q})$ (denoted $\mathbb{H}^{*}$ by the same equivalence relation as earlier; this new space, topologically equivalent to the sphere, is called the modular curve $X(1)$.
The functions and “differential forms” on the modular curve $X(1)$ are of special interest. They can be obtained from functions on the upper half plane (with the “point at infinity”) satisfying certain conditions related to the modular group. If they are holomorphic everywhere, including the “point at infinity”, they are called modular forms. Modular forms are an interesting object of study in themselves, and their generalizations, automorphic forms, are a very active part of modern mathematical research.
Moduli Space on Wikipedia
Elliptic Curve on Wikipedia
Weierstrass’s Elliptic Functions on Wikipedia
Fundamental Pair of Periods on Wikipedia
Modular Group on Wikipedia
Fundamental Domain on Wikipedia
Modular Form on Wikipedia
Automorphic Form on Wikipedia
Image by Alvano Lozano Robledo on Wikipedia
Image by Sam Derbyshire on Wikipedia
Image by User Fropuff of Wikipedia
Advanced Topics in the Arithmetic of Elliptic Curves by Joseph H. Silverman
A First Course in Modular Forms by Fred Diamond and Jerry Shurman
# Elliptic Curves
An elliptic curve (not to be confused with an ellipse) is a certain kind of polynomial equation which can usually be expressed in the form
$\displaystyle y^{2}=x^{3}+ax+b$
where $a$ and $b$ are numbers (more precisely, elements of some field) which satisfy the condition that the quantity
$\displaystyle 4a^{3}+27b^{2}$
is not equal to zero. This is not the most general form of an elliptic curve, as it will not hold for coefficients of “finite characteristic” equal to $2$ or $3$; however, for our present purposes, this definition will suffice.
Examples of elliptic curves are the following:
$\displaystyle y^{2}=x^{3}-x$
$\displaystyle y^{2}=x^{3}-x+1$
which, for real $x$ and $y$ may be graphed in the “Cartesian” or “$x$$y$” plane as follows (image courtesy of user YassineMrabet of Wikipedia):
This rather simple mathematical object has very interesting properties which make it a central object of study in many areas of modern mathematical research.
In this post we focus mainly on one of these many interesting properties, which is the following:
The points of an elliptic curve form a group.
A group is a set with a law of composition which is associative, and the set contains an “identity element” under this law of composition, and every element of this set has an “inverse” (see Groups). Now this law of composition applies whether the points of the elliptic curve have rational numbers, real numbers, or complex numbers for coordinates, and it is always given by the same formula. It is perhaps most visible if we consider real numbers, since in that case we can plot it on the $x$$y$ plane as we have done earlier. The law of composition is also often called the “tangent and chord” or “tangent and secant” construction.
We now expound on this construction. Given two points on the elliptic curve $P$ and $Q$ on the curve, we draw a line passing through both of them. In most cases, this line will pass through another point $R$ on the curve. Then we draw a vertical line that passes through the point $R$. This vertical line will pass through another point $R'$ on the curve. This gives us the law of composition of the points of the elliptic curve, and we write $P+Q=R'$. Here is an image courtesy of user SuperManu of Wikipedia:
The usual case that we have described is on the left; the other three images show other different cases where the line drawn does not necessarily go through three points. This happens, for example, when the line is tangent to the curve at some point $Q$, as in the second picture; in this case, we think of the line as passing through $Q$ twice. Therefore, when we compute $P+Q$, the third point is $Q$ itself, and it is through $Q$ that we draw our vertical line to locate $Q'$, which is equal to $P+Q$.
The second picture also shows another computation, that of $Q+Q$, or $2Q$. Again, since this necessitates taking a line that passes through the point $Q$ twice, this means that the line must be tangent to the elliptic curve at $Q$. The third point that it passes through is the point $P$, and we draw the vertical line through $P$ to find the point $P'$, which is equal to $2Q$.
Now we discuss the case described by the third picture, where the line going through the two points $P$ and $Q$ which we want to “add” is a vertical line. To explain what happens, we need the notion of a “point at infinity” (see Projective Geometry). We write the point at infinity as $0$, expressing the idea that it is the identity element of our group. We cannot find this point at infinity in the $x$$y$ plane, but we can think of it as the third point that the vertical line passes through aside from $P$ and $Q$. In this case, of course, there is no need to draw another vertical line – we simply write $P+Q=0$.
Finally we come to the case described by the fourth picture; this is simply a combination of the earlier cases we have described above. The vertical line is tangent to the curve at the point $P$, so we can think of it as passing through $P$ twice, and the third point is passes through is the point at infinity $0$, so we can write $2P=0$.
We will not prove explicitly that the points form a group under this law of composition, i.e. that the conditions for a set to form a group are satisfied by our procedure, but it is an interesting exercise to attempt to do so; readers may try it out for themselves or consult the references provided at the end of the post. It is worth mentioning that our group is also an abelian group, i.e. we have $P+Q=Q+P$, and hence we have written our law of composition “additively”.
Now, to make the group law apply even when $x$ and $y$ are not real numbers, we need to write this procedure algebraically. This is a very powerful approach, since this allows us to operate with mathematical concepts even when we cannot visualize them.
Let $x_{P}$ and $y_{P}$ be the $x$ and $y$ coordinates of a point $P$, and let $x_{Q}$ and $y_{Q}$ be the $x$ and $y$ coordinates of another point $Q$. Let
$\displaystyle m=\frac{y_{Q}-y_{P}}{x_{Q}-x_{P}}$
be the slope of the line that connects the points $P$ and $Q$. Then the point $P+Q$ has $x$ and $y$ coordinates given by the following formulas:
$\displaystyle x_{P+Q}=m^{2}-x_{P}-x_{Q}$
$\displaystyle y_{P+Q}=-y_{P}-m(x_{P+Q}-x_{P})$
In the case that $Q$ is the same point as $P$, then we define the slope of the tangent line to the elliptic curve at the point $P$ using the formula
$\displaystyle m=\frac{3x_{P}^{2}+a}{2y_{P}}$
where $a$ is the coefficient of $x$ in the formula, of the elliptic curve, i.e.
$\displaystyle y^{2}=x^{3}+ax+b$.
Then the $x$ and $y$ coordinates of the point $2P$ are given by the same formulas as above, appropriately modified to reflect the fact that now the points $P$ and $Q$ are the same:
$\displaystyle x_{2P}=m^{2}-2x_{P}$
$\displaystyle y_{2P}=-y_{P}-m(x_{2P}-x_{P})$
This covers the first two cases in the image above; for the third case, when $P$ and $Q$ are distinct points and $y_{P}=-y_{Q}$, we simply set $P+Q=0$. For the fourth case, when $P$ and $Q$ refer to the same point, and $y_{P}=0$, we set $2P=0$. The point at infinity itself can be treated as a mere point and play into our computations, by setting $P+0=P$, reflecting its role as the identity element of the group.
The group structure on the points of elliptic curves have practical applications in cryptography, which is the study of “encrypting” information so that it cannot be deciphered by parties other than the intended recipients, for example in military applications, or when performing financial transactions over the internet.
On the purely mathematical side, the study of the group structure is currently a very active field of research. An important theorem called the Mordell-Weil theorem states that even though there may be an infinite number of points whose coordinates are given by rational numbers (called rational points), these points may all be obtained by performing the “tangent and chord” or “tangent and secant” construction on a finite number of points. In more technical terms, the group of rational points on an elliptic curve is finitely generated.
There is a theorem concerning finitely generated abelian groups stating that any finitely generated abelian group $G$ is isomorphic to the direct sum of $r$ copies of the integers and a finite abelian group called the torsion subgroup of $G$. The number $r$ is called the rank of $G$. The famous Birch and Swinnerton-Dyer conjecture, which currently carries a million dollar prize for its proof (or disproof), concerns the rank of the finitely generated abelian group of rational points on an elliptic curve.
Another thing that we can do with elliptic curves is use them to obtain representations of Galois groups (see Galois Groups). A representation of a group $G$ on a vector space $V$ over a field $K$ is a homomorphism from $G$ to $GL(V)$, the group of bijective linear transformations of the vector space $V$ to itself. We know of course from Matrices that linear transformations of vector spaces can always be written as matrices (in our case the matrices must have nonzero determinant to ensure that the linear transformations are bijective). Representation theory allows us to study the objects of abstract algebra using the methods of linear algebra.
To any elliptic curve we can associate a certain algebraic number field (see Algebraic Numbers). The elements of these algebraic number fields are “generated” by the algebraic numbers that provide the coordinates of “$p$-torsion” points of the elliptic curve, i.e. those points $P$ for which $pP=0$ for some prime number $p$.
The set of $p$-torsion points of the elliptic curve is a $2$-dimensional vector space over the finite field $\mathbb{Z}/p\mathbb{Z}$ (see Modular Arithmetic and Quotient Sets), also written as $\mathbb{F}_{p}$. Among other things this means that we can choose two $p$-torsion points $P$ and $Q$ of the elliptic curve such that any other $p$-torsion point can be written as $aP+bQ$ for integers $a$ and $b$ between $0$ and $p-1$. When an element of the Galois group of the algebraic number field generated by the coordinates of the $p$-torsion points of the elliptic curve permutes the elements of the algebraic number field, it also permutes the $p$-torsion points of the elliptic curve. This permutation can then be represented by a $2\times 2$ matrix with coefficients in $\mathbb{F}_{p}$.
The connection between Galois groups and elliptic curves is a concept that is central to many developments and open problems in mathematics. It plays a part, for example in the proof of the famous problem called Fermat’s Last Theorem. It is also related to the open problem called the Kronecker Jugendtraum (which is German for Kronecker’s Childhood Dream, and named after the mathematician Leopold Kronecker), also known as Hilbert’s Twelfth Problem, which seeks a procedure for obtaining all field extensions of algebraic number fields whose Galois group is an abelian group. This problem has been solved only in the special case of imaginary quadratic fields, and the solution involves special kinds of “symmetries” of elliptic curves called complex multiplication (not to be confused with the multiplication of complex numbers). David Hilbert, who is one of the most revered mathematicians in history, is said to have referred to the theory of complex multiplication as “…not only the most beautiful part of mathematics but of all science.”
References:
Elliptic Curve on Wikipedia
Mordell-Weil Theorem on Wikipedia
Birch and Swinnerton-Dyer Conjecture on Wikipedia
Wiles’ Proof of Fermat’s Last Theorem on Wikipedia
Hilbert’s Twelfth Problem on Wikipedia
Complex Multiplication on Wikipedia
Image by User YassineMrabet of Wikipedia
Image by User SuperManu of Wikipedia
Fearless Symmetry: Exposing the Hidden Patterns of Numbers by Avner Ash and Robert Gross
Elliptic Tales: Curves, Counting, and Number Theory by Avner Ash and Robert Gross
Rational Points on Elliptic Curves by Joseph H. Silverman
# Basics of Arithmetic Geometry
Here is a mathematics problem well-known since ancient times: Find integers $a$, $b$, and $c$ that solve the famous equation in the Pythagorean theorem,
$\displaystyle a^{2}+b^{2}=c^{2}$
Examples are $a=3$, $b=4$, $c=5$, and $a=5$, $b=12$, $c=13$ ($a$ and $b$ are of course interchangeable).
The general solution was already known to the ancient Greek mathematician Euclid. Let $m$ and $n$ be integers; then $a$, $b$, and $c$ are given by
$\displaystyle a=m^{2}-n^{2}$
$\displaystyle b=2m^{2}n^{2}$
$\displaystyle c=m^{2}+n^{2}$
Direct substitution and a little algebra completes the proof.
Now for some geometry. If we divide both sides of the equation by $c^{2}$, and let $x=\frac{a}{c}$, and $y=\frac{b}{c}$, then the equation becomes
$x^{2}+y^{2}=1$
which is the equation of a circle of radius $1$ centered at the origin. The problem of finding integer solutions to the equation of the Pythagorean theorem now becomes the problem of finding points in the unit circle whose coordinates are rational numbers.
There are analogous problems of finding “rational points” in “shapes” other than circles (the technical term for shapes described by polynomial equations is “variety”). For the other quadratic equations like the conic sections (parabola, hyperbola, and ellipse) this problem has already been solved.
However for cubic equations (like the so-called “elliptic curves”) and equations with an even higher degree this is still a very fruitful area of research, part of a field of mathematics called arithmetic geometry (also called Diophantine geometry).
One famous theorem in this field is Faltings’ theorem (formerly the Mordell conjecture): The number of rational points on a curve (a curve is a one-dimensional variety – take note that over the complex numbers this is actually a surface) with rational coefficients and genus greater than one (the genus is a number related to the degree) is finite.
References:
Diophantine Geometry on Wikipedia
Diophantine Equation on Wikipedia
Elliptic Curve on Wikipedia
Faltings’s Theorem on Wikipedia
Rational Points on Elliptic Curves by Joseph H. Silverman
# Geometry on Curved Spaces
Differential geometry is the branch of mathematics used by Albert Einstein when he formulated the general theory of relativity, where gravity is the curvature of spacetime. It was originally invented by Carl Friedrich Gauss to study the curvature of hills and valleys in the Kingdom of Hanover.
From what I described, one may guess that differential geometry has something to do with curvature. The geometry we learn in high school only occurs on a flat surface. There we can put coordinates $x$ and $y$ and compute distances, angles, areas, and so on.
To imagine what geometry on curved spaces looks like, imagine a globe. Instead of $x$ and $y$ coordinates, we can use latitude and longitude. One can now see just how different geometry is on this globe. Vertical lines (the lines of constant $x$) on a flat surface are always the same distance apart. On a globe, the analogues of these vertical lines, the lines of constant longitude, are closer near the poles than they are near the equator.
Other weird things happen on our globe: One can have triangles with angles that sum to more than 180 degrees. Run two perpendicular line segments from the north pole to the equator. They will meet the equator at a right angle and form a triangle with three right angles for a total of 270 degrees. Also on the globe the ratio between the circumference of a circle to its diameter might no longer be equal to the number $\pi$.
To make things more explicit, we will introduce the concept of a metric (the word “metric” refers to a variety of mathematical concepts related to notion of distance – in this post we use it in the sense of differential geometry to refer to what is also called the metric tensor). The metric is an example of a mathematical object called a tensor, which we will not discuss much of in this post. Instead, we will think of the metric as expressing a kind of “distance formula” for our space, which may be curved. The part of differential geometry that makes use of the metric is called Riemannian geometry, named after the mathematician Bernhard Riemann, a student of Gauss who extended his results on curved spaces to higher dimensions.
We recall from From Pythagoras to Einstein several important versions of the “distance formula”, from the case of 2D space, to the case of 4D spacetime. We will focus on the simple case of 2D space in this post, since it is much easier to visualize; in fact, we have already given an example of a 2D space earlier, the globe, which we shall henceforth technically refer to as the $2$-sphere. As we have learned in From Pythagoras to Einstein, a knowledge of the most simple cases can go very far toward the understanding of more complicated ones.
We will make a little change in our notation so as to stay consistent with the literature. Instead of the latitude, we will make use of the colatitude, written using the symbol $\theta$, and defined as the complementary angle to the latitude, i.e. the colatitude is 90 degrees minus the latitude. We will keep using the longitude, and we write it using the symbol $\varphi$. Note that even though we colloquially express our angles in degrees, for calculations we will always use radians, as is usual practice in mathematics and physics.
On a flat 2D space, the distance formula is given by
$\displaystyle (\Delta x)^{2}+(\Delta y)^{2}=(\Delta s)^{2}$.
It will be productive for us to work with extremely small quantities for now; from them we can obtain larger quantities later on using the language of calculus (see An Intuitive Introduction to Calculus). Adopting the notation of this language, we write
$\displaystyle (dx)^{2}+(dy)^{2}=(ds)^{2}$
We now give the distance formula for a $2$-sphere:
$\displaystyle R^{2}(d\theta)^{2}+R^{2}\text{sin}(\theta)^{2}(d\varphi)^{2}=(ds)^{2}$
where $R$ is the radius of the $2$-sphere. This formula agrees with our intuition; the same difference in latitude and longitude result in a bigger distance for a bigger $2$-sphere than for a smaller one, and the same difference in longitude results in a bigger distance for points near the equator than for points near the poles.
The idea behind the concept of the metric is that it gives how the distance formula changes depending on the coordinates. It is often written as a matrix (see Matrices) whose entries are the “coefficients” of the distance formula. Hence, for a flat 2D space it is given by
$\displaystyle \left(\begin{array}{cc}1&0\\ 0&1\end{array}\right)$
while for a $2$-sphere it is given by
$\displaystyle \left(\begin{array}{cc}R^{2}&0\\ 0&R^{2}\text{sin}(\theta)^{2}\end{array}\right)$.
We have seen that the metric can express how a space is curved. There are several other quantities related to the metric (and which can be derived from it), such as the Christoffel symbol and the Riemann curvature tensor, which express ideas related to curvature – however, unlike the metric which expresses curvature in terms of the distance formula, the Christoffel symbol and the Riemann curvature tensor express curvature in terms of how vectors (see Vector Fields, Vector Bundles, and Fiber Bundles) change as they move around the space.
The main equations of Einstein’s general theory of relativity, called the Einstein equations, relate the Riemann curvature tensor of 4D spacetime to the distribution of mass (or, more properly, the distribution of energy and momentum), expressed via the so-called energy-momentum tensor (also known as the stress-energy tensor).
The application of differential geometry is not limited to general relativity of course, and its objects of study are not limited to the metric. For example, in particle physics, gauge theories such as electrodynamics (see Some Basics of (Quantum) Electrodynamics) use the language of differential geometry to express forces like the electromagnetic force as a kind of “curvature”, even though a metric is not used to express this more “abstract” kind of curvature. Instead, a generalization of the concept of “parallel transport” is used. Parallel transport is the idea behind objects like the Christoffel symbol and the Riemann curvature tensor – it studies how vectors change as they move around the space. To generalize this, we replace vector bundles by more general fiber bundles (see Vector Fields, Vector Bundles, and Fiber Bundles).
To give a rough idea of parallel transport, we give a simple example again in 2D space – this 2D space will be the surface of our planet. Now space itself is 3D (with time it forms a 4D spacetime). But we will ignore the up/down dimension for now and focus only on the north/south and east/west dimensions. In other words, we will imagine ourselves as 2D beings, like the characters in the novel Flatland by Edwin Abbott. The discussion below will not make references to the third up/down dimension.
Imagine that you are somewhere at the Equator, holding a spear straight in front of you, facing north. Now imagine you take a step forward with this spear. The spear will therefore remain parallel to its previous direction. You take another step, and another, walking forward (ignoring obstacles and bodies of water) until you reach the North Pole. Now at the North Pole, without turning, you take a step to the right. The spear is still parallel to its previous direction, because you did not turn. You just keep stepping to the right until you reach the Equator again. You are not at your previous location of course. To go back you need to walk backwards, which once again keeps the spear parallel to its previous direction.
When you finally come back to your starting location, you will find that you are not facing the same direction as when you first started. In fact, you (and the spear) will be facing the east, which is offset by 90 degrees clockwise from the direction you were facing at the beginning, despite the fact that you were keeping the spear parallel all the time.
This would not have happened on a flat space; this “turning” is an indicator that the space (the surface of our planet) is curved. The amount of turning depends, among other things, on the curvature of the space. Hence the idea of parallel transport gives us a way to actually measure this curvature. It is this idea, generalized to mathematical objects other than vectors, which leads to the abstract notion of curvature – it is a measure of the changes that occur in certain mathematical objects when you move around a space in a certain way, which would not have happened if you were on a flat space.
In closing, I would like to note that although differential geometry is probably most famous for its applications in physics (another interesting application in physics, by the way, is the so-called Berry’s phase in quantum mechanics), it is by no means limited to these applications alone, as already reflected in its historical origins, which barely have anything to do with physics. It has even found applications in number theory, via Arakelov theory. Still, it has an especially important role in physics, with much of modern physics written in its language, and many prospects for future theories depending on it. Whether in pure mathematics or theoretical physics, it is one of the most fruitful and active fields of research in modern times.
