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https://gmatclub.com/forum/each-of-60-cars-is-parked-in-one-of-three-empty-parking-lots-306824.html	GMAT Question of the Day: Daily via email \| Daily via Instagram New to GMAT Club? Watch this Video It is currently 27 Jan 2020, 20:38 ### 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 # Each of 60 cars is parked in one of three empty parking lots Author Message TAGS: ### Hide Tags EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 15994 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Each of 60 cars is parked in one of three empty parking lots [#permalink] ### Show Tags 30 Sep 2019, 20:27 00:00 Difficulty: 15% (low) Question Stats: 89% (01:53) correct 11% (01:27) wrong based on 62 sessions ### HideShow timer Statistics EMPOWERgmat PS Series: Block 1, Question 5 Each of 60 cars is parked in one of three empty parking lots. After all of the cars have been parked, the largest lot holds 8 more cars than the middle lot and 16 more cars than the smallest lot. How many cars are in the largest lot? A. 12 B. 20 C. 22 D. 28 E. 30 48 Hour Window To Win An $85 EMPOWERgmat Tuition Credit (1 Month Free!) Share your explanation! The GMAT Club member with the most verified Kudos in total on the 5 question SC Block 1 question pack will win an$85 EMPOWERgmat tuition credit, which will entitle the winner to a full month of complete access to the EMPOWERgmat system. Even if you're not sure about your answer or your rationale, share your explanation to help boost your learning and earn a chance to win. To be eligible, your explanation must be submitted within the 48 hour window after this post was created and should explain your reasoning why the answer you chose is correct The OA and official explanation will be held until the 48 hour window has lapsed. _________________ Contact Rich at: [email protected] The Course Used By GMAT Club Moderators To Earn 750+ souvik101990 Score: 760 Q50 V42 ★★★★★ ENGRTOMBA2018 Score: 750 Q49 V44 ★★★★★ VP Joined: 20 Jul 2017 Posts: 1266 Location: India Concentration: Entrepreneurship, Marketing WE: Education (Education) Re: Each of 60 cars is parked in one of three empty parking lots [#permalink] ### Show Tags 30 Sep 2019, 20:43 1 2 EMPOWERgmatRichC wrote: EMPOWERgmat PS Series: Block 1, Question 5 Each of 60 cars is parked in one of three empty parking lots. After all of the cars have been parked, the largest lot holds 8 more cars than the middle lot and 16 more cars than the smallest lot. How many cars are in the largest lot? A. 12 B. 20 C. 22 D. 28 E. 30 48 Hour Window To Win An $85 EMPOWERgmat Tuition Credit (1 Month Free!) Share your explanation! The GMAT Club member with the most verified Kudos in total on the 5 question SC Block 1 question pack will win an$85 EMPOWERgmat tuition credit, which will entitle the winner to a full month of complete access to the EMPOWERgmat system. Even if you're not sure about your answer or your rationale, share your explanation to help boost your learning and earn a chance to win. To be eligible, your explanation must be submitted within the 48 hour window after this post was created and should explain your reasoning why the answer you chose is correct The OA and official explanation will be held until the 48 hour window has lapsed. Let the number of cars parked in largest lot = x --> Middle lot = x - 8 & Smallest lot = x - 16 So, x + (x - 8) + (x - 16) = 60 --> 3x - 24 = 60 --> 3x = 84 --> x = 28 IMO Option D Senior Manager Joined: 28 Feb 2014 Posts: 280 Location: India GMAT 1: 570 Q49 V20 GPA: 3.97 WE: Engineering (Education) Re: Each of 60 cars is parked in one of three empty parking lots [#permalink] ### Show Tags 30 Sep 2019, 21:25 1 EMPOWERgmatRichC wrote: EMPOWERgmat PS Series: Block 1, Question 5 Each of 60 cars is parked in one of three empty parking lots. After all of the cars have been parked, the largest lot holds 8 more cars than the middle lot and 16 more cars than the smallest lot. How many cars are in the largest lot? A. 12 B. 20 C. 22 D. 28 E. 30 let a,b,c be lot size in ascending order a+b+c=60 c=b+8 c=a+16 Substituting c-16+c-8+c=60 c = 28 D is correct Intern Joined: 09 Jul 2018 Posts: 12 Re: Each of 60 cars is parked in one of three empty parking lots [#permalink] ### Show Tags 02 Oct 2019, 16:42 1 Let the number of cars parked in largest lot = L Middle lot = M & Smallest lot = S But M=L-8 & S=L-16 So, L + (L - 8) + (L - 16) = 60 3L - 24 = 60 3L = 84 L = 28 IMO Option D EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 15994 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: Each of 60 cars is parked in one of three empty parking lots [#permalink] ### Show Tags 03 Oct 2019, 11:34 OFFICIAL EXPLANATION Hi All, We're told that each of 60 cars is parked in one of three empty parking lots. After all of the cars have been parked, the largest lot holds 8 more cars than the middle lot and 16 more cars than the smallest lot. We're asked for the number of cars that are in the LARGEST lot. This is an example of a 'System' question; it can be solved Algebraically, but it can also be solved rather quickly by TESTing THE ANSWERS... Based on the information that we're given, the three parking lots clearly each end up holding a different number of cars. We're asked for the LARGEST number of the three, so we should look to TEST one of the larger answers first. Let's TEST Answer D... IF....the largest lot holds 28 cars.... then the middle lot holds 28 - 8 = 20 cars... and the smallest lot holds 28 - 16 = 12 cars... Total = 28 + 20 + 12 = 60 cars This is an exact MATCH for what we were told, so this MUST be the answer! GMAT assassins aren't born, they're made, Rich _________________ Contact Rich at: [email protected] The Course Used By GMAT Club Moderators To Earn 750+ souvik101990 Score: 760 Q50 V42 ★★★★★ ENGRTOMBA2018 Score: 750 Q49 V44 ★★★★★ GMAT Club Legend Joined: 12 Sep 2015 Posts: 4234 Re: Each of 60 cars is parked in one of three empty parking lots [#permalink] ### Show Tags 03 Oct 2019, 12:39 Top Contributor EMPOWERgmatRichC wrote: EMPOWERgmat PS Series: Block 1, Question 5 Each of 60 cars is parked in one of three empty parking lots. After all of the cars have been parked, the largest lot holds 8 more cars than the middle lot and 16 more cars than the smallest lot. How many cars are in the largest lot? A. 12 B. 20 C. 22 D. 28 E. 30 Let x = number of cars in the LARGEST lot Given: the largest lot holds 8 more cars than the middle lot. So, x - 8 = number of cars in the MIDDLE lot Given: the largest lot holds 16 more cars than the smallest lot. So, x - 16 = number of cars in the SMALLEST lot Given: Each of 60 cars is parked in one of three empty parking lots. We can write: (# of cars in LARGEST lot) + (# of cars in MIDDLE lot) + (# of cars in SMALLEST lot) = 60 Substitute values: (x) + (x - 8) + (x - 16) = 60 Simplify: 3x - 24 = 60 Add 24 to both sides: 3x = 84 Divide both sides by 3: x = 28 Cheers, Brent _________________ Test confidently with gmatprepnow.com Senior Manager Joined: 05 Jul 2018 Posts: 459 Location: India Concentration: General Management, Technology GMAT 1: 600 Q47 V26 GRE 1: Q162 V149 GPA: 3.6 WE: Information Technology (Consulting) Re: Each of 60 cars is parked in one of three empty parking lots [#permalink] ### Show Tags 10 Oct 2019, 22:39 1 The 3 slots A + B + C total to 60 cars The highest slot C has 8 more than B and 16 more than A Let us look at the second-largest answer choice which says C=28 As per question A=12 (16 less than C), B=20(8 less than C) ...This gives a total of 60. Hence Correct answer is D EMPOWERgmatRichC: This is the correct way to do TTA ...right EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 15994 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: Each of 60 cars is parked in one of three empty parking lots [#permalink] ### Show Tags 11 Oct 2019, 11:30 Hi saukrit, YES - that's a perfect use of TEST THE ANSWERS. In these sorts of situations, that Tactic (and some basic Arithmetic) is almost always faster and easier that a typical Algebraic approach - so it's something to keep in mind (especially if you have a pacing problem in the Quant section). GMAT assassins aren't born, they're made, Rich _________________ Contact Rich at: [email protected] The Course Used By GMAT Club Moderators To Earn 750+ souvik101990 Score: 760 Q50 V42 ★★★★★ ENGRTOMBA2018 Score: 750 Q49 V44 ★★★★★ Board of Directors Status: QA & VA Forum Moderator Joined: 11 Jun 2011 Posts: 4868 Location: India GPA: 3.5 Re: Each of 60 cars is parked in one of three empty parking lots [#permalink] ### Show Tags 12 Oct 2019, 00:09 EMPOWERgmatRichC wrote: EMPOWERgmat PS Series: Block 1, Question 5 Each of 60 cars is parked in one of three empty parking lots. After all of the cars have been parked, the largest lot holds 8 more cars than the middle lot and 16 more cars than the smallest lot. How many cars are in the largest lot? A. 12 B. 20 C. 22 D. 28 E. 30 48 Hour Window To Win An $85 EMPOWERgmat Tuition Credit (1 Month Free!) Share your explanation! The GMAT Club member with the most verified Kudos in total on the 5 question SC Block 1 question pack will win an$85 EMPOWERgmat tuition credit, which will entitle the winner to a full month of complete access to the EMPOWERgmat system. Even if you're not sure about your answer or your rationale, share your explanation to help boost your learning and earn a chance to win. To be eligible, your explanation must be submitted within the 48 hour window after this post was created and should explain your reasoning why the answer you chose is correct The OA and official explanation will be held until the 48 hour window has lapsed. $$L = M + 8 = S + 16$$ Again , L + M + S = 60 Now, Plug in the options and check (A) Can be straightaway rejected as value of L must be > 16 , Since in this case if L = 12 = S + 16 , Or, S = - 4 ( Not Possible ) (B) If L = 20 , M = 12 & S = 4 , L + M + S = 36 (C) If L = 22 , M = 14 & S = 6 , L + M + S = 42 (D) If L = 28 , M = 20 & S = 12 , L + M + S = 60 (E) If L = 30 , M = 22 & S = 14 , L + M + S = 66 _________________ Thanks and Regards Abhishek.... PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS How to use Search Function in GMAT Club \| Rules for Posting in QA forum \| Writing Mathematical Formulas \|Rules for Posting in VA forum \| Request Expert's Reply ( VA Forum Only ) Intern Joined: 06 Sep 2019 Posts: 2 Each of 60 cars is parked in one of three empty parking lots [#permalink] ### Show Tags 12 Oct 2019, 01:42 As the middle lot has 8 cars less than the larger it would be X-8 Smaller Lot is having 16 cars less than the larger that means X-16 As all the lots together having 60 it would be X+(X-8)+(X-16)=60 On solving this it becomes X= 28 Cars Hence Option D would be apt. Each of 60 cars is parked in one of three empty parking lots [#permalink] 12 Oct 2019, 01:42 Display posts from previous: Sort by	2020-01-28T03:38:21	{ "domain": "gmatclub.com", "url": "https://gmatclub.com/forum/each-of-60-cars-is-parked-in-one-of-three-empty-parking-lots-306824.html", "openwebmath_score": 0.2563014328479767, "openwebmath_perplexity": 3974.236522246419, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. Yes\n2. Yes", "lm_q1_score": 0.9759464499040092, "lm_q2_score": 0.8856314723088733, "lm_q1q2_score": 0.8643288913231058 }