Bonus:
Since we have restricted ourselves to 2D spaces in this post, here is an example of a metric in 4D spacetime – this is the Schwarzschild metric, which describes the curved spacetime around objects like stars or black holes (it makes use of spherical polar coordinates):
$\displaystyle \left(\begin{array}{cccc}-(1-\frac{2GM}{rc^{2}})&0&0&0\\0&(1-\frac{2GM}{rc^{2}})^{-1}&0&0\\0&0&r^{2}&0\\ 0&0&0&r^{2}\text{sin}(\theta)^{2}\end{array}\right)$
In other words, the “infinitesimal distance formula” for this curved spacetime is given by
$\displaystyle -(1-\frac{2GM}{rc^{2}})(d(ct))^{2}+(1-\frac{2GM}{rc^{2}})^{-1}(dr)^{2}+r^{2}(d\theta)^{2}+r^{2}\text{sin}(\theta)^{2}(d\varphi)^{2}=(ds)^{2}$
where $G$ is the gravitational constant and $M$ is the mass. Note also that as a matter of convention the time coordinate is “scaled” by the constant $c$ (the speed of light in a vacuum).
References:
Differential Geometry on Wikipedia
Riemannian Geometry on Wikipedia
Metric Tensor on Wikipedia
Parallel Transport on Wikipedia
Differential Geometry of Curves and Surfaces by Manfredo P. do Carmo
Geometry, Topology, and Physics by Mikio Nakahara
# Valuations and Completions
In ordinary everyday life, there are several notions of closeness. There is for example, a physical notion of distance, and we say, for instance, that we are close to our next-door neighbors. But there is another sense of closeness, such that we can say that we are “close” to our relatives, or to our friends, even though physically they may be far away.
There is also a similar notion of “closeness” between numbers. The most basic method is provided by the familiar “absolute value“. Given three numbers $x$, $x_{1}$, and $x_{2}$, to say that $x$ is closer to $x_{1}$ than to $x_{2}$ means that $|x-x_{1}|<|x-x_{2}|$. So for example, since $|(-1)-(2)|=3$ and $|(8)-(2)|=6$, we therefore say that the number $2$ is “closer” to the number $-1$ than to the number $8$. In other words, the smaller the value of $|x-y|$, the closer $x$ and $y$ are to each other.
But there are also other notions of “closeness” for numbers, just as we have explained above, that with our relatives or friends we may be “close” even if we are far away from each other. Consider the numbers $1$ and $10001$. Simply by looking, they can perhaps be said to be “relatives” or “friends”, which makes them in some way closer than, say, $1$ and $18$. The same may be said for $5$ and $3000005$, that they are perhaps members of the same “family”. This is, of course, because their difference is divisible by a large power of $10$, and since we use the decimal system to write our numbers, there is some sort of visual cue that these numbers are “family members”.
But in number theory, $10$ is not really very special. Perhaps it just so happens that we have $10$ fingers which we use for counting, so we used $10$ as a base for our number system. What is really special in number theory are the prime numbers. So for our notion of closeness we choose a prime, and define our measure of closeness so that two numbers are closer together whenever their difference is divisible by a large power of that prime number. For our chosen prime $p$, we want an analogue of the absolute value, which we will call the $p$-adic absolute value, and written $|x-y|_{p}$, which is smaller if the difference of $x$ and $y$ is divisible by a large power of $p$. The “ordinary” absolute value will now be denoted by $|x-y|_{\infty}$.
We want to define this for rational numbers as follows. Given a rational number $a$, we express it as
$a=p^{m}\frac{b}{c}$
such that $b$, $c$, and $p$ are mutually prime, i.e. they have no factors in common except $1$. Then we set
$|a|_{p}=\frac{1}{p^{m}}$.
We can see that this definition gives us the properties we are looking for – the value of $|a|_{p}$ is indeed smaller if $a$ is divisible by a large power of $p$.
The absolute value (both the “ordinary” absolute value and the $p$-adic absolute value) is also called the multiplicative valuation. There is also a related notion called the exponential valuation, which, in the $p$-adic case, we denote by $v_{p}(a)$ for a rational number $a$. The exponential valuation is obtained from the multiplicative valuation by setting
$v_{p}(a)=-\text{log}_{p}|a|_{p}$.
In the case above, where $a=p^{m}\frac{b}{c}$ and $b$, $c$, and $p$ are mutually prime, we simply have
$v_{p}(a)=m$.
For the ordinary absolute value, we just set
$v_{\infty}(a)=-\text{ln}|a|_{\infty}$
where $\text{ln }$ of course stands for the natural logarithm.
The concept of “closeness” between numbers, even just the “ordinary” one, was used to discover something interesting about the number line. If it was merely composed of the rational numbers, then there would be “gaps” in the line. To make a “true” number line, one must fill in these gaps, and this lead to the construction of the real numbers by the mathematician Richard Dedekind in the 19th century.
We elaborate on the nature of these “gaps”, following closely the idea behind Dedekind’s construction. Consider the real number $\sqrt{2}$. It is known from ancient times that this number cannot be written as a ratio of two integers and is therefore not a rational number. However, we can construct an infinite sequence of rational numbers such that every successive rational number in the sequence is “closer” to $\sqrt{2}$, compared to the one before it.
The mathematician Leopolod Kronecker once claimed, “God made the integers, all else is the work of man.” We know how to construct the rational numbers from the integers (for those who would like to think of the natural numbers as being even more basic than the integers, it is also easy to construct the integers from the natural numbers), by taking pairs of integers, and considering sets of equivalence classes (see Modular Arithmetic and Quotient Sets) of these pairs; for example, we set $\frac{1}{2}$ and $\frac{2}{4}$ as equivalent, because “cross multiplication” on the numerators and denominators gives us the same result. So the rational numbers are really equivalence classes of pairs of integers.
The problem we face now is how to construct the real numbers from the rational numbers. We have seen that we can construct sequences which “converge” in some sense to some value that is not a rational number. By “converge”, we mean that successive terms become closer and closer to each other late in the sequence. Technically, we do not refer to such a sequence as a convergent sequence, since it is a sequence of rational numbers but it does not converge to a rational number. Instead, we refer to it as a Cauchy sequence.
And this gives us a possible solution to our problem above – we could simply define the real numbers as the set of all Cauchy sequences. Those that converge to a rational number “represent” that rational number, and those that do not “represent” an irrational number such as $\sqrt{2}$. However, there is still one more problem that we have to take care of. There may be more than one Cauchy sequence that “represents” a certain rational or irrational number.
Consider, for instance, the sequence
$\displaystyle 5,5,5,5,5,...$
which obviously converges to the rational number $5$, and consider another sequence
$\displaystyle 6,5,5,5,5,...$
which is different in the first term but similarly converges to the rational number $5$. They are different sequences, but they “represent” the same rational number. We would like to have a method of “identifying” these two sequences under some equivalence relation. In order to do this, we consider the “difference” of these two sequences:
$\displaystyle 1,0,0,0,0,...$
We see that it converges to $0$. Such a sequence is called a nullsequence, and this gives us our equivalence relation – two Cauchy sequences are to be considered equivalent if they differ by a nullsequence. The set of real numbers $\mathbb{R}$ is then defined as the set of equivalence classes of Cauchy sequences under this equivalence relation.
The process of “filling in” the “gaps” between the rational numbers is called completion. Note that a notion of “closeness” is important in the process of completion. If we had a different notion of closeness, for example, by using the $p$-adic absolute value instead of the ordinary absolute value, we would obtain a different kind of completion. Instead of the real numbers $\mathbb{R}$, we would have instead the $p$-adic numbers $\mathbb{Q}_{p}$. The $p$-adic numbers play an important role in number theory, as they encode information related to primes.
References:
Valuation on Wikipedia
Complete Metric Space on Wikipedia
Algebraic Number Theory by Jurgen Neukirch
Algebraic Number Theory by J. W. S. Cassels and A. Frohlich
# Divisors and the Picard Group
In this post, once again focusing on the subject of algebraic geometry, we will consider a “curve”, which, confusingly, refers what we usually think of as a surface. The reason for this is that if we are considering complex numbers $x$ and $y$, an equation such as $y^{2}=x^{3}-x$, which we would normally think of as a “curve” if $x$ and $y$ were real numbers, actually refers to something that looks like a surface, in the same way the real numbers form a line and complex numbers form a plane. We will rely on this intuition and leave the more formal definitions of curves, surfaces, and dimension to the references for now.
A divisor is a finite “linear combination” of points on the curve, with integer coefficients. For example, if we have points $P_{1}$ and $P_{2}$ on the curve, we can have something like
$\displaystyle 5P_{1}-3P_{2}$.
The degree of a divisor is the sum of its coefficients. For the example above, the degree is equal to $2$.
A special kind of divisor called a principal divisor comes from so-called “rational functions” (which, despite the name, may not really be “functions” in the set-theoretic sense but merely expressions involving a “fraction” whose numerator and denominator are both polynomials) on the curve. We let the coefficients of each point denote the “order of vanishing” of the function. For instance, the function
$\displaystyle \frac{x(x-1)^{2}}{(x-3)^5}$
gives rise to the principal divisor
$(f)=P_{1}+2P_{2}-5P_{3}$
where $P_{1}$ is the point $x=0$$P_{2}$ is the point $x=1$, and $P_{3}$ is the point $x=3$.
The Picard group of a curve is a group (whose law of composition is given by addition – see also Groups) obtained from the divisors by considering two divisors $D$ and $D'$ equivalent (see Modular Arithmetic and Quotient Sets) if their difference $D-D'$ is a principal divisor. An element of the Picard group is also called a divisor class.
The Picard group of a curve can say a lot of things about the curve. For instance, it can be used to prove that on the curve $y^{2}=x^{3}-x$, which is an example of what is called an elliptic curve, the points form a group. The group structure on the elliptic curve, along with other properties such as its being a Riemann surface (a surface which “locally” looks like the complex plane), makes it one of the most interesting objects in mathematics.
The Picard group is also important because its elements, the divisor classes on the curve, correspond to line bundles (vector bundles of dimension $1$ – see Vector Fields, Vector Bundles, and Fiber Bundles – but do keep in mind our discussion earlier regarding complex numbers and how this changes our conventions regarding dimension, as in the case of the line and the plane, and curves and surfaces) on the curve. Line bundles are also related to sheaves, in particular those called “locally free sheaves of rank $1$” (more general vector bundles correspond to locally free sheaves of finite rank). There is, therefore, a relation between the concept of divisors, the concept of vector bundles, and the concept of sheaves.
We now relate the theory of divisors and the Picard group to number theory. We have mentioned in Localization that we can obtain a scheme out of the integers $\mathbb{Z}$; the points of this scheme are the prime ideals of $\mathbb{Z}$, and the set of all these points (prime ideals) we call $\text{Spec }\mathbb{Z}$. As we can make a scheme out of a more general ring, we can therefore make a scheme out of the ring of integers $\mathcal{O}_{K}$ of an algebraic number field $K$ (see Algebraic Numbers); its points will be the prime ideals of $\mathcal{O}_{K}$, and the “rational functions” on this scheme will be the elements of $K$.
In this case, the divisors are made up of “linear combinations” of prime ideals. The principal divisors, which come from rational functions, then correspond, accordingly, to principal fractional ideals, ideals which are generated by a single element of $K$, which as we have mentioned above correspond to the rational functions. Finally, the Picard group is none other than the ideal class group, which “measures” the failure of unique factorization in an algebraic number field!
More explicitly, an example of a divisor may be written in this way:
$\displaystyle 5\mathfrak{p}_{1}-3\mathfrak{p}_{2}$
for prime ideals $\mathfrak{p}_{1}$ and $\mathfrak{p}_{2}$, which as we have said correspond to points. For a principal divisor, we may have, for example, the following element of the rational numbers $\mathbb{Q}$
$\displaystyle \frac{63}{64}$
which generates the principal fractional ideal
$\displaystyle (\frac{63}{64})=\{...,-\frac{189}{64},-\frac{126}{64},-\frac{63}{64},0,\frac{63}{64},\frac{126}{64},\frac{189}{64},...\}$
which in turn gives us the principal divisor
$\displaystyle 2\mathfrak{p}_{1}+\mathfrak{p}_{2}-6\mathfrak{p}_{3}$
where $\mathfrak{p}_{1}=(3)$$\mathfrak{p}_{2}=(7)$, and $\mathfrak{p}_{3}=(2)$, the principal ideals generated by $3$, $7$, and $2$ respectively. Note that if we “factorize” the numerator and denominator of $\frac{63}{64}$, we obtain
$\displaystyle \frac{63}{64}=\frac{3^{2}\cdot 7}{2^{6}}$.
More generally, we should “factorize” in terms of ideals, in case we don’t have unique factorization:
$\displaystyle (\frac{63}{64})=\frac{(3)^{2}(7)}{(2)^{6}}$.
The coefficients of a principal divisor, measuring “how much” of a certain prime is in the factorization of the principal fractional ideal it corresponds to, are called valuations. The theory of valuations offers us another way to develop the entire field of algebraic number theory under a new perspective.
References:
Divisor on Wikipedia
Picard Group on Wikipedia
Algebraic Geometry by Robin Hartshorne
Algebraic Number Theory by Jurgen Neukirch
# Direct Images and Inverse Images of Sheaves
In this post we will be working with the etale topology once again, so we start by formalizing some concepts. We want to first define the category (see Category Theory) called $\text{Et}(X)$ or $\text{Et}/X$ in the literature. The objects of this category are etale morphisms (see Cohomology in Algebraic Geometry) $\varphi:U\rightarrow X$ of schemes to $X$, while the morphisms are etale morphisms $\psi:U\rightarrow U'$ such that if $\varphi ':U'\rightarrow X$ is another etale morphism to $X$, then $\varphi' \circ \psi=\varphi$.
Presheaves of sets, abelian groups, etc. on the category $\text{Et}(X)$ are defined as contravariant functors from the category $\text{Et}(X)$ to sets, abelian groups, etc. They are sheaves if they satisfy the sheaf conditions commonly referred to as local identity and gluing (see Even More Category Theory: The Elementary Topos). We will refer to presheaves (resp. sheaves) on $\text{Et}(X)$ as etale presheaves (resp. etale sheaves) or simply as presheaves (resp. sheaves) on $X$.
Let $f:X\rightarrow Y$ be a morphism of schemes, and let $\mathcal{F}$ be a sheaf on $X$. There is a sheaf on $Y$ determined by the $f$ and $\mathcal{F}$, called the direct image sheaf, written $f_{*}\mathcal{F}$ and defined by
$\displaystyle f_{*}\mathcal{F}(U)=\mathcal{F}(X\times_{Y}U)$
for an etale morphism $U\rightarrow Y$.
The direct image functor $f_{*}$ is the functor that assigns to a sheaf $\mathcal{F}$ the direct image sheaf $f_{*}\mathcal{F}$. The derived functor (see More on Chain Complexes and The Hom and Tensor Functors) of $f_{*}$ is called the higher direct image functor and written $R^{n}f_{*}$.
On the other hand, a morphism $f:X\rightarrow Y$ of schemes and a sheaf $\mathcal{G}$ on $Y$ also determine a sheaf on $X$ called the inverse image sheaf, written $f^{*}\mathcal{F}$ and obtained via the following construction:
Let $U\rightarrow X$ be an etale morphism of schemes. The presheaf $\mathcal{G}'$ is given by
$\mathcal{G}'(U)=\varinjlim \mathcal{G}(V)$
where the direct limit (see Etale Cohomology of Fields and Galois Cohomology) is taken over all $V\rightarrow Y$ such that the morphisms commute (i.e. the composition of morphisms $U\rightarrow V$ and $V\rightarrow Y$ is equal to the composition of morphisms $U\rightarrow X$ and $X\rightarrow Y$).
We then define the sheaf $f^{*}\mathcal{G}$ as the sheaf associated to the presheaf $\mathcal{G}'$ (the process of associating a sheaf to a presheaf, also known as sheafification, is left to the references for now).
We now introduce the notions of open subschemes and closed subschemes, and open immersions and closed immersions. We quote directly from the book Algebraic Geometry by Robin Hartshorne:
An open subscheme of a scheme $X$ is a scheme $U$ whose topological space is an open subset of $X$, and whose structure sheaf $\mathcal{O}_{U}$ is isomorphic to the restriction $\mathcal{O}_{X|U}$ of the structure sheaf of $X$. An open immersion is a morphism $f:X\rightarrow Y$ which induces an isomorphism of $X$ with an open subscheme of $Y$.
Note that every open subset of a scheme carries a unique structure of open subscheme.
A closed immersion is a morphism $f: Y\rightarrow X$ of schemes such that induces a homeomorphism of $\text{sp}(Y)$ onto a closed subset of $\text{sp}(X)$ and furthermore the induced map $f^{\#}:\mathcal{O}_{X}\rightarrow f_{*}\mathcal{O}_{Y}$ of sheaves on $X$ is surjective. A closed subscheme of a scheme $X$ is an equivalence class of closed immersions, where we say $f:Y\rightarrow X$ and $f':Y'\rightarrow X$ are equivalent if there is an isomorphism $i: Y'\rightarrow Y$ such that $f'=f\circ i$.
Now it may happen that given an open immersion $j:U\rightarrow X$ and a sheaf $\mathcal{F}$ on $U$, the stalks of $j_{*}\mathcal{F}$ may not be zero for points outside $U$. Therefore we define another sheaf $j_{!}$ on $X$, given by the following construction:
Given a sheaf $\mathcal{F}$ on $U$, and an etale morphism $\varphi:V\rightarrow X$, let
$\displaystyle \mathcal{F}_{!}(V)=\mathcal{F}(V)$ if $\varphi(V)\subseteq U$
$\displaystyle \mathcal{F}_{!}(V)=0$ if $\varphi(V)\nsubseteq U$
Once again, $\mathcal{F}_{!}$ is a presheaf on $X$, but it need not be a sheaf, therefore we define instead $j_{!}\mathcal{F}$ to be the sheaf associated to the presheaf $\mathcal{F}_{!}$.
One concept related to this “extension by zero” functor $j_{!}$ is cohomology with compact support, written $H_{c}^{n}(U,\mathcal{F})=H^{r}(X,j_{!}\mathcal{F})$.
The functors $f_{*}$, $f^{*}$, and the generalization of $j_{!}$, called the direct image with compact support and denoted $f_{!}$, are part of the so-called “six operations” which play an important role in modern algebraic geometry.
References:
Image Functors for Sheaves on Wikipedia
Direct Image Functor on Wikipedia
Inverse Image Functor on Wikipedia
Direct Image with Compact Support on Wikipedia
Six Operations on Wikipedia
Six Operations on the nLab
Lectures on Etale Cohomology by J.S. Milne
Algebraic Geometry by Robin Hartshorne
Etale Cohomology and the Weil Conjecture by Eberhard Freitag and Reinhardt Kiehl | 23,948 | 91,795 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 745, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.90625 | 4 | CC-MAIN-2023-14 | latest | en | 0.933966 |
https://brainly.ph/question/33735 | 1,484,808,268,000,000,000 | text/html | crawl-data/CC-MAIN-2017-04/segments/1484560280485.79/warc/CC-MAIN-20170116095120-00048-ip-10-171-10-70.ec2.internal.warc.gz | 788,543,691 | 9,761 | # A company charges P211.25 for 5 trees and 15 shrubs. The company charges P15.25 more for a tree than a shrub. How much does each shrub cost?
1
by MarvindelaReyes483
## Answers
2014-07-18T10:44:03+08:00
X=trees ; y=shrubs
211.25 = 5x + 15y
211.25 = 5(y+15.25) + 15y
211.25 = 5y + 76.25 +15y
211.25 = 20y + 76.25
135=20y
y=6.75
Each shrub costs P6.75.
While each tree (represented by x) costs 6.75+15.25 or P22.00
Check:
5 trees and 15 shrubs
= 5(22) + 15(6.75)
= 110 + 101.25
= 211.25 | 222 | 488 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.125 | 4 | CC-MAIN-2017-04 | latest | en | 0.837669 |
https://www.qsstudy.com/physics/how-to-measure-the-unit-of-electric-current | 1,568,733,241,000,000,000 | text/html | crawl-data/CC-MAIN-2019-39/segments/1568514573080.8/warc/CC-MAIN-20190917141045-20190917163045-00066.warc.gz | 1,005,177,121 | 15,379 | # How to Measure the Unit of Electric Current?
Unit: The unit of electric current is ampere. If an amount of charge 1 C flows in 1 second through any cross-section of a conductor, then the quantity of electric current produced is called 1 A.
I = 1 C / 1s = 1 Cs1
= 1 A
In an isolated charged conductor, the charge stays on its surface and do not move. Such type of charges is called an electrostatic charge. However, if we can provide a conducting path, the charges will flow instead of being bound on the conductor. When it happens, we say an electric current is produced.
How electric current is produced from moving charges is described in terms of the circuit as shown in the figure. At the start of the experiment, two plug keys Ig and are taken out and the two metal plates A and B are uncharged by touching with hand. Now, if the plug Ig is closed, the high voltage source will be connected to the two metal plates.
Next, switch on the high voltage source to charge up the two metal plates positively and negatively by an equal amount. Now, key Ig is removed and key Is is plugged in to provide a continuous conducting path linking the positively and negatively charged metal plates to the galvanometer. Here, the galvanometer is a device that can detect the existence of the flow of current. It would be observed that the pointer in the galvanometer is seen to deflect momentarily to one side and then quickly return to its initial position.
The galvanometer’s deflection shows that an electric current is produced. How is this electric current produced? The current is caused by the flow of electrons from the negatively charged plate B through the galvanometer and then to positively charged plate A. The positive charges of plate A are neutralized by the incoming negatively charged electrons. As a result, the transient current which is detected by the galvanometer is produced due to the discharge of the two metal plates. | 409 | 1,942 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.75 | 4 | CC-MAIN-2019-39 | longest | en | 0.952102 |
https://math.stackexchange.com/questions/4615591/understanding-detab | 1,713,677,416,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296817729.0/warc/CC-MAIN-20240421040323-20240421070323-00530.warc.gz | 357,793,999 | 34,011 | # Understanding $\,\det(A+B)$
From this paper on Determinant of sums, where $$$$\det(A+B) = \sum_{r} \sum_{\alpha,\beta} (-1)^{s(\alpha) + s(\beta)} \det(A[\alpha|\beta])\det(B[\alpha|\beta]),$$$$ the meaning of $$A[\alpha|\beta]$$, and how the sum runs over $$\alpha, \beta$$ is not clear to me. Would appreciate an explanation of this this result.
• Page $3$ (or page $131$) of your paper already explained what these symbols mean after display your posted equality (equality $(1)$ in the paper). Jan 10, 2023 at 14:08
• It talks of $\alpha, \beta$ being sequences! Could you illustrate it with an example, when say $n=4$? Jan 10, 2023 at 14:11
• $\alpha$ and $\beta$ are increasing sequences of length $r$: $\alpha=(\alpha_1,\alpha_2,...,\alpha_r)$ and $\beta=(\beta_1,...,\beta_r)$, with $a_i$ and $\beta_i$ strictly increasing and drawn from the set $\{1,2,...,n\}$. Mark the rows $\alpha_1,...,\alpha_r$ and the columns $\beta_1,...,\beta_r$ of $A$. Those rows and columns intersect at $r^2$ elements. The matrix formed by those elements is $A[\alpha|\beta]$. The matrix $B[\alpha|\beta]$ is similar, but using the rows that are not in $\alpha$ and the columns that are not in $\beta$.
– plop
Jan 10, 2023 at 14:13
• For $n=4$, $r=2$, the sum over $\alpha, \beta$ is the sum over ordered pairs of $(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)$, $30$ terms in total. Jan 10, 2023 at 14:14
• You mean, for $n=4$, we have $r=0,1,2,3,4$, and it is for $r=2$ we sum over the ordered pairs you mentioned? What about $r=0$ and $r=1$? Jan 10, 2023 at 14:21 | 524 | 1,547 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 3, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0} | 4.09375 | 4 | CC-MAIN-2024-18 | latest | en | 0.787775 |
https://docsbay.net/semester-exam-practice-materials | 1,713,230,496,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296817036.4/warc/CC-MAIN-20240416000407-20240416030407-00660.warc.gz | 188,481,009 | 10,311 | # Semester Exam Practice Materials
ALGEBRA II Honors/Algebra II
SEMESTER EXAM PRACTICE MATERIALS
SEMESTER 2
2014–2015
______
1. (6.1) What is an equation for the translation of that has asymptotes at and ?
(A)
(B)
(C)
(D)
1. (6.1) What is the equation of the vertical asymptote of ?
(A)
(B)
(C)
(D)
1. (6.1) Sketch the asymptotes and graph of . Identify the domain and range.
2. (6.3) A board of length cm was cut into two pieces. If one piece is cm, express the length of the other board as a rational expression.
(A)
(B)
(C)
(D)
1. (6.3, 6.4)Use the following expressions to answer the questions.
(a)Explain how to add two rational expressions. Simplify as an example. Be sure to give reasons for each step. Simplify completely.
(b)Explain how subtracting (such as simplifying) would be different from adding.
(c)Explain how to multiply rational expressions, simplifying as an example.
(d)Show how to divide by .
(e)Explain how you would solve a rational equation like . Then describe, in detail, the strategy for solving an equation like . Do not completely solve the equations. Instead, concentrate on the first two or three steps in solving; show what to do and explain why.
1. (6.6) The rate of heat loss from a metal object is proportional to the ratio of its surface area to its volume.
(a)What is the ratio of a steel sphere’s surface area to volume?
(b)Compare the rate of heat loss for two steel spheres of radius 2 meters and 3 meters, respectively.
(A)
(B)
(C)
(D)
1. (6.2) Which expression represents the quotient?
(A)
(B)
(C)
(D)
1. (6.6) Last week, Wendy jogged for a total of 10 miles and biked for a total of 10 miles. She biked at a rate that was twice as fast as her jogging rate.
(a)Suppose Wendy jogs at a rate of miles per hour. Write an expression that represents the amount of time she jogged last week and an expression that represents the amount to time she biked last week. (hint:)
(b)Write and simplify an expression for the total amount of time Wendy jogged and biked last week.
(c)Wendy jogged at a rate of 5 miles per hour. What was the total amount of time Wendy jogged and biked last week?
1. (6.2) Which expression is equivalent to for all ?
(A)
(B)
(C)
(D)
1. (6.2) Which set contains all the real numbers that are not part of the domain of
?
(A){8}
(B){-4}
(C){-4, 8}
(D){-8, 4}
1. (6.4) Solve the equation .
(A)-8
(B)7
(C)8
(D)No solution
1. (6.6) A sight-seeing boat travels at an average speed of 20 miles per hour in the calm water of a large lake. The same boat is also used for sight-seeing in a nearby river. In the river, the boat travels 2.9 miles downstream (with the current) in the same amount of time it takes to travel 1.8 miles upstream (against the current). Find the current of the river.
2. (6.6) A baseball player’s batting average is found by dividing the number of hits the player has by the number of at-bats the player has. Suppose a baseball player has 45 hits and 130 at-bats. Write and solve an equation to model the number of consecutive hits the player needs in order to raise his batting average to 0.400. Explain now you found your answer.
1. (6.1) Which intervals correctly define the domain of
(A)
(B)
(C)
(D)
1. (6.1) Which statement is true for the function ?
(A)4 is not in the range of the function.
(B)4 is not in the domain of the function.
(C)-4 is not in the range of the function.
(D)-4 is not in the domain of the function.
1. (7.1) What is the value of?
(A)-42
(B)-17
(C)88
(D)363
1. (7.2) Given the sequence 1, 2, 4, 8, ….
Find the sum of the infinite series.
(A)15
(B)18
(C)30
(D)
1. (7.2) During a flu outbreak, a hospital recorded 12 cases the first week, 54 cases the second week, and 243 cases the third week.
a)Write a geometric sequence to model the flu outbreak.
b)How many cases will occur in the sixth week if the hospital cannot stop the outbreak?
1. (7.2) Given the geometric sequence with common ratio , write a rule for the nth term of the sequence 4, -28, 196, -1372…
(A)
(B)
(C)
(D)
1. (7.1, 7.2) In a classic math problem a king wants to reward a knight who has rescued him from an attack. The king gives the knight a chessboard and plans to place money on each square. He gives the knight two options. Potion 1 is to place a thousand dollars on the first square, two thousand on the second square, three thousand on the third square, and so on. Option 2 is to place one penny on the first square, two pennies on the second, four on the third, and so on.
a) List the first five terms in the sequences formed by the given options. Identify each sequence as arithmetic, geometric, or neither.
Option 1
Option 2
b)For each option, write a rule that tells how much money is placed on the nth square of the chessboard and a rule that tells the total amount of money placed on squares one through .
Option 1
Option 2
c)Find the amount of money placed on the 20th square of the chessboard and the total amount placed on squares 1 through 20 for each option.
Option 1
Option 2
d)There are 64 squares on a chessboard. Find the total amount of money placed on the chessboard for each option.
Option 1
Option 2
e)Which gives the better reward, Option 1 or Option 2? Explain why.
1. (7.4) If, then which of the following is ?
(A)
(B)7
(C)
(D)
1. (7.4) Which is the inverse of ?
(A)
(B)
(C)
(D)
1. (7.5) Find the value of.
(A)5
(B)1024
(C)16
(D)4
1. (7.5) Consider the function.
a)Identify the transformation applied to to create.
b)Identify the transformation applied to to create.
c)Compare the graphs of and. What do you notice?
d)Use the properties of logarithms to explain your answer to part c.
1. (7.6) Which is the same function as?
(A)
(B)
(C)
(D)
1. (7.6) Rewrite in exponential form.
(A)
(B)
(C)
(D)
1. (7.6) Psychologists try to predict the activation of memory when a person is tested on a list of words they learned. The following model is used to make this prediction: where A is the number of words learned, n is the number of exercises, T is the amount of time between learning and testing and L is the length of the list that was tested.
a)Write the formula as the ln of a single expression.
b)Discuss the influence on A (going up or down) when increasing n, T, and L, according to the formula. Do these results make sense?
c)If you want A to be bigger than 0, what conditions must be placed on L, T, and n?
1. (7.8) If, then what is?
(A)81
(B)48
(C)27
(D)9
1. (7.8) Which equation has the same solution as?
(A)
(B)
(C)
(D)
1. (7.8) A biologist studying the relationship between the brain weight and body weight in mammals uses the formula:
Where =body weight in grams and=brain weight in grams. What is the formula for the body weight?
(A)
(B)
(C)
(D)
1. (7.9) Choose the function that describes the graph below:
(A)
(B)
(C)
(D)
1. (7.9) What function is represented by the following graph?
(A)
(B)
(C)
(D)
1. (7.9) The graph of the equation is translated right 3 units and down 3.5 units to form a new graph. Which equation bestrepresents the new graph?
(A)
(B)
(C)
(D)
1. (7.9) John graphs the equation. Lana graphs the equation. How does Lana’s graph compare to John’s graph?
(A)Lana’s graph shifts 2 units downward
(B)Lana’s graph shifts 2 units upward
(C)Lana’s graph shifts 2 units to the left
(D)Lana’s graph shifts 2 units to the left
1. (7.10) In 1950, the city of San Jose had a population of 95,000. Since then, on average, it grows 4% per year. What is the best formula to model San Jose’s growth?
(A)95,000(1.04)t
(B)95,000(0.96)t
(C)-.04t + 95,000
(D).04t + 95,000
1. (7.10) Sarai bought \$400 of Las Vegas Cellular stock in January 2005. The value of the stock is expected to increase by 6.5% per year.
a)Write a model to describe Sarai’s investment.
b)Use the graph to show when Sarai’s investment will reach \$1100?
1. (7.10) The loudness of sound is measured on a logarithmic scale according to the formula, where is the loudness of sound in decibels (), is the intensity of sound, and is the intensity of the softest audible sound.
a)Find the loudness in decibels of each sound listed in the table.
b)The sound at a rock concert is found to have a loudness of 110 decibels. Where should this sound be placed in the table in order to keep the sound intensities in order from least to greatest?
c)A decibel is of a bel. Is a jet plane louder than a sound that measures 20bels? Explain.
1. (7.10) Aaron invested \$4000 in an account that paid an interest rate compounded continuously. After 10 years he has \$5809.81. The compound interest formula is, where is the principal (the initial investment), is the total amount of money (principal plus interest), is the annual interest rate, and is the time in years.
a)Divide both sides of the formula by and then use logarithms to rewrite the formula without an exponent. Show your work.
b)Using your answer for part (a) as a starting point, solve the compound interest formula for the interest rate .
c)Use your equation from part (a) to determine the interest rate.
1. (7.10) Denise is reviewing the change in the value of an investment.
Which statement can Denise use to model the data? Why is this type of function a good model for the data?
(A); an exponential function is a good model because the value of the investment changes by a constant amount in each time period.
(B); an exponential function is a good model because the value of the investment changes by a constant factor in each time period.
(C); a linear function is a good model because the value of the investment changes by a constant amount in each time period.
(D); a linear function is a good model because the value of the investment changes by a constant factor in each time period.
1. (7.10) Amy recorded the total number of ladybugs observed in a garden over a 7-day period. The scatterplot below represents the data she collected.
Which type of function do these data points best fit?
(A)Cubic
(B)Exponential
(C)Linear
1. (7.10) Public Service Utilities uses the equation to determine the cost of electricity where represents the time in hours and represents the cost. The first hour of use costs \$6.66 and three hours cost \$18.11.
a)Determine the value of and in the model.
b)What is the of the graph of the model? What is the real world meaning of the ?
c)Use the model to find the cost for 65 hours of electricity use.
d)If a customer can afford \$40 per month for electricity, how long can he or she have the electricity turned on?
1. (7.10) On an the earth is located at (2, -1) and an asteroid is traveling on the path of .
a)Write an equation representing the distance from the earth to the asteroid.
b)If the asteroid is currently located at , what is the distance from the earth to the asteroid?
c)Sketch a graph of.
d)Find the point when the asteroid is closest to the earth.
1. (7.10) Rashid is in Biology class and has gathered data on fruit flies. The table below shows the number of fruit flies in his sample at the end of each day for a week.
If the population continues to grow in this manner, which function will Rashid use to predict the population of fruit flies on any given day?
(A)
(B)
(C)
(D)
1. (7.10) Which function best fits the data shown in this scatter plot?
(A)
(B)
(C)
(D)
1. (7.10) The graph below shows the change in temperature of a burning house over time.
a)Describe the graph.
b)This graph was found in an old math
book and next to it was written:
Rise of temperature = t0.25
Show that this function does not
describe the graph correctly.
c)Assume that the power function
is a good description of the
graph. Find a reasonable value for .
Graph the new function.
d)Compare the graph in part (c) to the original one.
Do you think that a different power of might result in a better model? Would a larger or smaller power produce a better fit? Explain.
e)Use the original graph to find data. Carry out a power regression on the data to find a function that would produce a better fit.
Use for questions 47 and 48.
1. (8.1) Which of the following is equal to?
(A)
(B)
(C)
(D)
1. (8.1) Which expression represents the length of ?
(A)
(B)
(C)
1. (8.1) If , what is ?
(A)
(B)
(C)
(D)
1. (8.1) Find when and is in Quadrant I.
(A)
(B)
(C)
(D)
1. (8.3) An analog watch had been running fast and needed to be set back. In resetting the watch, the minute hand on the watch subtended an arc of radians.
Part A: Suppose the radius of the watch is 1 unit. What is the length of the arc on the outside of the watch that the angle subtends?
Part B: If the watch was at 10:55 before being reset, what is the new time on the watch?
(A)Part A: units
Part B: 9:05
(B)Part A: units
Part B: 10:05
(C)Part A: units
Part B: 9:20
(D)Part A: units
Part B: 8:40
1. (8.2) Convert radians to degrees.
2. (8.2) Convert radians to degrees.
3. (8.3) Suppose each paddle on the wall of a clothes dryer makes 80 revolutions per minute.
Part B: Write an algebraic expression to determine the measure in radians of the subtended angle after x seconds. Show how the units simplify in your expression.
Part C: You are interested in determining the total distance a point on the drum travels in a 20-minute drying cycle. Can you use your expression from Part B? What other information, if any, is needed? Explain.
1. (8.3) What is the exact value of ?
(A)
(B)
(C)
(D)
1. (8.3) Which expression has the same value as ?
(A)
(B)
(C)
(D)
1. (8.3) What is the reference angle corresponding to ?
(A)
(B)
(C)
(D)
1. (8.3) A regular hexagon is inscribed in the unit circle. One vertex of the hexagon is at the point . A diameter of the circle starts from that vertex and ends on another vertex of the hexagon. What are the coordinates of the other vertex?
(A)
(B)
(C)
(D)
1. (8.3) For what angles x in does the have the same value as ?
(A)and
(B)and
(C)and
(D)and
1. (8.3) For which radian measures x will tan x be negative?
(A)
(B)
(C)
(D)
1. (8.3) The diameter of a bicycle tire is 20 in. A point on the outer edge of the tire is marked with a white dot. The tire is positioned so that the white dot is on the ground, then the bike is rolled so that the dot rotates clockwise through an angle of radians.
Part A: To the nearest tenth of an inch, how high off the ground is the dot when the wheel stops? Show your work.
Part B: What distance was the bicycle pushed? Round your answer to the nearest foot.
Part C: Would changing the size of the tire (value of r) change either of the answers found in Parts A or B? Explain your reasoning.
1. (8.3) A ribbon is tied around a bicycle tire at the standard position . The diameter of the wheel is 26 inches. The bike is then pushed forward 20 feet from the starting point. In what quadrant is the ribbon? Explain how you obtained your answer.
2. (8.3) Find when and is in Quadrant IV.
(A)
(B)
(C)1
(D)
1. (8.3) Two friends counted 24 evenly spaced seats on a Ferris wheel. As they boarded one of the seats, they noticed the edge of the wheel was 1 meter off the ground. They learned from the operator that the diameter of the wheel was 28 meters. After they got seated and started moving, in a counter-clockwise direction, they counted 13 chairs pass the operator, and then the Ferris wheel was stopped on the fourteenth chair to load another passenger.
Part A: Design a representation of the Ferris wheel and locate where the friends were when the wheel stopped to load the next passenger.
Part B: How many radians had they rotated through in the time before they stopped?
Part C: To the nearest tenth of a meter, how far above the ground were they? Show your work.
1. (8.4) If and , then what is the value of ?
(A)
(B)
(C)
(D)
1. (8.5) In , , , , and . Which expression can be used to find the length of side t?
(A)
(B)
(C)
(D)
1. (8.5) Solve , given that , and .
2. (8.5) Given with , and , find c. Round your answer to two decimal places.
3. (8.5) Solve with , , and .
4. (8.5) A 50 foot ramp makes an angle of with the horizontal. To meet new accessibility guidelines, a new ramp must be built so it makes an angle of with the horizontal. What will be the length of the new ramp?
1. (8.6) Which equation would you use to find ?
(A)
(B)
1. (8.6) Which expression can be used to find ?
(A)
(B)
(C)
(D)
1. (8.7) Give an expression for the height hof , and use the expression to write a formula for the height of the triangle in terms of the variables shown by replacing h in the formula .
(A)
(B)
(C)
(D)
1. (8.7) is an isosceles right triangle.
Part A: Determine the exact value of t. Use radical notation if necessary, and do not approximate. Show your work.
Part B:Use to determine the exact value of . Use radical notation if necessary, and do not approximate. Show your work.
Refer to to answer Parts C and D.
Part C:Use your answer to Part B to determine the exact value for the area of .
Part D:Using a calculator, determine the area of to the nearest tenth of a cm2.
1. (8.7) Which expression represents the area of the triangle in square feet?
(A)
(B)
(C)
(D)
1. (8.9) Which is the equation of the graph shown below?
(A)
(B)
(C)
(D)
1. (8.9) The graph of which function has a period of and an amplitude of ?
(A)
(B)
(C)
(D)
1. (8.9) Which function has an amplitude of 2 and a period of ?
(A)
(B)
(C)
(D)
1. (8.9) Which function has an amplitude of and a period of ?