https://math.stackexchange.com/questions/903613/fibonacci-sequence-given-n-and-mathrmfibn-is-it-possible-to-calculate	# Fibonacci sequence: Given $n$ and $\mathrm{Fib}(n)$, is it possible to calculate $\mathrm{Fib}(n-1)$? Given $n$ and $\newcommand{\Fib}{\mathrm{Fib}} \Fib(n)$, is it possible to calculate the previous number in the Fibonacci sequence - $\Fib(n-1)$ using integer math in constant time? In other words, I believe I'm looking for a closed form formula. For example: Given $n=10$ and $\Fib(10)=55$, I'd like to determine that $\Fib(9)=34$ without using $\Fib(7) + \Fib(8)$. • Even output of $F_{n-1}$ cannot be done in constant time. (The length of binary representation of $F_{n-1}$ is linear in $n$.) But based on the wording in original post I would guess that you mean using constantly many operations with integers. Aug 20 '14 at 8:14 • @Martin Sleziak - you are correct that I meant "constantly many operations with integers" Aug 20 '14 at 13:23 By Binet's Formula, we know that $$F_n = \dfrac{1}{\sqrt{5}}\left(\phi^n-(-\phi)^{-n}\right) \approx \dfrac{1}{\sqrt{5}}\phi^n$$ where $$\phi = \dfrac{1+\sqrt{5}}{2}$$. Using this formula, we can get that $$F_{n-1}$$ is the nearest integer to $$\dfrac{1}{\phi} F_n$$ for large $$n$$ More rigorously, we have $$F_{n-1} - \dfrac{1}{\phi} F_n = \dfrac{1}{\sqrt{5}}\left(\phi^{n-1}-(-\phi)^{-(n-1)}\right) - \dfrac{1}{\phi\sqrt{5}}\left(\phi^n-(-\phi)^{-n}\right)$$ $$= -\dfrac{(-\phi)^{-(n-1)}}{\sqrt{5}} - \dfrac{(-\phi)^{-(n+1)}}{\sqrt{5}}$$ $$= \dfrac{(-\phi)^{-n}}{\sqrt{5}}\left(\phi + \phi^{-1}\right) = (-\phi)^{-n}$$. Thus, for $$n \ge 2$$, we have $$\left\|F_{n-1} - \dfrac{1}{\phi} F_n \right\| = \phi^{-n} < \frac{1}{2}$$, and so, $$F_{n-1}$$ is the nearest integer to $$\dfrac{1}{\phi}F_n$$. Dividing $$F_n$$ by $$\phi$$ and rounding the result to the nearest integer shouldn't be too computational. • Here is a table that shows the pattern:$$\begin{array} {c\|c\|c} n & a_n & a_n / a_{n-1} \\ \hline 0 & 0 & \\ \hline 1 & 1 & \\ \hline 2 & 1 & 1.0000 \\ \hline 3 & 2 & 2.0000 \\ \hline 4 & 3 & 1.5000 \\ \hline 5 & 5 & 1.6667 \\ \hline 6 & 8 & 1.6000 \\ \hline 7 & 13 & 1.6250 \\ \hline 8 & 21 & 1.6154 \\ \hline 9 & 34 & 1.6190 \\ \hline 10 & 55 & 1.6176 \\ \hline 11 & 89 & 1.6182 \\ \hline 12 & 144 & 1.6180 \\ \hline 13 & 233 & 1.6181 \\ \hline 14 & 377 & 1.6180 \\ \hline \end{array}$$ Aug 20 '14 at 2:25 • @DanielV - agreed, this will work but as marty cohen points out in his anti-answer below, for large values of n simply rounding to the nearest integer will not necessarily yield the correct answer. Aug 20 '14 at 13:34 This is a dissenting note on some of these answers. I am purposly making this an answer, though it really is an anti-answer. The problem with any answer that uses $\phi$ is that, as $n$ gets larger, $\phi$ has to be computed with increasingly great precision. This will not take constant time, either for the computation of $\phi$ or the multiplication by $\phi$. Hint $\rm\ n\:$ is a Fibonacci number iff the interval $\rm\ [\phi\ n - 1/n,\ \phi\ n + 1/n]\$ contains a positive