(A)
(B)
(C)
(D)
1. (8.9) Which of the following is a vertical asymptote of the graph of ?
(A)
(B)
(C)
(D)
(E)
1. (8.9) What is the equation for the graph shown?
(A)
(B)
(C)
(D)
(E)
1. (8.9) Which function has an amplitude of 3 and a period of ?
(A)
(B)
(C)
(D)
1. (8.9) Which function is represented by the graph shown?
(A)
(B)
(C)
(D)
1. (8.9) Write an equation of the form, where and , with amplitude and period 12.
1. (8.9) Write a function for the sinusoid. | 4,786 | 17,998 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.65625 | 5 | CC-MAIN-2024-18 | latest | en | 0.94278 |
https://gmatclub.com/forum/neither-a-rising-standard-of-living-nor-balanced-trade-by-46653.html?kudos=1 | 1,496,060,308,000,000,000 | text/html | crawl-data/CC-MAIN-2017-22/segments/1495463612283.85/warc/CC-MAIN-20170529111205-20170529131205-00006.warc.gz | 940,904,879 | 44,634 | Check GMAT Club Decision Tracker for the Latest School Decision Releases https://gmatclub.com/AppTrack
It is currently 29 May 2017, 05:18
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
# Events & Promotions
###### Events & Promotions in June
Open Detailed Calendar
# Neither a rising standard of living nor balanced trade, by
new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
Senior Manager
Joined: 04 Mar 2007
Posts: 435
Followers: 2
Kudos [?]: 44 [0], given: 0
Neither a rising standard of living nor balanced trade, by [#permalink]
### Show Tags
05 Jun 2007, 10:08
00:00
Difficulty:
(N/A)
Question Stats:
0% (00:00) correct 0% (00:00) wrong based on 0 sessions
### HideShow timer Statistics
Neither a rising standard of living nor balanced trade, by itself, establishes a country’s ability to compete in the international marketplace. Both are required simultaneously since standards of living can rise because of growing trade deficits and trade can be balanced by means of a decline in a country’s standard of living.
If the facts stated in the passage above are true, a proper test of a country’s ability to be competitive is its ability to
(A) balance its trade while its standard of living rises
(B) balance its trade while its standard of living falls
(C) increase trade deficits while its standard of living rises
(D) decrease trade deficits while its standard of living falls
(E) keep its standard of living constant while trade deficits rise
Director
Joined: 17 Jul 2006
Posts: 707
Followers: 1
Kudos [?]: 13 [0], given: 0
### Show Tags
05 Jun 2007, 11:12
It's A.
Senior Manager
Joined: 04 Jun 2007
Posts: 367
Followers: 1
Kudos [?]: 75 [0], given: 0
### Show Tags
05 Jun 2007, 13:50
I just did this question on the OG last night, and got A.
Don't remember what the OA is though.
05 Jun 2007, 13:50
Similar topics Replies Last post
Similar
Topics:
1 Neither rising industrial output nor a high national minimum 9 17 Aug 2010, 14:03
1 Reducing speed limits neither saves lives nor protects the 2 01 Jul 2009, 06:47
Reducing speed limits neither saves lives nor protects the 0 15 Aug 2012, 23:18
Neither a rising standard of living nor balanced trade, by 10 13 Jun 2008, 02:21
Reducing speed limits neither saves lives nor protects the 4 18 Oct 2007, 20:21
Display posts from previous: Sort by
# Neither a rising standard of living nor balanced trade, by
new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Powered by phpBB © phpBB Group and phpBB SEO Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®. | 796 | 3,178 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.734375 | 4 | CC-MAIN-2017-22 | latest | en | 0.898804 |
https://ch.mathworks.com/matlabcentral/answers/57970-how-can-i-compute-the-length-of-an-integer | 1,575,957,733,000,000,000 | text/html | crawl-data/CC-MAIN-2019-51/segments/1575540525821.56/warc/CC-MAIN-20191210041836-20191210065836-00006.warc.gz | 300,200,793 | 20,365 | ## how can i compute the length of an integer?
on 4 Jan 2013
if i have
int = 12345;
length_int = 5;
???
Jan
### Jan (view profile)
on 4 Jan 2013
What is the "length" of -1 and 0?
### Azzi Abdelmalek (view profile)
on 4 Jan 2013
Edited by Azzi Abdelmalek
### Azzi Abdelmalek (view profile)
on 4 Jan 2013
int=-123456789
max(ceil(log10(abs(int))),1)
### Sean de Wolski (view profile)
on 4 Jan 2013
Edited by Sean de Wolski
### Sean de Wolski (view profile)
on 4 Jan 2013
Edit
nnz(num2str(int) - '-')
Show 1 older comment
Sean de Wolski
### Sean de Wolski (view profile)
on 4 Jan 2013
Friedrich, neither work actually, consider -12345.
Friedrich
### Friedrich (view profile)
on 4 Jan 2013
Good point Sean. But then
numel(num2str(abs(int)))
should do ;)
Sean de Wolski
on 4 Jan 2013
arghh, you win!
### Davide Ferraro (view profile)
on 4 Jan 2013
Casting the variable into a string may be risky because you may get to "unexpected" cases such as:
int = 12345678901234567890123
numel(num2str(int))
ans =
12
You may consider a numeric approach using LOG10: floor(log10(int))+1 all numbers between 10 and 100 will have a LOG10 between 1 and 2 so you can use FLOOR to get the lower value (1 in this case) and then you need to add the value 1 cause you are trying to compute the number of digits and not the power of ten.
G A
### G A (view profile)
on 4 Jan 2013
why, it works:
>> int = 12345678901234567890123
numel(num2str(int))
int =
1.2346e+22
ans =
23 | 479 | 1,472 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.53125 | 4 | CC-MAIN-2019-51 | latest | en | 0.626544 |
http://antoine.frostburg.edu/chem/senese/101/measurement/faq/print-why-273.15-kelvin.shtml | 1,537,361,477,000,000,000 | text/html | crawl-data/CC-MAIN-2018-39/segments/1537267156224.9/warc/CC-MAIN-20180919122227-20180919142227-00032.warc.gz | 16,764,673 | 3,931 | # Why is the difference between Celsius and Kelvin temperatures 273 units?
John 10/13/00
Vocabulary absolute temperature absolute zero Celsius heat kelvin phase second law of thermodynamics SI temperature thermodynamic temperature triple point universal gas law constant work
According to NIST's Historical Perspective on SI, in 1954 the SI unit for temperature was defined as exactly 1/273.16 of the thermodynamic temperature of the triple point of water. This unit was officially named the "kelvin" (with symbol "K") in 1967.
What's a thermodynamic temperature? A thermodynamic temperature scale is consistent with all laws of thermodynamics. As the "true" temperature scale, the thermodynamic temperature scale is independent of the properties of any particular substance. This requirement is important for high-accuracy temperature measurements. For example, the temperatures read from a mercury thermometer and an alcohol thermometer don't quite agree with each other. The trouble is that the rate at which liquid volume changes with temperature isn't a constant- at higher temperatures, the liquids expand more per degree temperature rise than they do at lower temperatures. The degree marks would have to be placed farther apart on the barrel of the thermometer for higher temperatures than for lower ones. The degree spacing would also depend on the liquid used to fill the thermometer.
How are thermodynamic temperature scales defined? One solution to the problem of defining a substance-independent temperature scale came from theoretical studies of heat engines. The maximum efficiency for converting heat into work using a heat engine is equal to 1 - T/Thot, where the heat engine draws heat from a source at temperature Thot and dumps heat to a cold sink at temperature T.
Lord Kelvin defined the first thermodynamic temperature scale as follows:
T = (1 - e) Thot
where e is the maximum efficiency of the heat engine. The engine can't be more than 100% efficient (e=1), so the lowest temperature possible should be zero on this scale. The committee responsible for maintaining the SI used Kelvin's definition to develop an international temperature scale. When one of the temperatures is fixed at 273.16 K, the other is the thermodynamic temperature in kelvins.
Why choose the "triple point of water" as a reference point? At a temperature of exactly 273.16 K, and a pressure of about 610.5 Pa, the solid, liquid, and gaseous forms of water are at equilibrium. This set of conditions is called a "triple point" because at this point three phases can coexist indefinitely. Even a tiny change in temperature and/or pressure from these values will cause one of the phases to disappear, so it's easy to tell when you have a temperature of exactly 273.16 K and when you don't.
Why do you add 273.15 to convert Celsius to Kelvin, then? Why not 273.16? Because the Celsius scale uses the "ice point" (the point at which liquid water and ice can coexist indefinitely) as its reference point, not the triple point. The ice point is 0.01°C below the triple point of water. By definition, Celsius degrees and kelvins are exactly the same size, so absolute zero is -273.15°C.
Basic Unit Definitions: The Kelvin (NIST)
Milestones leading to the 1990 International Temperature Scale. Links identify the international committees responsible for maintaining the International System of Units (SI).http://physics.nist.gov/cuu/Units/kelvin.html (10/13/00)
Calibration of Liquid-in-glass Thermometers (NIST)
Jaquelyn Wise of NIST's Thermometry Group explains the procedure for recalibrating liquid-in-glass thermometers. Adobe Acrobat plugin required.http://www.cstl.nist.gov/div836/836.05/Thermometry%20site/Papers/Lig_recal.pdf (10/13/00)
Lord Kelvin
A brief bio of Lord Kelvin (William Thomson), with a portrait.http://www.phy.bg.ac.yu/web_projects/giants/kelvin.html (10/13/00)
The Origin of the Celsius Temperature Scale (Johan and Ann Santesson)
A summary of Anders Celsius's 1742 paper "Observations on two persistent degrees on a thermometer", which describes the rationale and origins of the Celsius temperature scale.http://www.santesson.com/engtemp.html (11/14/98, 10/13/00)
Author: Fred Senese [email protected]
General Chemistry Online! Why is the difference between Celsius and Kelvin temperatures 273 units? | 946 | 4,347 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.609375 | 4 | CC-MAIN-2018-39 | longest | en | 0.906352 |
https://www.cuemath.com/questions/i-am-the-smallest-number-having-four-different-prime-factors-can-you-find-me/ | 1,620,597,718,000,000,000 | text/html | crawl-data/CC-MAIN-2021-21/segments/1620243989018.90/warc/CC-MAIN-20210509213453-20210510003453-00477.warc.gz | 750,329,203 | 155,711 | # I am the Smallest Number Having Four Different Prime Factors. Can You Find me?
## Question: I am the smallest number having four different prime factors. Can you find me?
Prime factors of a number are the prime numbers that are multiplied to get the original number
## Answer: 210 is the smallest number having four different prime factors
The smallest number having four different prime factors will be the product of the first four prime numbers
## Explanation:
The first four prime numbers are 2, 3, 5, and 7
The required number will be the product of these numbers
2 × 3 × 5 × 7 = 210
So, 210 is the smallest number having four different prime factors | 154 | 666 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.21875 | 4 | CC-MAIN-2021-21 | longest | en | 0.889241 |
https://chouprojects.com/intercept-excel/ | 1,718,434,679,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861584.65/warc/CC-MAIN-20240615062230-20240615092230-00138.warc.gz | 151,818,298 | 21,822 | Published on
Written by Jacky Chou
# Intercept: Excel Formulae Explained
## Key Takeaway:
• The INTERCEPT function in Excel is used for linear regression analysis to determine the point where the linear equation crosses the Y-axis.
• Using the INTERCEPT function requires meeting certain requirements such as having a set of known X and Y values and ensuring that the data is linear.
• The INTERCEPT function can be used in various applications such as trendline analysis, forecasting, and other statistical analysis.
Are you looking to become an Excel pro? Then you’ve come to the right place! In this article you’ll find simple tips and tricks to help you understand the most important formulae in Excel. Stop worrying, start learning and take control of your data today!
## Understanding the INTERCEPT Function in Excel
In Excel, the INTERCEPT function calculates the y-value at which a straight line intersects the y-axis by finding the value of the intercept of two variables. This function is commonly used in linear regression analysis for predicting values based on a given set of data. By using this formula, you can calculate the point at which a line crosses the y-axis even if you don’t have any data points that intersect it directly.
When working with the INTERCEPT function, it is important to note that the values involved must be continuous and linear. That is, the values must be plotted on a straight line. The formula also assumes that the relationship between the two variables is statistically significant.
It’s worth mentioning that the INTERCEPT function is just one of the many formulae available in Excel for analyzing and interpreting data. By becoming familiar with these formulae, you can make more informed decisions based on the data you have.
Don’t fall behind your competitors by ignoring the importance of the INTERCEPT function in your Excel data analysis. Master this formula and take your data interpretation skills to the next level.
## How to Use the INTERCEPT Function
The Efficient Use of INTERCEPT Function in Excel
To make the most of your Excel experience, a fundamental function to master is the INTERCEPT function. It is a statistical function that finds the intercept of the simple regression line. Here is a concise guide to aid you in using it.
1. Select a cell to display the result of the function.
2. Type ‘`=INTERCEPT(`’ with reference to the range of X values followed by a comma.
3. Enter the range of Y values where the reference to the range of X values ends.
4. Press enter to calculate and see the value of the intercept.
5. Alternatively, click on the function button and select ‘Statistical Functions’ and scroll to find the ‘INTERCEPT’ function.
6. Follow Steps 2 to 4 to get the intercept.
It is vital to note that an error pops up when the ranges of X and Y values contain different lengths, blanks, or non-numeric data.
Further, to minimize errors, we suggest double-checking the parameters of the function and using parentheses to ensure calculations are executed correctly. Using the function in conjunction with Microsoft Excel’s other formulas like LINEST offers immense flexibility.
By understanding how to use the INTERCEPT function, calculating intercepts in your Excel sheet becomes hassle-free, leading to quicker analyses and better decision-making.
## Examples of Using the INTERCEPT Function
When it comes to the INTERCEPT function in Excel, there are several ways to use it effectively. Here are some tips for making the most of this useful tool:
1. Start with two sets of data: one for the x-axis and one for the y-axis. These could be any set of related data points, such as sales figures over time or population growth rates for different countries.
2. Make sure your data is organized into columns or rows, depending on your preference. You’ll need to know which column or row corresponds to each axis when using the INTERCEPT function.
3. Type `=INTERCEPT(` into the formula bar and select the range of cells containing your x-axis data.
4. Enter a comma and then select the range of cells containing your y-axis data.
5. Close the parentheses and press Enter. The result should be the intercept value for the two data sets.
6. Repeat this process for any other sets of data you want to analyze.
One important thing to keep in mind is that the INTERCEPT function is just one tool for analyzing data in Excel. To get a more complete picture of your data, you may also want to use other functions like SLOPE or TREND.
Pro Tip: When working with the INTERCEPT function, it’s always a good idea to double-check your data sets to make sure they’re accurate and complete. Small errors or omissions in your data can lead to big mistakes in your analysis.
## Five Facts About “INTERCEPT: Excel Formulae Explained”:
• ✅ “INTERCEPT” is an Excel formula that calculates the point where a linear regression line intersects the y-axis. (Source: Excel Easy)
• ✅ The formula is used in data analysis to find the relationship between two variables. (Source: Vertex42)
• ✅ “INTERCEPT” calculates the y-value of a point on the regression line based on the x-value. (Source: Spreadsheet Planet)
• ✅ “INTERCEPT” is commonly used in finance and economics for trend analysis. (Source: Investopedia)
• ✅ Understanding “INTERCEPT” is essential for anyone working with data in Excel. (Source: Skillshare)
## FAQs about Intercept: Excel Formulae Explained
### What is INTERCEPT: Excel Formulae Explained?
INTERCEPT is an Excel formula that calculates the intercept point at which a line crosses the y-axis. It is used in statistical analysis and forecasting.
### How do I use the INTERCEPT function in Excel?
To use the INTERCEPT function, select the cell where you want the result to appear and then type in “=INTERCEPT(arrayX, arrayY)”. Replace “arrayX” and “arrayY” with the arrays that contain your x and y data points.
### What is the difference between INTERCEPT and SLOPE?
SLOPE is another Excel formula that calculates the slope of a line. The difference between SLOPE and INTERCEPT is that SLOPE calculates the slope of the line that best fits the data points, while INTERCEPT calculates the point where that line crosses the y-axis.
### Can INTERCEPT be used for non-linear data?
No, INTERCEPT can only be used for linear data.
### What does the output of the INTERCEPT formula represent?
The output of the INTERCEPT formula represents the y-intercept of the line that best fits the data points.
### What is the purpose of using INTERCEPT in financial analysis?
INTERCEPT is commonly used in financial analysis to estimate the fair value of a company’s stock based on its historical prices.
## Related Articles
### Max: Excel Formulae Explained
Key Takeaway: The MAX function in Excel is used to ...
### Lower: Excel Formulae Explained
Key Takeaway: The LOWER formula in Excel allows users to ...
### Match: Excel Formulae Explained
Key Takeaway: The MATCH function in Excel is used to ... | 1,464 | 6,999 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.03125 | 4 | CC-MAIN-2024-26 | latest | en | 0.885468 |
http://math.stackexchange.com/questions/220317/an-interesting-problem-of-tossing-a-fair-die-successively | 1,466,964,368,000,000,000 | text/html | crawl-data/CC-MAIN-2016-26/segments/1466783395546.12/warc/CC-MAIN-20160624154955-00167-ip-10-164-35-72.ec2.internal.warc.gz | 191,644,057 | 18,808 | # An interesting problem of tossing a fair die successively
A fair die is tossed successively. Let $X$ denote the number of tosses until each of the six possible outcomes occurs at least once. Find the probability mass function of $X$. I'm also given this $hint$: For $1\leq i \le6$ let $E_i$ be the event that the outcome $i$ does not occur during the first $n$ tosses of the die. First calculate $P(X>n)$ by writing the event $X>n$ in terms of $E_1, E_2,...E_6$.
I know that $P(X>n)=1-P(X<n)$ and from $P(X<n)$ we can find the probability mass function. But I dont know how to find $P(X<n)$.
I looked and the answer is $$(\frac56)^{n-1}-5(\frac46)^{n-1}+10(\frac36)^{n-1}-10(\frac26)^{n-1}+5(\frac16)^{n-1}\quad for \quad n\ge6$$ I tried to derive how this was found but I found the alternating signs to be tricky and I'm also confused with why the coefficients are what they are.
-
Note that $X=\sum\limits_{k=1}^6T_k$ where $T_k$ is the number of tosses between the appearances of the $k-1$th new result and the $k$th new result. Thus, $T_1=1$, $T_2$ is geometric with parameter $\frac56$, and so on until $T_6$ which is geometric with parameter $\frac16$.
The generating function of a random variable $T$ geometric with parameter $p$ is $\mathrm E(s^T)=ps/(1-(1-p)s)$ hence $$\mathrm E(s^X)=\prod_{k=1}^6\frac{\frac{k}6s}{1-(1-\frac{k}6)s}=\frac{5!}{6^5}s^6\cdot\prod_{k=1}^5\frac1{1-\frac{k}6s}.$$ The decomposition of the last product in simple fractions is $$\prod_{k=1}^5\frac1{1-\frac{k}6s}=\sum_{k=1}^5\frac{c_k}{1-\frac{k}6s},\qquad c_k=\prod_{1\leqslant i\leqslant 5}^{i\ne k}\frac1{1-\frac{i}k},$$ hence $$\mathrm E(s^X)=\frac{5!}{6^5}s^6\cdot\sum_{k=1}^5c_k\sum_{n\geqslant0}\left(\frac{k}6s\right)^n.$$ The coefficients of each power of $s$ must coincide hence, for every $n\geqslant0$, $$\mathbb P(X=n+6)=\frac{5!}{6^5}\cdot\sum_{k=1}^5c_k\left(\frac{k}6\right)^n.$$
-
Let $Y_j$ be the number of tosses after there have been $j-1$ distinct outcomes until there have been $j$ distinct outcomes. Thus $Y_1 = 1$ (i.e. the first toss always is one of the six outcomes), $X = Y_1 + Y_2 + \ldots + Y_6$, and $Y_1, \ldots, Y_6$ are independent. $Y_j$ for $j = 2$ to $6$ is a (shifted) geometric random variable, so its probability generating function is easy to find, ...
-
Another way: Let $A_{i,n}$ be the event that the number $i$ is not rolled after $n$ rolls. Then the probability you're looking for is $1-P(A_{1,n} \cup ... \cup A_{6,n})$
You can compute the second term using inclusion-exclusion.
Edit: In case this isn't clear, this gives you $P(X \leq n)$
Then do $P(X=n)= P(X \leq n) - P(X \leq n-1)$
- | 934 | 2,636 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.09375 | 4 | CC-MAIN-2016-26 | longest | en | 0.74186 |
https://math.stackexchange.com/questions/176589/taylor-expansion-on-interval-or-at-infinity | 1,721,130,422,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514745.49/warc/CC-MAIN-20240716111515-20240716141515-00746.warc.gz | 331,263,451 | 35,707 | # Taylor expansion on interval or at infinity
I'm trying to figure out how to get this result via taylor expansion as $x,y \rightarrow \infty$:
$f(x) = \sqrt{(x-1)y} = \sqrt{xy} - \frac{1}{2}\sqrt{\frac{y}{x}} + ...$
I've been told (on yahoo answers), that since you can't take taylor expansions at infinity, you do it on intervals (what does that mean?). Specifically, I was told:
You can write
f(x) = f(a) + (x-a) f'(a) + error term
or
f(a) = f(x) -(x-a)f'(x) + error term
Here your 2 ends are X and X - 1
I'm not sure what that means =P.
I am looking for a step-by-step taylor series calculation up to the first 2 terms. I've never approximated a function "at infinity" nor am I familiar with the generalized formula for taylor expansion.
So, any help would be great.
Thanks!
• To take a Taylor expansion of $f$ "at infinity", write the Taylor expansion of $f(\frac{1}{x})$ at $0$. Commented Jul 29, 2012 at 20:27
• Yes. It's actually a Laurent series. Commented Jul 29, 2012 at 20:33
• ... or it would be if $f(1/x)$ was analytic at $0$. In this case it's a Puiseux series. Commented Jul 29, 2012 at 20:35
In your case, you can just ignore the common factor $\sqrt{y}$ and concentrate on $\sqrt{x-1}$. To take the Taylor expansion at infinity is to replace $x$ by $1/u$ and take the Taylor expansion of at $u=0$ instead. Naïvely: $$\sqrt{x-1}=\sqrt{\frac1u-1}$$ which is of course singular at $u=0$, so you take out a troublesome factor: $$\sqrt{x-1}=\frac{\sqrt{1-u}}{\sqrt{u}}.$$ Now you expand the numerator as $\sqrt{1-u}=1-\frac12u+\cdots$, replace $u$ by $1/x$, and put the pieces together. The approximation will be good when $\lvert u\rvert$ is small, i.e., when $\lvert x\rvert$ is large. | 535 | 1,715 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.828125 | 4 | CC-MAIN-2024-30 | latest | en | 0.897136 |
http://blog.learningresources.com/math-madness/ | 1,556,161,250,000,000,000 | text/html | crawl-data/CC-MAIN-2019-18/segments/1555578678807.71/warc/CC-MAIN-20190425014322-20190425040322-00157.warc.gz | 22,768,260 | 32,183 | March Madness is here and it’s time for math! What? March Madness is about college men’s basketball, right? What does that have to do with math?
If you look at it as just basketball, it doesn’t have much to do with math. But if you look closer, you’ll find many fun opportunities to help children learn math. Here are some ideas.
## Counting
• Discuss how many people on one team can play at one time. (5) How many teams are playing at once? (2) How many people in all are on the court when playing the game? (10) Practice counting by fives and tens.
• How many points can you score with one basket? (2 or 3) Practice counting by twos and threes up to 90/100.
## More Than/Less Than
• Team A has 8 points. Team B has 5 points. Which team has the greater number of points? The fewer number of points?
• Player A is 6 feet tall. Player B is 7 feet tall. Who is taller? Shorter?
• Team A has 3 fouls. Team B has 1 foul. Who has more fouls? Less?
• Team A has 10 points. Team B has 6 points. How many points does Team B have to get to catch up to team A?
• When a team has the ball, they have 30 seconds to make a basket. If 10 seconds has gone by, how much time do they have left?
• There are 16 teams in the beginning of the tournament. If there are 4 teams left, how many teams are out?
## Converting Inches and Feet
• Basketball nets are 10 feet above the ground. If there are 12 inches in a foot, how many inches high is the net? (120 inches)
• Player A is 72 inches tall. You know that there are 12 inches in a foot. How many feet tall is Player A? (6 feet)
## Other fun ideas
• If you know how tall a basketball player is, try and figure out how much taller he is than you. How many of you could fit into one of him?
• Knowing that the dimensions of a college basketball court are 94 feet long and 50 feet wide, what is the perimeter (all sides added together)? (288 feet) The area (length x width)? (4700 feet)
• When the stats come up on the screen, you can add up the assists, points, and rebounds.
Have your kids do the math on paper or have them use our Primary Calculator!
What are some other ideas you can come up with to integrate math and basketball? Let us know at [email protected]! | 565 | 2,220 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.15625 | 4 | CC-MAIN-2019-18 | longest | en | 0.970151 |
https://www.o.vg/unit/mass-flow-rate/gram-second-to-kilogram-second.php | 1,721,415,103,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514917.3/warc/CC-MAIN-20240719170235-20240719200235-00462.warc.gz | 817,899,365 | 9,975 | Convert Gram/Second to Kilogram/Second | kg/s to g/s
# document.write(document.title);
## g/s to kg/s Converter
From g/s to kg/s: 1 g/s = 0.001 kg/s;
From kg/s to g/s: 1 kg/s = 1000 g/s;
## How to Convert Gram/Second to Kilogram/Second?
As we know One g/s is equal to 0.001 kg/s (1 g/s = 0.001 kg/s).
To convert Gram/Second to Kilogram/Second, multiply your g/s figure by 0.001.
Example : convert 25 g/s to kg/s:
25 g/s = 25 × 0.001 kg/s = kg/s
To convert Kilogram/Second to Gram/Second, divide your kg/s figure by 0.001.
Example : convert 25 kg/s to g/s:
25 kg/s = 25 ÷ 0.001 g/s = g/s
## How to Convert Kilogram/Second to Gram/Second?
As we know One kg/s is equal to 1000 g/s (1 kg/s = 1000 g/s).
To convert Kilogram/Second to Gram/Second, multiply your kg/s figure by 1000.
Example : convert 45 kg/s to g/s:
45 kg/s = 45 × 1000 g/s = g/s
To convert Gram/Second to Kilogram/Second, divide your g/s figure by 1000.
Example : convert 45 g/s to kg/s:
45 g/s = 45 ÷ 1000 kg/s = kg/s
## Convert Gram/Second or Kilogram/Second to Other Mass Flow Rate Units
Gram/Second Conversion Table
g/s to kg/s 1 g/s = 0.001 kg/s g/s to g/min 1 g/s = 60 g/min g/s to g/h 1 g/s = 3600 g/h g/s to g/d 1 g/s = 86400 g/d g/s to mg/min 1 g/s = 60000 mg/min g/s to mg/h 1 g/s = 3600000 mg/h g/s to mg/d 1 g/s = 86400000 mg/d g/s to kg/min 1 g/s = 0.06 kg/min g/s to kg/h 1 g/s = 3.6 kg/h g/s to kg/d 1 g/s = 86.4 kg/d g/s to Eg/s 1 g/s = 1.E-18 Eg/s g/s to Pg/s 1 g/s = 1.E-15 Pg/s g/s to Tg/s 1 g/s = 1.E-12 Tg/s g/s to Gg/s 1 g/s = 1.E-9 Gg/s g/s to Mg/s 1 g/s = 0.000001 Mg/s g/s to hg/s 1 g/s = 0.01 hg/s
Gram/Second Conversion Table
g/s to dag/s 1 g/s = 0.1 dag/s g/s to dg/s 1 g/s = 10 dg/s g/s to cg/s 1 g/s = 100 cg/s g/s to mg/s 1 g/s = 1000 mg/s g/s to µg/s 1 g/s = 1000000 µg/s g/s to t/s 1 g/s = 0.000001 t/s g/s to t/min 1 g/s = 0.00006 t/min g/s to t/h 1 g/s = 0.0036 t/h g/s to t/d 1 g/s = 0.0864 t/d g/s to ton (US)/h 1 g/s = 0.0039683207 ton (US)/h g/s to lb/s 1 g/s = 0.0022046226 lb/s g/s to lb/min 1 g/s = 0.1322773573 lb/min g/s to lb/h 1 g/s = 7.9366414387 lb/h g/s to lb/d 1 g/s = 190.47939453 lb/d Created @ o.vg Free Unit Converters
## FAQ
### What is 9 Gram/Second in Kilogram/Second?
kg/s. Since one g/s equals 0.001 kg/s, 9 g/s in kg/s will be kg/s.
### How many Kilogram/Second are in a Gram/Second?
There are 0.001 kg/s in one g/s. In turn, one kg/s is equal to 1000 g/s.
### How many g/s is equal to 1 kg/s?
1 kg/s is approximately equal to 1000 g/s.
### What is the g/s value of 8 kg/s?
The Gram/Second value of 8 kg/s is g/s. (i.e.,) 8 x 1000 = g/s.
### g/s to kg/s converter in batch
Cite this Converter, Content or Page as:
"" at https://www.o.vg/unit/mass-flow-rate/gram-second-to-kilogram-second.php from www.o.vg Inc,07/20/2024. https://www.o.vg - Instant, Quick, Free Online Unit Converters | 1,116 | 2,844 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.8125 | 4 | CC-MAIN-2024-30 | latest | en | 0.725239 |
https://electronics.stackexchange.com/questions/524079/simple-question-about-the-difference-between-thevenin-resistance-and-non-theveni | 1,716,996,069,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971059246.67/warc/CC-MAIN-20240529134803-20240529164803-00152.warc.gz | 198,352,361 | 42,485 | # Simple question about the difference between Thevenin resistance and non-Thevenin resistance
In a lecture for an Electrical Engineering course I was taking, the professor did an example on solving a Thevenin circuit.
When he solves the circuit for Thevenin resistance he adds two resistors in parallel between the open circuit "a" port, however when solving to make an equivalent resistor to find the voltage he goes and adds two resistors in series between the same "a" open circuit. I am confused as too how this is allowed.
I would think that for both cases either it could only be added in parallel or series however this seems to not be the case. The only thing that changes is that now there is a voltage source and I am confused on how that has an effect on the calculation of resistance.
Picture for reference of solution:
Edit: Added full solution of the problem using Thevenin:
• yes, I did as I feel that it wasn't relevant to the question but I will edit it into the post if that is helpful to the question. Oct 1, 2020 at 2:44
• the circuit does not change in the second solution ... the resistors can be oriented any way you like ... note: between the open circuit "a" ... you cannot have anything between only one point ... when something is between, it is always between two or more points Oct 1, 2020 at 2:59
When reading what follows, keep in mind that you are always allowed to select one of the wires (nodes) and designate it as $$\0\:\text{V}\$$ (or ground.) That is your prerogative. In my work below, I've chosen to make b the same as ground.
## 1
Just redraw the first circuit:
simulate this circuit – Schematic created using CircuitLab
Now it is completely obvious how it is that $$\R_1\$$ and $$\R_2\$$ can be taken in-parallel and Thevenized that way into $$\\frac{R_1\: R_2}{R_1+R_2}=2\:\Omega\$$. That value is then in series with $$\R_4\$$, so just add the two values together. And that result is in-parallel with $$\R_3\$$. So it's pretty easy.
One of the things that you need to learn early is to realize that our brains are often easily misled. Just get into the practice of patiently sitting down, getting out paper and pencil, and re-drawing what you see in a completely different way. Do this with intention. Make it radically different. But make sure that all the connections (nodes) are correct, of course.
You'll begin to see in a new way, soon enough.
## 2
Again, re-draw:
simulate this circuit
You can now Thevenize $$\R_1\$$ and $$\R_2\$$ with $$\V_1\$$'s voltage. That resulting Thevenin resistance will be in series with $$\R_4\$$, so just add them. Now you have your $$\V_1\$$ source at one end of a voltage divider and your Thevenin source, $$\V_x\$$, at the other end. That means the final resulting Thevenin resistance will be those two values, the earlier result and $$\R_3\$$, taken in-parallel again. You can also easily compute the new Thevenin voltage from the divider pair, as well.
## Redrawing Schematic Appendix
One of the better ways to try and understand a circuit that at first appears to be confusing is to redraw it. There are some rules you can follow that will help get a leg-up on learning that process. But there are also some added personal skills that gradually develop over time, too.
I first learned these rules in 1980, taking a Tektronix class that was offered only to its employees. This class was meant to teach electronics drafting to people who were not electronics engineers, but instead would be trained sufficiently to help draft schematics for their manuals.
The nice thing about the rules is that you don't have to be an expert to follow them. And that if you follow them, even blindly almost, that the resulting schematics really are easier to figure out.
The rules are:
• Arrange the schematic so that conventional current appears to flow from the top towards the bottom of the schematic sheet. I like to imagine this as a kind of curtain (if you prefer a more static concept) or waterfall (if you prefer a more dynamic concept) of charges moving from the top edge down to the bottom edge. This is a kind of flow of energy that doesn't do any useful work by itself, but provides the environment for useful work to get done.
• Arrange the schematic so that signals of interest flow from the left side of the schematic to the right side. Inputs will then generally be on the left, outputs generally will be on the right.
• Do not "bus" power around. In short, if a lead of a component goes to ground or some other voltage rail, do not use a wire to connect it to other component leads that also go to the same rail/ground. Instead, simply show a node name like "Vcc" and stop. Busing power around on a schematic is almost guaranteed to make the schematic less understandable, not more. (There are times when professionals need to communicate something unique about a voltage rail bus to other professionals. So there are exceptions at times to this rule. But when trying to understand a confusing schematic, the situation isn't that one and such an argument "by professionals, to professionals" still fails here. So just don't do it.) This one takes a moment to grasp fully. There is a strong tendency to want to show all of the wires that are involved in soldering up a circuit. Resist that tendency. The idea here is that wires needed to make a circuit can be distracting. And while they may be needed to make the circuit work, they do NOT help you understand the circuit. In fact, they do the exact opposite. So remove such wires and just show connections to the rails and stop.
• Try to organize the schematic around cohesion. It is almost always possible to "tease apart" a schematic so that there are knots of components that are tightly connected, each to another, separated then by only a few wires going to other knots. If you can find these, emphasize them by isolating the knots and focusing on drawing each one in some meaningful way, first. Don't even think about the whole schematic. Just focus on getting each cohesive section "looking right" by itself. Then add in the spare wiring or few components separating these "natural divisions" in the schematic. This will often tend to almost magically find distinct functions that are easier to understand, which then "communicate" with each other via relatively easier to understand connections between them.
The above rules aren't hard and fast. But if you struggle to follow them, you'll find that it does help a lot.
You can read a snippet of my own education by those schematic draftsmen at Tektronix who trained me by reading here.
• Nice answer. +1 Oct 1, 2020 at 3:10
• @relayman357 Thanks! How are you doing with LambertW?
– jonk
Oct 1, 2020 at 3:11
• Haven’t messed with it. Didn’t know it would be research project. :-) Oct 1, 2020 at 3:16
• @relayman357 Okay. Got it. Wolfram has a nice page on the topic. Worth a read, I think. It's how you solve equations that include exponential or logarithm product components to them. Every well-vetted mathematician is well-familiar with the idea and its application.
– jonk
Oct 1, 2020 at 3:17
• @HirokazuMiyashita It's truly my pleasure. Help others when you may be able to do so and pass it along. We all need each other in different ways and times and I owe a debt I cannot ever repay for those who were there for me. I'm just glad I was helpful at all. Thanks for letting me know.
– jonk
Oct 1, 2020 at 10:42 | 1,741 | 7,434 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 13, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.03125 | 4 | CC-MAIN-2024-22 | latest | en | 0.965576 |
https://studybay.com/math/foundations-of-mathematics/set-theory/ | 1,604,082,813,000,000,000 | text/html | crawl-data/CC-MAIN-2020-45/segments/1603107911229.96/warc/CC-MAIN-20201030182757-20201030212757-00633.warc.gz | 526,937,870 | 36,766 | # Set theory
## Set theory Topics
Sort by:
### Empty set
The set containing no elements, commonly denoted or , the former of which is used in this work. These correspond to Wolfram Language and TeX characters summarized in the table below.symbolTeXWolfram Language\varnothing\[Diameter]\emptyset\[EmptySet]Unfortunately, some authors use the notation 0 instead of for the empty set (Mendelson 1997). The empty set is generally designated using (i.e., the empty list) in the Wolfram Language.A set that is not the empty set is called a nonemptyset. The empty set is sometimes also known as the null set (Mendelson 1997).The complement of the empty set is the universal set.Strangely, the empty set is both open and closed for any set and topology.A groupoid, semigroup, quasigroup, ringoid, and semiring can be empty. Monoids, groups, and rings must have at least one element, while division algebras and fields must have at least two elements...
### Element
If is a member of a set , then is said to be an element of , written . If is not an element of , this is written .The term element also refers to a particular member of a group, or entry in a matrix or unevaluated determinant .
### Disjoint sets
Two sets and are disjoint if their intersection , where is the empty set. sets , , ..., are disjoint if for . For example, and are disjoint, but and are not. Disjoint sets are also said to be mutually exclusive or independent.
### Superset
A set containing all elements of a smaller set. If is a subset of , then is a superset of , written . If is a proper superset of , this is written .
### Directed set
A set together with a relation which is both transitive and reflexive such that for any two elements , there exists another element with and . In this case, the relation is said to "direct" the set.
### Subset
A subset is a portion of a set. is a subset of (written ) iff every member of is a member of . If is a proper subset of (i.e., a subset other than the set itself), this is written . If is not a subset of , this is written . (The notation is generally not used, since automatically means that and cannot be the same.)The subsets (i.e., power set) of a given set can befound using Subsets[list].An efficient algorithm for obtaining the next higher number having the same number of 1 bits as a given number (which corresponds to computing the next subset) is given by Gosper (1972) in PDP-10 assembler.The set of subsets of a set is called the power set of , and a set of elements has subsets (including both the set itself and the empty set). This follows from the fact that the total number of distinct k-subsets on a set of elements is given by the binomial sumFor sets of , 2, ... elements, the numbers of subsets are therefore 2, 4, 8, 16, 32, 64, ... (OEIS A000079). For example, the set..
### Poretsky's law
The theorem in set theory and logic that for all sets and ,(1)where denotes complement set of and is the empty set. The set is depicted in the above Venn diagram and clearly coincides with iff is empty.The corresponding theorem in a Boolean algebra states that for all elements of ,(2)The version of Poretsky's Law for logic can be derived from (2) using the rules of propositional calculus, namely for all propositions and ,(3)where "is equivalent to" means having the same truth table. In fact, in the following table, the values in the second and in the third column coincide if and only if the value in the first column is 0.( and not ) or (not and )000011101110
### Complete product
The complete products of a Boolean algebra of subsets generated by a set of cardinal number are the Boolean functions(1)where each may equal or its complement . For example, the complete products of are(2)Each Boolean function has a unique representation(up to order) as a union of complete products. For example,(3)(4)(5)(Comtet 1974, p. 186).
### Matroid
Roughly speaking, a matroid is a finite set together with a generalization of a concept from linear algebra that satisfies a natural set of properties for that concept. For example, the finite set could be the rows of a matrix, and the generalizing concept could be linear dependence and independence of any subset of rows of the matrix.Formally, a matroid consists of a finite set of elements together with a family of nonempty subsets of , called circuits, which satisfy the axioms 1. No proper subset of a circuit is a circuit, 2. If and , then contains a circuit. (Harary 1994, p. 40).An equivalent definition considers a matroid as a finite set of elements together with a family of subsets of , called independent sets, such that 1. The empty set is independent, 2. Every subset of an independent set is independent,3. For every subset of , all maximum independent sets contained in have the same number of elements. (Harary 1994, pp. 40-41).The number..