integer (the next Fibonacci number for $\rm\ n > 1)$. This follows from basic properties of continued fractions. For example, $\rm\ 2 \to [2.7,\ 3.7],\ \ 3\to [4.5,\ 5.2],\ \ 5 \to [7.9,\ 8.3]\ \ldots$ For a proof see e.g. T. Komatsu: The interval associated with a fibonacci number. $\$	2022-01-29T02:15:37	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/903613/fibonacci-sequence-given-n-and-mathrmfibn-is-it-possible-to-calculate", "openwebmath_score": 0.9017306566238403, "openwebmath_perplexity": 180.5184766596439, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9759464453565514, "lm_q2_score": 0.8856314723088732, "lm_q1q2_score": 0.8643288872957339 }
http://mathhelpforum.com/algebra/70782-geometric-series-problem.html	# Thread: Geometric series problem . 1. ## Geometric series problem . A house buyer borrows RM 50000 from a bank to buy a house which costs RM 70000 .The rate of interest charged by the bank is 9% per annum ,and is calculated based on the amount outstanding at the begining of each year . The house buyer is required to repay his loan in monthly installments for a period of 15 years . Assuming that the rate of interest is fixed for the entire duration of the loan , find the amount per month he has to repay the bank ? My attempt : It forms a sequence as follows : $\displaystyle 50000,50000(1.09),50000(1.09)^2,...,50000(1.09)^{n-1}$ The 15th term , $\displaystyle T_15=50000(1.09)^14=167086.35$ which is the amount he has to pay by the end of 15 years . Thus , every month he has to pay $\displaystyle \frac{167086.35}{15\times12}=928.26$ My answer is obviously wrong . I wonder where my mistake is .. Thanks to the one who help me out .. 2. ## Repayments Originally Posted by mathaddict A house buyer borrows RM 50000 from a bank to buy a house which costs RM 70000 .The rate of interest charged by the bank is 9% per annum ,and is calculated based on the amount outstanding at the begining of each year . The house buyer is required to repay his loan in monthly installments for a period of 15 years . Assuming that the rate of interest is fixed for the entire duration of the loan , find the amount per month he has to repay the bank ? My attempt : It forms a sequence as follows : $\displaystyle 50000,50000(1.09),50000(1.09)^2,...,50000(1.09)^{n-1}$ The 15th term , $\displaystyle T_15=50000(1.09)^14=167086.35$ which is the amount he has to pay by the end of 15 years . Thus , every month he has to pay $\displaystyle \frac{167086.35}{15\times12}=928.26$ My answer is obviously wrong . I wonder where my mistake is .. Thanks to the one who help me out .. Thanks for showing us your working here. What you're forgetting is that each year he pays not only the interest owing that year, but he also pays back some of the amount borrowed. Are you allowed to use a spreadsheet to work out this answer? It's the easiest way. If not, you'll have to let the monthly payment be x, and then say that, at the start of each new year, the amount owing = amount owed at the start of the last year + interest payable during that year - 12x Then work out how much interest he'll have to pay in the coming year to cover this new amount owing. Then repeat for the following year, and so on, up to the start of year 16. Then say that at the start of year 16 he owes nothing. This is an Amortization problem and requires a special formula. I've seen this type of problem posted many times before. It seem that the students are never given the formula, . . nor are they prepared for the (very) long derivation. Could it be that all those teachers are ignorant of the difficulties . . involved with time-payment problems? I'm beginning to suspect that this is so . . . A house buyer borrows $50,000 from a bank to buy a house. The rate of interest 9% per annum. The buyer is required to repay his loan in monthly installments for 15 years. Assuming that the rate of interest is fixed for the entire duration of the loan, find the amount per month he has to repay the bank? The formula is: .$\displaystyle A \;=\;P\,\frac{i(1+i)^n}{(1+i)^n-1}$. . where: .$\displaystyle \begin{Bmatrix}P &=& \text{principal} \\ i &=& \text{periodic interst rate} \\ n &=& \text{number of periods} \\ A &=& \text{periodic payment} \end{Bmatrix}$We have: .$\displaystyle \begin{array}{ccc}P \:=\:50,\!000 \\ i \:=\:\frac{9\%}{12} \:=\:0.0075 \\ n \:=\:15\!\cdot\!12 \:=\:180 \end{array}$Hence: .$\displaystyle A \;=\;50,\!000\,\frac{0.0075(1.0075)^{180}}{1.0075^ {180}-1} \;=\;507.1332921$Therefore, the monthly payment is: .$\displaystyle \boxed{\$507.13}$ 4. ## Re : Very big thank you to both of you !!! Soroban , just out of curioscity , how did the formula come or perhaps where did it come from ? 5. Originally Posted by mathaddict Very big thank you to both of you !!! Soroban , just out of curioscity , how did the formula come or perhaps where did it come from ? Check this and especially this out. It is one way of deriving it. There are or course other ways. Go through page 70 up to page 99. Or you can just order this book if get tired of reading it online. 6. ## Repayments In fact, the interest is calculated at the start of each year for the whole year, as if the whole amount is owing for the whole year. So you have to use the values P = 50000 i = 0.09 n = 15 to get the total payment each year, and then divide by 12, to get a monthly payment of 516.91. Incidentally, I don't think it's really that hard to work out the formula for yourself. After $\displaystyle n$ years you'll find that the amount owing is $\displaystyle P(1+i)^n - [1 + (1+i) + (1+i)^2 + ... + (1+i)^{n-1}]x$, where $\displaystyle x$ is the annual payment Inside the square brackets, it's a GP whose sum is $\displaystyle \frac{(1+i)^n - 1}{i}$ Set this sum equal to zero and solve for $\displaystyle x$ to obtain the formula. 7. Thanks all but i am having some problems with the series . [tex] [1 + (1+i) + (1+i)^2 + ... + (1+i)^{n-1}]x[/math $\displaystyle T_1=x$ $\displaystyle T_2=xi+x=x(1+i)$ $\displaystyle T_3=[x(1+i)]\cdot{i}+x$ but this is not equal to $\displaystyle x(1+i)^2$ Why is it so ?? 8. ## GP Originally Posted by mathaddict Thanks all but i am having some problems with the series . [tex] [1 + (1+i) + (1+i)^2 + ... + (1+i)^{n-1}]x[/math $\displaystyle T_1=x$ $\displaystyle T_2=xi+x=x(1+i)$ $\displaystyle T_3=[x(1+i)]\cdot{i}+x$ but this is not equal to $\displaystyle x(1+i)^2$ Why is it so ?? Set your working out like this: At the end of year 1, the amount owed is the original sum borrowed + the interest for the year - the total payments during the year $\displaystyle = P + Pi - x$ $\displaystyle = P(1 + i) - x$ This, then is the amount owing at the start of year 2: the 'new $\displaystyle P$' if you like. So at the end of year 2, we'll replace $\displaystyle P$ by $\displaystyle P(1+i) - x$, to get that the amount owing is: $\displaystyle [P(1 +i) - x](1+i) - x$ $\displaystyle = P(1+i)^2 - x[1 + (1 + i)]$ Continue like this, multiplying by $\displaystyle (1+i)$ and subtracting $\displaystyle x$ each time. So at the end of the year 3, the amount owing is: $\displaystyle P(1+i)^3 - x[1 + (1 + i) + (1 +i)^2]$ And, in a similar way, at the end of year $\displaystyle n$, it is: $\displaystyle P(1+i)^n - x[1 + (1 + i) + ...