### Set class
A class is a generalized set invented to get around Russell's antinomy while retaining the arbitrary criteria for membership which leads to difficulty for sets. The members of classes are sets, but it is possible to have the class of "all sets which are not members of themselves" without producing a paradox (since is a proper class (and not a set), it is not a candidate for membership in ).The distinction between classes and sets is a concept from vonNeumann-Bernays-Gödel set theory.
### Inductive set
A set-theoretic term having a number of different meanings. Fraenkel (1953, p. 37) used the term as a synonym for "finite set." However, according to Russell's definition (Russell 1963, pp. 21-22), an inductive set is a nonempty partially ordered set in which every element has a successor. An example is the set of natural numbers , where 0 is the first element, and the others are produced by adding 1 successively.Roitman (1990, p. 40) considers the same construction in a more abstract form: the elements are sets, 0 is replaced by the empty set , and the successor of every element is the set . In particular, every inductive set contains a sequence of the formFor many other authors (e.g., Bourbaki 1970, pp. 20-21; Pinter 1971, p. 119), an inductive set is a partially ordered set in which every totally ordered subset has an upper bound, i.e., it is a set fulfilling the assumption of Zorn's lemma.The versions..
### Set
A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is generally also ignored (unlike a list or multiset). Members of a set are often referred to as elements and the notation is used to denote that is an element of a set . The study of sets and their properties is the object of set theory.Older words for set include aggregate and set class. Russell also uses the unfortunate term manifold to refer to a set.Historically, a single horizontal overbar was used to denote a set stripped of any structure besides order, and hence to represent the order type of the set. A double overbar indicated stripping the order from the set and hence represented the cardinal number of the set. This practice was begun by set theory founder Georg Cantor.Symbols used to operate on sets include (which means "and" or intersection), and (which means "or" or union). The symbol is used to denote the set containing..
### Proper subset
A proper subset of a set , denoted , is a subset that is strictly contained in and so necessarily excludes at least one member of . The empty set is therefore a proper subset of any nonempty set.For example, consider a set . Then and are proper subsets, while and are not.
### Finite set
A set whose elements can be numbered through from 1 to , for some positive integer . The number is called the cardinal number of the set, and is often denoted or . In other words, is equipollent to the set . We simply say that has elements. The empty set is also considered as a finite set, and its cardinal number is 0.A finite set can also be characterized as a set which is not infinite, i.e., as a set which is not equipollent to any of its proper subsets. In fact, if , and , a certain number of elements of do not belong to , so that .For all , the number of subsets having exactly elements (the so-called k-subsets, or combinations of elements out of ) is equal to the binomial coefficient(1)(2)Hence the number of subsets of (i.e., the cardinal number of its power set) is(3)(4)(5)by virtue of the binomial theorem.Assigning to each k-subset of its complement set defines a one-to-one correspondence between the set of k-subsets and the set of -subsets of . This proves the..
### Rank
The word "rank" refers to several related concepts in mathematics involving graphs, groups, matrices, quadratic forms, sequences, set theory, statistics, and tensors.In graph theory, the graph rank of a graph is defined as , where is the number of vertices on and is the number of connected components (Biggs 1993, p. 25).In set theory, rank is a (class) function from sets to ordinal numbers. The rank of a set is the least ordinal number greater than the rank of any member of the set (Mirimanoff 1917; Moore 1982, pp. 261-262; Rubin 1967, p. 214). The proof that rank is well-defined uses the axiom of foundation.For example, the empty set has rank 0 (since it has no members and 0 is the least ordinal number), has rank 1 (since , its only member, has rank 0), has rank 2, and has rank . Every ordinal number has itself as its rank.Mirimanoff (1917) showed that, assuming the class of urelements is a set, for any ordinal number , the class..
### Nowhere dense
A set is said to be nowhere dense if the interior of the set closure of is the empty set. For example, the Cantor set is nowhere dense.There exist nowhere dense sets of positive measure. For example, enumerating the rationals in as and choosing an open interval of length containing for each , then the union of these intervals has measure at most 1/2. Hence, the set of points in but not in any of has measure at least 1/2, despite being nowhere dense.
### Complement set
Given a set with a subset , the complement (denoted or ) of with respect to is defined as(1)Using set difference notation, the complement isdefined by(2)If , then(3)where is the empty set. The complement is implemented in the Wolfram Language as Complement[l, l1, ...].Given a single set, the second probabilityaxiom gives(4)Using the fact that ,(5)(6)This demonstrates that(7)Given two sets,(8)(9)
### Disjoint union
The disjoint union of two sets and is a binary operator that combines all distinct elements of a pair of given sets, while retaining the original set membership as a distinguishing characteristic of the union set. The disjoint union is denoted(1)where is a Cartesian product. For example, the disjoint union of sets and can be computed by finding(2)(3)so(4)(5)
### Set difference
The set difference is defined byHere, the backslash symbol is defined as Unicode U+2216.The set difference is therefore equivalent to the complement set, and is implemented in the Wolfram Language as Complement[A, B].The symbol is sometimes also used to denote a set difference (Smith et al. 1997, p. 68).
### Concurrent relation
Let and be sets, and let be a relation on . Then is a concurrent relation if and only if for any finite subset of , there exists a single element of such that if , then . Examples of concurrent relations include the following: 1. The relation on either the natural numbers, the integers, the rational numbers, or the real numbers. 2. The relation between elements of an extension of a field , defined by3. The containment relation between open neighborhoods of a given point of a topological space .
### Reflexive reduction
The reflexive reduction of a binary relation on a set is the minimum relation on with the same reflexive closure as . Thus for any elements and of , provided that and are distinct and .
### Transitive reduction
The transitive reduction of a binary relation on a set is the minimum relation on with the same transitive closure as . Thus for any elements and of , provided that and there exists no element of such that and .The transitive reduction of a graph is the smallest graph such that , where is the transitive closure of (Skiena 1990, p. 203).
### Reflexive closure
The reflexive closure of a binary relation on a set is the minimal reflexive relation on that contains . Thus for every element of and for distinct elements and , provided that .
### Partial order ideal
An ideal of a partial order is a subset of the elements of which satisfy the property that if and , then . For disjoint chains in which the th chain contains elements, there are ideals. The number of ideals of a -element fence poset is the Fibonacci number .
### Ground set
A partially ordered set is defined as an ordered pair . Here, is called the ground set of and is the partial order of .
### Well ordered set
A totally ordered set is said to be well ordered (or have a well-founded order) iff every nonempty subset of has a least element (Ciesielski 1997, p. 38; Moore 1982, p. 2; Rubin 1967, p. 159; Suppes 1972, p. 75). Every finite totally ordered set is well ordered. The set of integers , which has no least element, is an example of a set that is not well ordered.An ordinal number is the ordertype of a well ordered set.
### Partial order
A relation "" is a partial order on a set if it has: 1. Reflexivity: for all . 2. Antisymmetry: and implies . 3. Transitivity: and implies .For a partial order, the size of the longest chain (antichain) is called the partial order length (partial order width). A partially ordered set is also called a poset.A largest set of unrelated vertices in a partial order can be found using MaximumAntichain[g] in the Wolfram Language package Combinatorica$. MinimumChainPartition[g] in the Wolfram Language package Combinatorica$ partitions a partial order into a minimum number of chains.
### Fence poset
A partial order defined by (, ), (, ) for odd .
### Duality law
A metatheorem stating that every theorem on partially ordered sets remains true if all inequalities are reversed. In this operation, supremum must be replaced by infimum, maximum with minimum, and conversely. In a lattice, this means that meet and join must be interchanged, and in a Boolean algebra, 1 and 0 must be switched.Each of de Morgan's two laws can be derived from the other by duality.
### Totally ordered set
A total order (or "totally ordered set," or "linearly ordered set") is a set plus a relation on the set (called a total order) that satisfies the conditions for a partial order plus an additional condition known as the comparability condition. A relation is a total order on a set (" totally orders ") if the following properties hold. 1. Reflexivity: for all . 2. Antisymmetry: and implies . 3. Transitivity: and implies . 4. Comparability (trichotomy law): For any , either or . The first three are the axioms of a partial order, while addition of the trichotomy law defines a total order.Every finite totally ordered set is well ordered. Any two totally ordered sets with elements (for a nonnegative integer) are order isomorphic, and therefore have the same order type (which is also an ordinal number)...
### Dominance
The dominance relation on a set of points in Euclidean -space is the intersection of the coordinate-wise orderings. A point dominates a point provided that every coordinate of is at least as large as the corresponding coordinate of .A partition dominates a partition if, for all , the sum of the largest parts of is the sum of the largest parts of . For example, for , dominates all other partitions, while is dominated by all others. In contrast, and do not dominate each other (Skiena 1990, p. 52).The dominance orders in are precisely the partially ordered sets of dimension at most .
### Dilworth's lemma
The partial order width of a set is equal to the minimum number of chains needed to cover . Equivalently, if a set of elements is partially ordered, then contains a chain of size or an antichain of size . Letting be the cardinal number of , the partial order width, and the partial order length, this last statement says . Dilworth's lemma is a generalization of the Erdős-Szekeres theorem. Ramsey's theorem generalizes Dilworth's lemma.
### Tarski's fixed point theorem
Let be any complete lattice. Suppose is monotone increasing (or isotone), i.e., for all , implies . Then the set of all fixed points of is a complete lattice with respect to (Tarski 1955)Consequently, has a greatest fixed point and a least fixed point . Moreover, for all , implies , whereas implies .Consider three examples: 1. Let satisfy , where is the usual order of real numbers. Since the closed interval is a complete lattice, every monotone increasing map has a greatest fixed point and a least fixed point. Note that need not be continuous here. 2. For declare to mean that , , (coordinatewise order). Let satisfy . Then the set(1)(2)is a complete lattice (with respect to the coordinatewise order). Hence every monotone increasing map has a greatest fixed point and a least fixed point. 3. Let and be injections. Then there is a bijection (Schröder-Bernstein theorem), which can be constructed as follows. The power set of ordered by set inclusion,..
### Order isomorphic
Two totally ordered sets and are order isomorphic iff there is a bijection from to such that for all ,(Ciesielski 1997, p. 38). In other words, and are equipollent ("the same size") and there is an order preserving mapping between the two.Dauben (1990) and Suppes (1972) call this property "similar." The definitionworks equally well on partially ordered sets.
### Cover relation
The transitive reflexive reduction of a partial order. An element of a partially ordered set covers another element provided that there exists no third element in the poset for which . In this case, is called an "upper cover" of and a "lower cover" of .
### Strictly between
Let be a partially ordered set, and let . If , then is said to be between and . If is between and and , then is strictly between and .
### Maximal element
Let be a partially ordered set. Then an element is said to be maximal if, for all , . Alternatively, an element is maximal such that if for any , then .Note that the definition for a maximal element above is true for any two elements of a partially ordered set that are comparable. However, it may be the case that two elements of a given partial ordering are not comparable.
### Complete lattice
A partially ordered set (or ordered set or poset for short) is called a complete lattice if every subset of has a least upper bound (supremum, ) and a greatest lower bound (infimum, ) in .Taking shows that every complete lattice has a greatest element (maximum, ) and a least element (minimum, ).Of course, every complete lattice is a lattice. Moreover, every lattice with a finite set is a complete lattice.
### Realizer
A set of linear extensions of a partially ordered set is a realizer of (and is said to realize ) provided that for all , iff is below in every member of .
### Linear extension
A linear extension of a partially ordered set is a permutation of the elements , , ... of such that implies . For example, the linear extensions of the partially ordered set are 1234, 1324, 1342, 3124, 3142, and 3412, all of which have 1 before 2 and 3 before 4.
### Comparable elements
Suppose is a partial ordering on a nonempty set . Then the elements are said to be comparable provided or .Because two elements in a partially ordered set need not be comparable, it is possible for a partially ordered set to have more than one maximal element. For example, suppose we have a nonempty partially ordered set in which every element is incomparable to every other element, i.e., is totally unordered. It follows that every element of is maximal.
### Preorder
A relation "" is called a preorder (or quasiorder) on a set if it satisfies: 1. Reflexivity: for all . 2. Transitivity: and implies . A preorder that also has antisymmetry is a partial order.
### Chain
Let be a finite partially ordered set. A chain in is a set of pairwise comparable elements (i.e., a totally ordered subset). The partial order length of is the maximum cardinal number of a chain in . For a partial order, the size of the longest chain is called the partial order length.
### Poset dimension
The dimension of a partially ordered set is the size of the smallest realizer of . Equivalently, it is the smallest integer such that is isomorphic to a dominance order in .
### Isomorphic posets
Two partially ordered sets are said to be isomorphic if their "structures" are entirely analogous. Formally, partially ordered sets and are isomorphic if there is a bijection from to such that precisely when .
### Partially ordered set
A partially ordered set (or poset) is a set taken together with a partial order on it. Formally, a partially ordered set is defined as an ordered pair , where is called the ground set of and is the partial order of .An element in a partially ordered set is said to be an upper bound for a subset of if for every , we have . Similarly, a lower bound for a subset is an element such that for every , . If there is an upper bound and a lower bound for , then the poset is said to be bounded.
### Interval order
A partially ordered set is an interval order if it is isomorphic to some set of intervals on the real line ordered by left-to-right precedence. Formally, is an interval order provided that one can assign to each an interval such that in the real numbers iff in .
### Ordinal number
In common usage, an ordinal number is an adjective which describes the numerical position of an object, e.g., first, second, third, etc.In formal set theory, an ordinal number (sometimes simply called an "ordinal" for short) is one of the numbers in Georg Cantor's extension of the whole numbers. An ordinal number is defined as the order type of a well ordered set (Dauben 1990, p. 199; Moore 1982, p. 52; Suppes 1972, p. 129). Finite ordinal numbers are commonly denoted using arabic numerals, while transfinite ordinals are denoted using lower case Greek letters.It is easy to see that every finite totally ordered set is well ordered. Any two totally ordered sets with elements (for a nonnegative integer) are order isomorphic, and therefore have the same order type (which is also an ordinal number). The ordinals for finite sets are denoted 0, 1, 2, 3, ..., i.e., the integers one less than the corresponding nonnegative..
### Ordinal comparison
Let and be well ordered sets with ordinal numbers and . Then iff is order isomorphic to an initial segment of (Dauben 1990, p. 199). From this, it can easily be shown that the ordinal numbers are totally ordered by the relation. In fact, they are well ordered by the relation.
### Order type
Every totally ordered set is associated with a so-called order type. Two sets and are said to have the same order type iff they are order isomorphic (Ciesielski 1997, p. 38; Dauben 1990, pp. 184 and 199; Moore 1982, p. 52; Suppes 1972, pp. 127-129). Thus, an order type categorizes totally ordered sets in the same way that a cardinal number categorizes sets. The term is due to Georg Cantor, and the definition works equally well on partially ordered sets.The order type of the negative integers is called (Moore 1982, p. 62), although Suppes (1972, p. 128) calls it . The order type of the rationals is called (Dauben 1990, p. 152; Moore 1982, p. 115; Suppes 1972, p. 128). Some sources call the order type of the reals (Dauben 1990, p. 152), while others call it (Suppes 1972, p. 128).In general, if is any order type, then is the same type ordered backwards (Dauben 1990, p. 153)...
### Limit ordinal
An ordinal number is called a limit ordinal iff it has no immediate predecessor, i.e., if there is no ordinal number such that (Ciesielski 1997, p. 46; Moore 1982, p. 60; Rubin 1967, p. 182; Suppes 1972, p. 196). The first limit ordinal is .
### Model theory
Model theory is a general theory of interpretations of axiomatic set theory. It is the branch of logic studying mathematical structures by considering first-order sentences which are true of those structures and the sets which are definable in those structures by first-order formulas (Marker 1996).Mathematical structures obeying axioms in a system are called "models" of the system. The usual axioms of analysis are second order and are known to have the real numbers as their unique model. Weakening the axioms to include only the first-order ones leads to a new type of model in what is called nonstandard analysis.
### Uniquely complemented lattice
A uniquely complemented lattice is a complemented lattice that satisfiesThe class of uniquely complemented lattices is not a subvariety of the class of complemented lattices. On the other hand, there is a well-known class of uniquely complemented lattices that is a subvariety of the variety of complemented lattices, namely the class of Boolean algebras. They form a variety because they are the distributive complemented lattices, and one can prove that any distributive complemented lattice is uniquely complemented.
### Locally bounded lattice
A lattice is locally bounded if and only if each of its finitely generated sublattices is bounded.Every locally bounded lattice is locally subbounded, and every locally bounded lattice has a bounded hyperfinite extension in any nonstandard enlargement . This latter nonstandard property characterizes locally subbounded lattices.A locally bounded lattice is locally tight if and only if each of its hyperfinitely generated extensions is internally tight. One can also prove the following result, using nonstandard characterizations of these notions: Let be a locally finite lattice with at least one strictly increasing meet endomorphism and at least one strictly decreasing join endomorphism. If is locally tight, then it is bounded.
### Local polarity
Let be a lattice, and let . Then the pair is a local polarity if and only if for each finite set , there is a finitely generated sublattice of that contains and on which the restriction is a lattice polarity.Using nonstandard methods, one may show that the following result holds: Let be a locally finite lattice. Then the set of local polarities of is a relation which is a one-to-one correspondence between its domain and range.
### Independent vertex set
An independent vertex set of a graph is a subset of the vertices such that no two vertices in the subset represent an edge of . The figure above shows independent sets consisting of two subsets for a number of graphs (the wheel graph , utility graph , Petersen graph, and Frucht graph).The polynomial whose coefficients give the number of independent vertex sets of each cardinality in a graph is known as its independence polynomial.A set of vertices is an independent vertex set iff its complement forms a vertex cover (Skiena 1990, p. 218). The counts of vertex covers and independent vertex sets in a graph are therefore the same.The empty set is trivially an independent vertex setsince it contains no vertices, and therefore no edge endpoints.A maximum independent vertex set is an independent vertex set of a graph containing the largest possible number of vertices for the given graph, and the cardinality of this set is called the independence number..
### Semialgebraic set
A semialgebraic set is a subset of which is a finite Boolean combination of sets of the form and , where and are polynomials in , ..., over the reals.By Tarski's theorem, the solution set of a quantified system of real algebraic equations and inequalities is a semialgebraic set (Strzebonski 2000).
### Antichain
Let be a finite partially ordered set, then an antichain in is a set of pairwise incomparable elements. Antichains are also called Sperner systems in older literature (Comtet 1974).For example, consider to be a family of subsets together with the subset relation (i.e., if is a subset of ). The following table gives the antichains on the set of subsets (i.e., the power set) of the -set for small .antichains123The number of antichains on the -set for , 1, 2, ..., are 1, 2, 5, 19, 167, ... (OEIS A014466). If the empty set is not considered a valid antichain, then these reduce to 0, 1, 4, 18, 166, ... (OEIS A007153; Comtet 1974, p. 273). The numbers obtained by adding one to OEIS A014466, 2, 3, 6, 20, 168, 7581, 7828354, ... (OEIS A000372), are also frequently encountered (Speciner 1972).The number of antichains on the -set are equal to the number of monotonic increasing Boolean functions of variables, and also the number of free distributive lattices..
### Cardinal exponentiation
Let and be any sets, and let be the cardinal number of a set . Then cardinal exponentiation is defined by(Ciesielski 1997, p. 68; Dauben 1990, p. 174; Moore 1982, p. 37; Rubin 1967, p. 275, Suppes 1972, p. 116).It is easy to show that the cardinal number of the power set of is , since and there is a natural bijection between the subsets of and the functions from into .
### Infinite set
A set of elements is said to be infinite if the elements of a proper subset can be put into one-to-one correspondence with the elements of . An infinite set whose elements can be put into a one-to-one correspondence with the set of integers is said to be countably infinite; otherwise, it is called uncountably infinite.
### Cardinal comparison
For any sets and , their cardinal numbers satisfy iff there is a one-to-one function from into (Rubin 1967, p. 266; Suppes 1972, pp. 94 and 116). It is easy to show this satisfies the reflexive and transitive axioms of a partial order. However, it is difficult to show the antisymmetry property, whose proof is known as the Schröder-Bernstein theorem. To show the trichotomy property, one must use the axiom of choice.Although an order type can be defined similarly, it does not seem usual to do so.