+ (1 +i)^{n-1}]$ So, inside the square brackets, [...], we have a GP whose first term is $\displaystyle 1$, common ratio $\displaystyle (1+i)$ and number of terms = $\displaystyle n$, and whose sum is $\displaystyle \frac{(1+i)^n - 1}{(1+i) - 1}= \frac{(1+i)^n - 1}{i}$. Now if the whole sum is paid of after $\displaystyle n$ years, this total is zero. I.e. $\displaystyle P(1+i)^n - x\frac{(1+i)^n - 1}{i} = 0$ Hence $\displaystyle x = \frac{iP(1+i)^n}{(1+i)^n - 1}$ OK now?	2018-03-25T05:42:55	{ "domain": "mathhelpforum.com", "url": "http://mathhelpforum.com/algebra/70782-geometric-series-problem.html", "openwebmath_score": 0.9452170133590698, "openwebmath_perplexity": 1966.3116111501001, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9759464429079201, "lm_q2_score": 0.885631470799559, "lm_q1q2_score": 0.8643288836541391 }
https://math.stackexchange.com/questions/868400/showing-that-y-has-a-uniform-distribution-if-y-fx-where-f-is-the-cdf-of-contin/868405	# Showing that Y has a uniform distribution if Y=F(X) where F is the cdf of continuous X Let $$X$$ be a random variable with a continuous and strictly increasing c.d.f. $$F$$ (so that the quantile function $$F^{−1}$$ is well-deﬁned). Deﬁne a new random variable $$Y$$ by $$Y = F(X)$$. Show that $$Y$$ follows a uniform distribution on the interval $$[0, 1]$$. My initial thought is that $$Y$$ is distributed on the interval $$[0,1]$$ because this is the range of $$F$$. But how do you show that it is uniform? • This is not true in cases where there's a discrete component. For example, suppose $X=\left.\begin{cases} 1/2 & \text{with probability }1/2, \\ W & \text{with probability }1/2,\end{cases}\right\}$ and $W$ is uniformly distributed on $[0,1]$, and that the choice between whether $X=1/2$ or not is independent of $W$. Then the cdf of $X$ has no values between $1/4$ and $3/4$, so it cannot be uniformly distributed on $[0,1]$. It is, however, true of continuous distributions. Jul 15, 2014 at 21:41 • see the text of the question. X is continuous! Jul 15, 2014 at 21:44 • By the way, it is not necessary that $F$ is a strictly increasing CDF, continuity is sufficient. Just define the quantile function the usual way as a generalized inverse via $F^-(y)=inf\{x\in\mathbb{R}: F(x)\geq y\}$. See the proof of Proposition 3.1 in Embrechts, P., Hofert, M.: A note on generalized inverses. Mathematical Methods of Operations Research 77(3), 423-432 link for a very careful and detailed explanation. Jul 16, 2014 at 14:38 • Thanks @binkyhorse - that reference is really good. Apr 5, 2015 at 22:33 • @s0ulr3aper07 By Proposition 3.1 in the paper I linked above, yes. Prop. 3.1: Let F be a distribution function and X ~ F. (a) If F is continuous, F(X)∼U[0,1]. The paper includes a detailed proof. Mar 27, 2019 at 20:10 Let $F_Y(y)$ be the CDF of $Y = F(X)$. Then, for any $y \in [0,1]$ we have: $F_Y(y) = \Pr[Y \le y] = \Pr[F(X) \le y] = \Pr[X \le F^{-1}(y)] = F(F^{-1}(y)) = y$. What distribution has this CDF? • Are all CDF's of continuous densities invertible? Aug 22, 2017 at 7:43 • @tintinthong: not always completely, but enough. If you define $G(y)= \inf\{x:F_X(x) \ge y\}$ then $F_X(G(y))=y$ when $y \in (0,1)$ Aug 22, 2017 at 22:12 • Strictly increasing and continuous CDF is needed. Feb 20, 2019 at 15:36 $$Prob(Y\leq x)=P(F(X)\leq x)=P(X\leq F^{-1}(x))=x \\$$ The last equality is from the definition of the quantile function. Let $y=g(x)$ be a mapping of the random variable $x$ distributed according to $f(x)$. In the mapping $y=g(x)$ you preserve the condition of probability