Let and be any sets with empty intersection, and let denote the cardinal number of a set . Then(Ciesielski 1997, p. 68; Dauben 1990, p. 173; Rubin 1967, p. 274; Suppes 1972, pp. 112-113).It is an interesting exercise to show that cardinal addition is well-defined. The main steps are to show that for any cardinal numbers and , there exist disjoint sets and with cardinal numbers and , and to show that if and are disjoint and and disjoint with and then . The second of these is easy. The first is a little tricky and requires an appeal to the axioms of set theory. Also, one needs to restrict the definition of cardinal to guarantee if is a cardinal, then there is a set satisfying .
### Set partition
A set partition of a set is a collection of disjoint subsets of whose union is . The number of partitions of the set is called a Bell number.
### Axiom of the sum set
The axiom of Zermelo-Fraenkel set theory which asserts the existence for any set of the sum (union) of all sets that are elements of . The axiom may be stated symbolically as
### Surreal number
Surreal numbers are the most natural collection of numbers which includes both the real numbers and the infinite ordinal numbers of Georg Cantor. They were invented by John H. Conway in 1969. Every real number is surrounded by surreals, which are closer to it than any real number. Knuth (1974) describes the surreal numbers in a work of fiction.The surreal numbers are written using the notation , where , is the simplest number greater than 0, is the simplest number greater than 1, etc. Similarly, is the simplest number less than 0, etc. However, 2 can also be represented by , , , etc.Some simple games have abbreviated names that can be expressed in terms of surreal numbers. For example, , , for an integer , , , and . Most surreal numbers can be represented as hackenbush positions.
### Join
In a lattice, any two elements and have a least upper bound. This least upper bound is often called the join of and , and is denoted by .One can also speak of the join operation in a general partially ordered set. If and are two elements in some partially ordered set , and if there is a smallest element (with respect to the given order) with the property that and , then is said to be the join of and .
### Distributive lattice
A lattice which satisfies the identitiesis said to be distributive.
### Complemented lattice
A complemented lattice is an algebraic structure such that is a bounded lattice and for each element , the element is a complement of , meaning that it satisfies 1. 2. . A related notion is that of a lattice with complements. Such a structure is a bounded lattice such that for each , there is such that and .One difference between these notions is that the class of complemented lattices forms a variety, whilst the class of lattices with complements does not. (The class of lattices with complements is a subclass of the variety of lattices, but it is not a subvariety of the class of lattices.) Every lattice with complements is a reduct of a complemented lattice, by the axiom of choice. To see this, let be a lattice with complements. For each , let denote the set of complements of . Because is a lattice with complements, for each , is nonempty, so by the axiom of choice, we may choose from each collection a distinguished complement for . This defines a function which..
### Lattice homomorphism
Let and be lattices, and let . Then is a lattice homomorphism if and only if for any , and . Thus a lattice homomorphism is a specific kind of structure homomorphism. In other words, the mapping is a lattice homomorphism if it is both a join-homomorphism and a meet-homomorphism.If is a one-to-one lattice homomorphism, then it is a lattice embedding, and if a lattice embedding is onto, then it is a lattice isomorphism.An example of an important lattice isomorphism in universal algebra is the isomorphism that is guaranteed by the correspondence theorem, which states that if is an algebra and is a congruence on , then the mapping that is defined by the formulais a lattice isomorphism.
### Meet
In a lattice, any two elements and have a greatest lower bound. This greatest lower bound is often called the meet of and , and is denoted by .One can also speak of the meet operation in a general partially ordered set. If and are two elements in some partially ordered set , and if there is a greatest element (with respect to the given order) with the property that and , then is said to be the meet of and .
### Ordinal multiplication
Let and be totally ordered sets. Let be the Cartesian product and define order as follows. For any and , 1. If , then , 2. If , then and compare the same way as (i.e., lexicographical order) (Ciesielski 1997, p. 48; Rubin 1967; Suppes 1972). However, Dauben (1990, p. 104) and Moore (1982, p. 40) define multiplication in the reverse order.Like addition, multiplication is not commutative, but it is associative,(1)An inductive definition for ordinal multiplication states that for any ordinal number ,(2)(3)If is a limit ordinal, then is the least ordinal greater than any ordinal in the set (Suppes 1972, p. 212).
### Dedekind cut
A set partition of the rational numbers into two nonempty subsets and such that all members of are less than those of and such that has no greatest member. Real numbers can be defined using either Dedekind cuts or Cauchy sequences.
### Zermelo set theory
The version of set theory obtained if Axiom 6 of Zermelo-Fraenkelset theory is replaced by 6'. Selection axiom (or "axiom of subsets"): for any set-theoretic formula , , which can be deduced from Axiom 6. However, there seems to be some disagreement in the literature about just which axioms of Zermelo-Fraenkel set theory constitute "Zermelo Set Theory." Mendelson (1997) does not include the axioms of choice, foundation, replacement In Zermelo set theory, but does includes 6'. However, Enderton (1977) includes the axioms of choice and foundation, but does not include the axioms of replacement or Selection.
### Ordinal exponentiation
Let and be any ordinal numbers, then ordinal exponentiation is defined so that if then . If is not a limit ordinal, then choose such that ,If is a limit ordinal, then if , . If then, is the least ordinal greater than any ordinal in the set (Rubin 1967, p. 204; Suppes 1972, p. 215).Note that this definition is not analogous to the definition for cardinals, since may not equal , even though and . Note also that .A familiar example of ordinal exponentiation is the definition of Cantor's first epsilon number. is the least ordinal such that . It can be shown that it is the least ordinal greater than any ordinal in .
Let and be disjoint totally ordered sets with order types and . Then the ordinal sum is defined at set where, if and are both from the same subset, the order is the same as in the subset, but if is from and is from , then has order type (Ciesielski 1997, p. 48; Dauben 1990, p. 104; Moore 1982, p. 40).One should note that in the infinite case, order typeaddition is not commutative, although it is associative. For example,(1)In addition, , with the least element, is order isomorphic to , but not to , with the greatest element, since it has a greatest element and the other does not.An inductive definition for ordinal addition states that for any ordinal number ,(2)and(3)If is a limit ordinal, then is the least ordinal greater than any ordinal in the set (Rubin 1967, p. 188; Suppes 1972, p. 205)...
### Counting number
A positive integer: 1, 2, 3, 4, ... (OEIS A000027), also called a natural number. However, zero (0) is sometimes also included in the list of counting numbers. Due to lack of standard terminology, the following terms are recommended in preference to "counting number," "natural number," and "whole number."setnamesymbol..., , , 0, 1, 2, ...integersZ1, 2, 3, 4, ...positive integersZ-+0, 1, 2, 3, 4, ...nonnegative integersZ-*0, , , , , ...nonpositive integers, , , , ...negative integersZ--
For a countable set of disjoint events , , ...,
### Natural number
The term "natural number" refers either to a member of the set of positive integers 1, 2, 3, ... (OEIS A000027) or to the set of nonnegative integers 0, 1, 2, 3, ... (OEIS A001477; e.g., Bourbaki 1968, Halmos 1974). Regrettably, there seems to be no general agreement about whether to include 0 in the set of natural numbers. In fact, Ribenboim (1996) states "Let be a set of natural numbers; whenever convenient, it may be assumed that ."The set of natural numbers (whichever definition is adopted) is denoted N.Due to lack of standard terminology, the following terms and notations are recommended in preference to "counting number," "natural number," and "whole number."setnamesymbol..., , , 0, 1, 2, ...integersZ1, 2, 3, 4, ...positive integersZ-+0, 1, 2, 3, 4, ...nonnegative integersZ-*0, , , , , ...nonpositive integers, , , , ...negative integersZ--..
### Urelement
An urelement contains no elements, belongs to some set, and is not identical with the empty set (Moore 1982, p. 3; Rubin 1967, p. 23). "Ur" is a German prefix which is difficult to translate literally, but has a meaning close to "primeval." Urelements are also called "atoms" (Rubin 1967, Moore 1982) or "individuals" (Moore 1982).In "pure" set theory, all elements are sets and there are no urelements. Often, the axioms of set theory are modified to allow the presence of urelements for ease in representing something. In fact, before Paul Cohen developed the method of forcing, some of the independence theorems in set theory were shown if urelements were allowed.
### Internally extendable homomorphism
Let be an infinite set of urelements, and let be an enlargement of the superstructure . Let be finitary algebras with finitely many operations, and let and be their extension monads in . Let be a homomorphism. Then is internally extendable provided that there is an internal subalgebra of which contains and there is a homomorphism such that if , then .For a homomorphism , the following are equivalent: 1. is internally extendable and is a subalgebra of , 2. For some homomorphism , is the restriction to of .
### Successor
For any ordinal number , the successor of is (Ciesielski 1997, p. 46). The successor of an ordinal number is therefore the next ordinal, .
### Initial ordinal
An ordinal number is called an initial ordinal if every smaller ordinal has a smaller cardinal number (Moore 1982, p. 248; Rubin 1967, p. 271). The s ordinal numbers are just the transfinite initial ordinals (Rubin 1967, p. 272).This proper class can be well ordered and put into one-to-one correspondence with the ordinal numbers. For any two well ordered sets that are order isomorphic, there is only one order isomorphism between them. Let be that isomorphism from the ordinals to the transfinite initial ordinals, thenwhere .
### Independence complement theorem
If sets and are independent, then so are and , where is the complement of (i.e., the set of all possible outcomes not contained in ). Let denote "or" and denote "and." Then(1)(2)where is an abbreviation for . But and are independent, so(3)Also, since and are complements, they contain no common elements, which means that(4)for any . Plugging (4) and (3) into (2) then gives(5)Rearranging,(6)Q.E.D.
### Inclusion map
Given a subset of a set , the injection defined by for all is called the inclusion map.
### Simple function
A simple function is a finite sum , where the functions are characteristic functions on a set . Another description of a simple function is a function that takes on finitely many values in its range.The collection of simple functions is closed under addition and multiplication. In addition, it is easy to integrate a simple function. By approximating a given function by simple functions, the Lebesgue integral of can be calculated.
### Set theory
The mathematical theory of sets. Set theory is closely associatedwith the branch of mathematics known as logic.There are a number of different versions of set theory, each with its own rules and axioms. In order of increasing consistency strength, several versions of set theory include Peano arithmetic (ordinary algebra), second-order arithmetic (analysis), Zermelo-Fraenkel set theory, Mahlo, weakly compact, hyper-Mahlo, ineffable, measurable, Ramsey, supercompact, huge, and -huge set theory.
### Hall's theorem
There exists a system of distinct representatives for a family of sets , , ..., iff the union of any of these sets contains at least elements for all from 1 to (Harary 1994, p. 53).
### Saturated enlargement
Let be a set of urelements, and let be the superstructure with as its set of individuals. Let be a cardinal number. An enlargement is -saturated provided that it satisfies the following:For each internal binary relation , and each set , if is contained in the domain of and the cardinality of is less than , then there exists a in the range of such that if , then .If is -saturated for some cardinal that is greater than or equal to the cardinality of , then we just say that is saturated. If it is -saturated for some cardinal that is greater than or equal to the cardinality of , then we say it is polysaturated.Let be the set of real numbers, as urelements. Let be a cardinal number that is larger than the cardinality of the power set of , and let be a -saturated enlargement of . Let be an internal subset of , and let . Then is a closed subset of (in the usual topology of the real numbers).Using saturated enlargements, one may prove the following result in universal algebra:Let..
### Finite
A set which contains a nonnegative integral number of elements is said to be finite. A set which is not finite is said to be infinite. A finite or countably infinite set is said to be countable. While the meaning of the term "finite" is fairly clear in common usage, precise definitions of finite and infinite are needed in technical mathematics and especially in set theory.
### Space join
Let and be topological spaces. Then their join is the factor space(1)where is the equivalence relation(2)
### Complex projective plane
The set is the set of all equivalence classes of ordered triples under the equivalence relation if for some nonzero complex number .
### Hilbert hotel
Let a hotel have a denumerable set of rooms numbered 1, 2, 3, .... Then any finite number of guests can be accommodated without evicting the current guests by moving the current guests from room to room . Furthermore, a denumerable number of guests can be similarly accommodated by moving the existing guests from to , freeing up the denumerable number of rooms .
### Countably infinite
Any set which can be put in a one-to-one correspondence with the natural numbers (or integers) so that a prescription can be given for identifying its members one at a time is called a countably infinite (or denumerably infinite) set. Once one countable set is given, any other set which can be put into a one-to-one correspondence with is also countable. Countably infinite sets have cardinal number aleph-0.Examples of countable sets include the integers, algebraic numbers, and rational numbers. Georg Cantor showed that the number of real numbers is rigorously larger than a countably infinite set, and the postulate that this number, the so-called "continuum," is equal to aleph-1 is called the continuum hypothesis. Examples of nondenumerable sets include the real, complex, irrational, and transcendental numbers...
### Cantor's equation
where is an ordinal number and is an inaccessible cardinal.
### Continuum hypothesis
The proposal originally made by Georg Cantor that there is no infinite set with a cardinal number between that of the "small" infinite set of integers and the "large" infinite set of real numbers (the "continuum"). Symbolically, the continuum hypothesis is that . Problem 1a of Hilbert's problems asks if the continuum hypothesis is true.Gödel showed that no contradiction would arise if the continuum hypothesis were added to conventional Zermelo-Fraenkel set theory. However, using a technique called forcing, Paul Cohen (1963, 1964) proved that no contradiction would arise if the negation of the continuum hypothesis was added to set theory. Together, Gödel's and Cohen's results established that the validity of the continuum hypothesis depends on the version of set theory being used, and is therefore undecidable (assuming the Zermelo-Fraenkel axioms together with the axiom of choice).Conway..
### Cantor diagonal method
The Cantor diagonal method, also called the Cantor diagonal argument or Cantor's diagonal slash, is a clever technique used by Georg Cantor to show that the integers and reals cannot be put into a one-to-one correspondence (i.e., the uncountably infinite set of real numbers is "larger" than the countably infinite set of integers). However, Cantor's diagonal method is completely general and applies to any set as described below.Given any set , consider the power set consisting of all subsets of . Cantor's diagonal method can be used to show that is larger than , i.e., there exists an injection but no bijection from to . Finding an injection is trivial, as can be seen by considering the function from to which maps an element of to the singleton set . Suppose there exists a bijection from to and consider the subset of consisting of the elements of such that does not contain . Since is a bijection, there must exist an element of such that . But by the..
### Transfinite number
One of Cantor's ordinal numbers , , , ..., , , ... which is "larger" than any whole number.
### Sperner's theorem
The maximum cardinal number of a collection of subsets of a -element set , none of which contains another, is the binomial coefficient , where is the floor function.
### Cardinal number
In common usage, a cardinal number is a number used in counting (a countingnumber), such as 1, 2, 3, ....In formal set theory, a cardinal number (also called "the cardinality") is a type of number defined in such a way that any method of counting sets using it gives the same result. (This is not true for the ordinal numbers.) In fact, the cardinal numbers are obtained by collecting all ordinal numbers which are obtainable by counting a given set. A set has (aleph-0) members if it can be put into a one-to-one correspondence with the finite ordinal numbers. The cardinality of a set is also frequently referred to as the "power" of a set (Moore 1982, Dauben 1990, Suppes 1972).In Georg Cantor's original notation, the symbol for a set annotated with a single overbar indicated stripped of any structure besides order, hence it represented the order type of the set. A double overbar then indicated stripping the order from the set and thus indicated..
### Cardinal multiplication
Let and be any sets. Then the product of and is defined as the Cartesian product(Ciesielski 1997, p. 68; Dauben 1990, p. 173; Moore 1982, p. 37; Rubin 1967, p. 274; Suppes 1972, pp. 114-115).
### Graphoid
A graphoid consists of a set of elements together with two collections and of nonempty subsets of , called circuits and cocircuits respectively, such that 1. For any and , , 2. No circuit properly contains another circuit and no cocircuit properly contains another cocircuit, 3. For any painting of with colors exactly one element green and the rest either red or blue, there exists either (a) a circuit containing the green element and no red elements, or (b) a cocircuit containing the green element and no blue elements.
### Kneser's conjecture
A combinatorial conjecture formulated by Kneser (1955). It states that whenever the -subsets of a -set are divided into classes, then two disjoint subsets end up in the same class.Lovász (1978) gave a proof based on graph theory. In particular, he showed that the Kneser graph, whose vertices represent the -subsets, and where each edge connects two disjoint subsets, is not -colorable. More precisely, his results says that the chromatic number is equal to , and this implies that Kneser's conjecture is always false if the number of classes is increased to .An alternate proof was given by Bárány (1978).
### Independent set
Two sets and are said to be independent if their intersection , where is the empty set. For example, and are independent, but and are not. Independent sets are also called disjoint or mutually exclusive.An independent vertex set of a graph is a subset of the vertices such that no two vertices in the subset represent an edge of . The figure above shows independent vertex sets consisting of two subsets for a number of graphs (the wheel graph , utility graph , Petersen graph, and Frucht graph).An independent edge set can be defined similarly(Skiena 1990, p. 219).
### Zorn's lemma
If is any nonempty partially ordered set in which every chain has an upper bound, then has a maximal element. This statement is equivalent to the axiom of choice.Renteln and Dundes (2005) give the following (bad) mathematical jokes about Zorn's lemma:Q: What's sour, yellow, and equivalent to the axiomof choice? A: Zorn's lemon.Q: What is brown, furry, runs to the sea, and is equivalent to the axiomof choice? A: Zorn's lemming.
### Characteristic function
Given a subset of a larger set, the characteristic function , sometimes also called the indicator function, is the function defined to be identically one on , and is zero elsewhere. Characteristic functions are sometimes denoted using the so-called Iverson bracket, and can be useful descriptive devices since it is easier to say, for example, "the characteristic function of the primes" rather than repeating a given definition. A characteristic function is a special case of a simple function.The term characteristic function is used in a different way in probability, where it is denoted and is defined as the Fourier transform of the probability density function using Fourier transform parameters ,(1)(2)(3)(4)(5)where (sometimes also denoted ) is the th moment about 0 and (Abramowitz and Stegun 1972, p. 928; Morrison 1995).A statistical distribution is not uniquely specified by its moments, but is by its characteristic..
### Power set
Given a set , the power set of , sometimes also called the powerset, is the set of all subsets of . The order of a power set of a set of order is . Power sets are larger than the sets associated with them. The power set of is variously denoted or .The power set of a given set can be found in the Wolfram Language using Subsets[s].
### Unsorted union
The unsorted union of a list is a list containing the same elements as but with the second and subsequent occurrence of any given element removed. For example, the unsorted union of the set is . The unsorted union differs from the usual union in that the elements of the unsorted union are not necessarily ordered.The unsorted union is implemented in the WolframLanguage as DeleteDuplicates[list].It can be implemented in the Wolfram Languagetop-level code as: UnsortedUnion1[x_] := Tally[x][[All, 1]]or UnsortedUnion2[x_] := Reap[Sow[1, x], _, #1&][[2]]or UnsortedUnion3[x_] := Module[{f}, f[y_] := (f[y] = Sequence[]; y); f /@ x ]Depending on the nature of the list to be unioned, different implementations above may be more efficient, although in general, the first is the fastest.
### Symmetric difference
The set of elements belonging to one but not both of two given sets. It is therefore the union of the complement of with respect to and with respect to , and corresponds to the XOR operation in Boolean logic. The symmetric difference can be implemented in the Wolfram Language as: SymmetricDifference[a_, b_] := Union[Complement[a, b], Complement[b, a]]The symmetric difference of sets and is variously written as , , (Borowski and Borwein 1991) or (Harris and Stocker 1998, p. 3). All but the first notation should probably be deprecated since each of the other symbols has a common meaning in other areas of mathematics.For example, for and , , since 2, 3, and 5 are each in one, but not both, sets.
### Lattice
An algebra is called a lattice if is a nonempty set, and are binary operations on , both and are idempotent, commutative, and associative, and they satisfy the absorption law. The study of lattices is called lattice theory.Note that this type of lattice is distinct from the regular array of points known as a point lattice (or informally as a mesh or grid). While every point lattice is a lattice under the ordering inherited from the plane, many lattices are not point lattices.Lattices offer a natural way to formalize and study the ordering of objects using a general concept known as the partially ordered set. A lattice as an algebra is equivalent to a lattice as a partially ordered set (Grätzer 1971, p. 6) since 1. Let the partially ordered set be a lattice. Set and . Then the algebra is a lattice. 2. Let the algebra be a lattice. Set iff . Then is a partially ordered set, and the partially ordered set is a lattice. 3. Let the partially ordered set be..
### Continuum
The term "continuum" has (at least) two distinct technical meanings in mathematics.The first is a compact connected metric space (Kuratowski 1968; Lewis 1983, pp. 361-394; Nadler 1992; Prajs and Charatonik).The second is the nondenumerable set of real numbers, denoted . The continuum satisfies(1)and(2)where is aleph0 (Aleph-0) and is a positive integer. It is also true that(3)for . However,(4)is a set larger than the continuum. Paradoxically, there are exactly as many points on a line (or line segment) as in a plane, a three-dimensional space, or finite hyperspace, since all these sets can be put into a one-to-one correspondence with each other.The continuum hypothesis, first proposed by Georg Cantor, holds that the cardinal number of the continuum is the same as that of aleph1. The surprising truth is that this proposition is undecidable, since neither it nor its converse contradicts the tenets of set theory...
### Union
The union of two sets and is the set obtained by combining the members of each. This is written , and is pronounced " union " or " cup ." The union of sets through is written . The union of a list may be computed in the Wolfram Language as Union[l].Let , , , ... be sets, and let denote the probability of . Then(1)Similarly,(2)(3)(4)(5)(6)The general statement of this property for sets is known as the inclusion-exclusion principle.If and are disjoint sets, then by definition , so(7)Continuing, for a set of disjoint elements , , ..., (8)which is the countable additivityprobability axiom. Now let(9)then(10)
### Conjunction
A product of ANDs, denotedThe conjunctions of a Boolean algebra of subsets of cardinality are the functionswhere . For example, the 8 conjunctions of are , , , , , , , and (Comtet 1974, p. 186).A literal is considered a (degenerate) conjunction (Mendelson1997, p. 30).The Wolfram Language command Conjunction[expr, a1, a2, ...] gives the conjunction of expr over all choices of the Boolean variables .
### Intersection
The intersection of two sets and is the set of elements common to and . This is written , and is pronounced " intersection " or " cap ." The intersection of sets through is written .The intersection of two lines and is written . The intersection of two or more geometric objects is the point (points, lines, etc.) at which they concur.
### Infinity
Infinity, most often denoted as , is an unbounded quantity that is greater than every real number. The symbol had been used as an alternative to M () in Roman numerals until 1655, when John Wallis suggested it be used instead for infinity.Infinity is a very tricky concept to work with, as evidenced by some of the counterintuitive results that follow from Georg Cantor's treatment of infinite sets.Informally, , a statement that can be made rigorous using the limit concept,Similarly,where the notation indicates that the limit is taken from the positive side of the real line.In the Wolfram Language, is represented using the symbol Infinity.
### Ultrafilter
Let be a nonempty set, then an ultrafilter on is a nonempty collection of subsets of having the following properties: 1. . 2. If then . 3. If and then . 4. For any subset of , either or its complement . An ultrafilter on is said to be free if it contains the cofinite filter of .
### Filter
Let be a nonempty set, then a filter on is a nonempty collection of subsets of having the following properties: 1. , 2. If , then , 3. If and then If is an infinite set, then the collection is a filter called the cofinite (or Fréchet) filter on .In signal processing, a filter is a function or procedure which removes unwanted parts of a signal. The concept of filtering and filter functions is particularly useful in engineering. One particularly elegant method of filtering Fourier transforms a signal into frequency space, performs the filtering operation there, then transforms back into the original space (Press et al. 1992). | 13,782 | 59,021 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.640625 | 4 | CC-MAIN-2020-45 | latest | en | 0.93524 |
https://mathinfocusanswerkey.com/math-in-focus-grade-6-chapter-4-review-test-answer-key/ | 1,702,144,954,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100942.92/warc/CC-MAIN-20231209170619-20231209200619-00377.warc.gz | 422,211,467 | 42,858 | # Math in Focus Grade 6 Chapter 4 Review Test Answer Key
Go through the Math in Focus Grade 6 Workbook Answer Key Chapter 4 Review Test to finish your assignments.
## Math in Focus Grade 6 Course 1 A Chapter 4 Review Test Answer Key
Concepts and Skills
Write each ratio in simplest form.
Question 1.
8 : 24
1 : 3
Explanation:
The simplest form of 8 : 24 is 1 : 3
Divide each side by the highest common factor.
That is the highest number which divides evenly into both numbers.
G.C.M for 8 and 24 is 8
Question 2.
6 : 20
3 : 10
Explanation:
The simplest form of 6 : 20 is 3 : 10
Divide each side by the highest common factor.
That is the highest number which divides evenly into both numbers.
G.C.M for 6 and 20 is 2
Question 3.
14 : 49
2 : 7
Explanation:
The simplest form of 14 : 49 is 2 : 7
Divide each side by the highest common factor.
That is the highest number which divides evenly into both numbers.
G.C.M for 14 and 49 is 7
Question 4.
27 : 72
3 : 8
Explanation:
The simplest form of 27 : 72 is 3 : 8
Divide each side by the highest common factor.
That is the highest number which divides evenly into both numbers.
G.C.M for 27 and 72 is 9
Question 5.
14 : 49
2 : 7
Explanation:
The simplest form of 14 : 49 is 2 : 7
Divide each side by the highest common factor.
That is the highest number which divides evenly into both numbers.
G.C.M for 14 and 49 is 7
Question 6.
45 : 30
3 : 2
Explanation:
The simplest form of 45 : 30 is 3 : 2
Divide each side by the highest common factor.
That is the highest number which divides evenly into both numbers.
G.C.M for 45 and 30 is 15
Question 7.
27 : 72
3 : 8
Explanation:
The simplest form of 27 : 72 is 3 : 8
Divide each side by the highest common factor.
That is the highest number which divides evenly into both numbers.
G.C.M for 27 and 72 is 9
Question 8.
64 : 56
8 : 7
Explanation:
The simplest form of 64 : 56 is 8 : 7
Divide each side by the highest common factor.
That is the highest number which divides evenly into both numbers.
G.C.M for 64 and 56 is 8
Find the missing term in each pair of equivalent ratios.
Question 9.
1 : 3 = 6 :
Explanation:
product of extremes = product of means
a:b = c:d
$$\frac{a}{b}$$ = $$\frac{c}{d}$$
a = 1, b = 3, c = 6, d = x
1 x x = 3 x 6
x = 18
Question 10.
4 : 7 = : 21
Explanation:
product of extremes = product of means
a:b = c:d
$$\frac{a}{b}$$ = $$\frac{c}{d}$$
a = 4, b = 7, c = x, d = 21
4 x 21 = 7 x x
84 = 7x
x = $$\frac{84}{7}$$
x = 12
Question 11.
25 : 15 = : 3
Explanation:
product of extremes = product of means
a:b = c:d
$$\frac{a}{b}$$ = $$\frac{c}{d}$$
a = 25, b = 15, c = x, d = 3
25 x 3 = 15 x x
15x = 75
x = $$\frac{75}{15}$$
x = 5
Question 12.
54 : 36 = 18 :
Explanation:
product of extremes = product of means
a:b = c:d
$$\frac{a}{b}$$ = $$\frac{c}{d}$$
a = 54, b = 36, c = 18, d = x
54 x x = 36 x 18
54x = 648
x = $$\frac{648}{54}$$
x = 12
Question 13.
4 : = 20 : 25
Explanation:
product of extremes = product of means
a:b = c:d
$$\frac{a}{b}$$ = $$\frac{c}{d}$$
a = 4, b = x, c = 20, d = 25
4 x 25 = x x 20
100 = 20x
x = $$\frac{100}{20}$$
x = 5
Question 14.
: 9 = 48 : 72
Explanation:
product of extremes = product of means
a:b = c:d
$$\frac{a}{b}$$ = $$\frac{c}{d}$$
a = x, b = 9, c = 48, d = 72
x x 72 = 9 x 48
72x = 432
x = $$\frac{432}{72}$$
x = 6
Question 15.
28 : = 4 : 6
Explanation:
product of extremes = product of means
a:b = c:d
$$\frac{a}{b}$$ = $$\frac{c}{d}$$
a = 28, b = x, c = 4, d = 6
28 x 6 = x x 4
168 = 4x
x = $$\frac{168}{4}$$
x = 42
Question 16.
: 36 = 21 : 12
Explanation:
product of extremes = product of means
a:b = c:d
$$\frac{a}{b}$$ = $$\frac{c}{d}$$
a = x, b = 36, c = 21, d = 12
x x 12 = 36 x 21
12x = 756
x = $$\frac{756}{12}$$
x = 63
Problem Solving
Question 17.
The city has 28 fire engines and 36 fire trucks. Find the ratio of the number of fire engines to the number of fire trucks in simplest form.
7 : 9
Explanation:
The city has 28 fire engines and 36 fire trucks.
Ratio is 28 : 36
The ratio of the number of fire engines to the number of fire trucks in simplest form.
7 : 9
Question 18.
Of the 80 students who signed up for after school clubs, 16 students signed up for the art club, and the rest signed up for other clubs. Find the ratio of the number of students who signed up for the art club to the number of students who signed up for other dubs. Give your answer in simplest form.
1 : 4
Explanation:
80 students who signed up for after school clubs,
16 students signed up for the art club,
the rest signed up for other clubs = 80 – 16 = 64
The ratio of the number of students who signed up for the art club to the number of students who signed up for other dubs are 16 : 64
simplest form of ratio is 1 : 4
Question 19.
On Saturday, Alison used her cell phone for 36 minutes. On Sunday, she used her cell phone for 18 minutes more than on Saturday. Find the ratio of the number of minutes Alison used on Saturday to the total number of minutes on Saturday and Sunday. Give your answer in simplest form.
2 : 5
Explanation:
On Saturday 36 min
On Sunday 36 + 18 = 54 min
the number of minutes Alison used on Saturday to the total number of minutes on Saturday and Sunday is
36 + 54 =90 min
the ratio is 36 : 90 = 2 : 5
Question 20.
Daniel is 12 years old. Elliot is 15 years older than Daniel. Frank is 3 years younger than Elliot. Find the ratio of Daniel’s age to Frank’s age. Give your answer in simplest form.
1: 2
Explanation:
Daniel’s age is 12 years
Elliots age is 15 + 12 = 27 years
Franks age is 27 -3 = 24 years
the ratio of Daniel’s age to Frank’s age
12 : 24
1 : 2
Question 21.
The ratio of the number of left-handed batters to the number of right-handed batters is 5 : 8. There are 45 left-handed batters.
a) How many right-handed batters are there?
Explanation:
45 : 5 = x : 8
x =(45 x 8) /5 = 72
b) Find the ratio of the number of left-handed batters to the total number of batters. Give your answer in simplest form.
5 : 13
Explanation:
45 + 72 = 117 total number of batters
45 : 117
5 : 13
Question 22.
Mrs. Johnson gave a sum of money to her son and daughter in the ratio 5 : 6. Her daughter received $2,400. How much did Mrs. Johnson give away in all? Answer:$2000
Explanation:
son and daughter in the ratio 5 : 6
6 : x = 5 x 2400
x = (5x 2400)/6 = 2000
Question 23.
The ratio of the number of boys to the number of girls in a school is 5 : 7. If there are 600 students in the school, how many girls are there?
350 girls
Explanation:
ratio of boys and girls = 5 + 7 = 12
total students = 600
number of girls =7 x 600/12 = 50 x 7 = 350
Question 24.
The murals in a school are painted by its grade 6 and grade 7 students. The number of mural painters from grade 6 and the number of mural painters from grade 7 is the same for each mural in the school. Copy and complete the table.
Explanation:
The number of mural painters from grade 6 and the number of mural painters from grade 7 is the same for each mural in the school
Each time the number of Murals in Garde 6 increased with 5 multiple.
At the same time in Grade 7 also increased with 7 multiple.
Question 25.
A sum of money was shared among Aaron, Ben, and Charles in the ratio 2 : 5 : 7. If Charles’s share was $1,180 more than Aaron’s share, what was the original sum of money shared? Answer: Explanation: Aaron, Ben, and Charles in the ratio 2 : 5 : 7 2+5+7= 14 Aaron: 2x Ben: 5x Charles:7x 7x = 2x + 1180 5x = 1180 x = 236 2x + 5x + 7x = 14x 14x = 14*236 =$3304
Question 26.
The ratio of the number of beads Karen had to the number of beads Patricia had was 2 : 5. After Patricia bought another 75 beads, the ratio became 4 : 15. How many beads did each girl have at first?
Explanation
2:5
2x : 5x
2x : 5x+75 = 4:15
2x X 15 = 4 ( 5x + 75)
30x = 20x + 180
30x-20x = 180
10x=180
x=180/10 = 18
2x:5x
18 x2 : 5×18
36 : 90
Question 27.
Mr. Young had some bottles of apple juice and orange juice. The ratio of the number of bottles of apple juice to the number of bottles of orange juice was 3 : 2. After he sold 64 bottles of apple juice, the ratio became 1 : 6. How many bottles of apple juice and orange juice did Mr. Young have altogether in the end? | 2,711 | 8,119 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.53125 | 5 | CC-MAIN-2023-50 | longest | en | 0.92048 |
https://www.meritnation.com/ask-answer/question/calculate-the-mass0-4-mole-of-co2-at-mass-of-c-12-o-16-a-m-u/atoms-and-molecules/8846243 | 1,639,029,223,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964363659.21/warc/CC-MAIN-20211209030858-20211209060858-00578.warc.gz | 964,742,212 | 8,155 | # calculate the mass0.4 mole of CO2 (At Mass of C = 12 &O=16 a.m.u) (17.6g)
Molar mass of CO2 = 44 g mol-1
Mass of 1 mole of CO2 = 44 g
• 10
the mass of 1 mole of CO2= 12(mass of C) + 16x2(mass of O2)=12+32=44g
therefore,
0.4 moles of CO2= 44x0.4=17.6g
• 4
What are you looking for? | 134 | 285 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.734375 | 4 | CC-MAIN-2021-49 | latest | en | 0.741799 |
https://bcts.bergen.org/isoborneol-vs-kuac/uyt19.php?tag=2aed20-multiple-linear-regression-example | 1,611,015,282,000,000,000 | text/html | crawl-data/CC-MAIN-2021-04/segments/1610703517159.7/warc/CC-MAIN-20210118220236-20210119010236-00376.warc.gz | 249,474,411 | 4,550 | Open the Multiple Regression dialog box. Every value of the independent variable x is associated with a value of the dependent variable y. Fit full multiple linear regression model of Height on LeftArm, LeftFoot, HeadCirc, and nose. Our equation for the multiple linear regressors looks as follows: Nonlinear regression analysis is commonly used for more complicated data sets in which the dependent and independent variables show a … Fit reduced multiple linear regression model of Height on LeftArm and LeftFoot. Calculate SSE for the full and reduced models. Multiple Linear Regression Example Suppose you have a data set consisting of the gender, height and age of children between 5 and 10 years old. The following example illustrates XLMiner's Multiple Linear Regression method using the Boston Housing data set to predict the median house prices in housing tracts. The most common models are simple linear and multiple linear. In addition to these variables, the data set also contains an additional variable, Cat. A multiple linear regression model based on a sample of 17 weeks is developed to predict standby hours based on the total staff present and remote hours. However, in practicality, most regression problems have more than one independent variable that determines/influences the value of the dependent variable. The multiple linear regression equation is as follows:, where is the predicted or expected value of the dependent variable, X 1 through X p are p distinct independent or predictor variables, b 0 is the value of Y when all of the independent variables (X 1 through X p) are equal to zero, and b 1 through b p are the estimated regression coefficients. Regression analysis includes several variations, such as linear, multiple linear, and nonlinear. You could use multiple linear regression to predict the height of a child (dependent variable) using both age and gender as predictors (i.e., two independent variables). Create a residual plot. Perform a linear regression analysis of Rating on Moisture and Sweetness. Multiple Linear Regression Multiple linear regression attempts to model the relationship between two or more explanatory variables and a response variable by fitting a linear equation to observed data. A description of each variable is given in the following table. Multiple Linear Regression is an extension of Simple Linear regression where the model depends on more than 1 independent variable for the prediction results. Multiple Regression: Example . Open the sample data, WrinkleResistance.MTW. The SSR is 20,868.57 and the SSE is 30,102.09. From a marketing or statistical research to data analysis, linear regression model have an important role in the business. The fact that this is statistically significant indicates that the association between treatment and outcome differs by sex. Calculate the general linear F statistic by hand and find the p-value. Consider an analyst who wishes to establish a linear relationship between the daily change in … The chemist performs a multiple regression analysis to fit a model with the predictors and eliminate the predictors that do not have a statistically significant relationship with the response. This data set has 14 variables. Click "Storage" in the regression dialog and check "Fits" to store the fitted (predicted) values. Complete parts (a) through (d) below A. Linear Regression vs. In the previous lesson, we learned about Simple Linear Regression where we modeled the relationship between a target variable and an independent variable. The multiple regression model is: The details of the test are not shown here, but note in the table above that in this model, the regression coefficient associated with the interaction term, b 3, is statistically significant (i.e., H 0: b 3 = 0 versus H 1: b 3 ≠ 0). Many of simple linear regression examples (problems and solutions) from the real life can be given to help you understand the core meaning. To create a scatterplot of the data with points marked by Sweetness and two lines representing the fitted regression … | 784 | 4,075 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.8125 | 4 | CC-MAIN-2021-04 | latest | en | 0.915679 |
https://www.physicsforums.com/threads/differential-equation-confused-on-directions-f-x-y.107246/ | 1,477,240,421,000,000,000 | text/html | crawl-data/CC-MAIN-2016-44/segments/1476988719286.6/warc/CC-MAIN-20161020183839-00114-ip-10-171-6-4.ec2.internal.warc.gz | 982,905,556 | 16,261 | # Differential equation, confused on directions, F(x,y)?
1. Jan 18, 2006
### mr_coffee
Hello everyone, I am confused on what they are wanting me to do here on this problem:
has an implicit general solution of the form
In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form
Find such a solution and then give the related functions requested.
here is my work, i think they wanted me to do, they gave no intial condition so i solved for C:
but it was incorrect
2. Jan 18, 2006
### Benny
Well what is supposed to be the correct answer? I can't see much wrong with what you've done.
Edit: You seem to have replaced the y with x after the integration of the exponential.
Last edited: Jan 18, 2006
3. Jan 18, 2006
### mr_coffee
thats a good question, it doesn't tell me. it will tell me if i get it right, not if i get it wrong. Whats this implicit general solution stuff do i have to take partial derivatives or somthing?
4. Jan 18, 2006
### HallsofIvy
Staff Emeritus
WHY do you keep writing $\frac{1}{e^{-2x}}$? Surely you know that $\frac{1}{e^{-2x}}= e^{2x}$
Your instructions were to write the answer as F(x,y)= K and then write it as F(x,y)= G(x,y)+ H(x,y)= K
You already have 5e2x- arcsin(x/2)= K. Can't you just look at that and pick G and H?
5. Jan 18, 2006
### benorin
Going from the second line to the third line your "y" became an "x". As far as what they want:
They asked for $$F(x,y)=G(x)+H(y)=K$$
So "Find such a solution" is satisfied by $$F(x,y)=-\mbox{arcsin}\left( \frac{x}{2}\right) + 5e^{2y} = K$$
but the answer they want is "give the related functions requested"
so try $$G(x)= -\mbox{arcsin}\left( \frac{x}{2}\right) \mbox{ and }H(y)= 5e^{2y}$$
6. Jan 18, 2006
### mr_coffee
Ahhh, thanks so much everyone it was correct. weee | 557 | 1,844 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.828125 | 4 | CC-MAIN-2016-44 | longest | en | 0.936691 |
Subsets and Splits