density (namely you counts the same number of events in the respective bins) $$h(y)dy=f(x)dx$$ where h(y) is the probability distribution of $y$ if $h(y)=1$ (uniform distribution) we have $$dy=g'(x)dx=f(x)dx$$ This means that $$g(x)=\int f(x)dx$$ namely the function $g(x)$ that maps the random variable $x$ distributed according $f(x)$, into a random variable $y$ distributed uniformly is his own cumulative distribution function $\int f(x) dx$. Here is an approach that does not use the quantile function whatsoever - the only property used is that independent copies of $$X$$ have zero probability of being equal. (The main ingredient in my argument is conditional expectation.) Consider the cumulative distribution function of $$X$$, namely $$F(t)=\mathbb P(X\leq t).$$ Your random variable - which I will suggestively call $$U$$ instead of $$Y$$ - can be described by starting with two independent and identically distributed random variables $$X,Z$$ and considering the conditional probability $$U=\mathbb P(X\leq Z\mid Z).$$ Then, for all integers $$n\geq 1$$, we can represent $$U^n$$ as follows. Let $$X_1,X_2,\ldots,X_n,Z$$ be independent and identically distributed. By independence, $$\mathbb P\bigl(X_1\leq Z,X_2\leq Z,\ldots, X_n\leq Z\bigm\vert Z\bigr)=U^n,$$ and thus by the tower property $$\mathbb EU^n=\mathbb P(X_1\leq Z,X_2\leq Z,\ldots, X_n\leq Z)=\mathbb P\bigl(Z=\max(X_1,X_2,\ldots,X_n,Z)\bigr).$$ Since $$X_1,\ldots,X_n,Z$$ are iid, each of them is equally likely to be the maximum and therefore $$\mathbb EU^n=\frac{1}{n+1}.$$ Thus $$U$$ has the same moments as a uniformly distributed random variable on $$[0,1]$$. Since $$U$$ is supported in $$[0,1]$$ as well, it follows (by the uniqueness of the Hausdorff moment problem) that $$U$$ is uniformly distributed, as desired. Let $$y\in(0,1)$$. Since $$F$$ is continuous, there exists $$x\in\mathbb{R}$$ s.t. $$F(x)=y$$. Thus, $$\mathsf{P}(Y\le y)=\mathsf{P}(F(X)\le F(x))=F(x)=y,$$ i.e., $$Y\sim\text{U}[0,1]$$. In order to see the first equality we don't need continuity. Specifically, since any cdf is right-continuous, \begin{align} \{F(X)\le F(x)\}&=\{\{F(X)\le F(x)\}\cap\{X\le x\}\}\cup \{\{F(X)\le F(x)\}\cap\{X>x\}\} \\ &=\{X\le x\}\cup \{\{F(X)=F(x)\}\cap\{X>x\}\}, \end{align} and $$\mathsf{P}(\{F(X)=F(x)\}\cap\{X>x\})=0$$. For a proof of this problem when $$F_X(x)$$ is strictly increasing, refer to JimmyK4542's answer. Let's assume $$F_X(x)$$ is just non-decreasing (there are intervals such as $$[a,b]$$ where $$F_X(x') = c$$ for $$x'\in[a,b]$$). We define $$G(y)$$ similar to what Henry's comment suggests: $$G(y)=\inf\{x:F_X(x)\gt y\}$$ Now substituting this expression in what Jimmy has written will give us: $$F_Y(y) = \Pr[Y \le y] = \Pr[F_X(X) \le y] = \Pr[X \le G(y)] = F_X(G(y))= y \label{eq:I}\tag{I}$$ We need to show that: 1. $$F_X(x)\le y \rightarrow x \le G(y)$$ 2. $$F_X(G(y))=y$$ The second argument is easier to prove. We have the following expression almost according to definitions: $$F_X(G(y))= F_X(\inf\{x:F_X(x)\gt y\})= y$$ Now for the first argument, we can still use what $$G(y)=\inf\{\cdots\}$$ implies; if $$F_X(x)\le y$$, then $$x\le \inf\{x:F_X(x)\gt y\};$$ hence $$x\le G(y)$$. With the two arguments proved and a substitution in \ref{eq:I}, we have proved the main argument. • I personally believe this problem is a severe case for abuse of notation, and a bad professor's problem. Feb 19 at 1:46	2022-06-30T14:40:05	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/868400/showing-that-y-has-a-uniform-distribution-if-y-fx-where-f-is-the-cdf-of-contin/868405", "openwebmath_score": 0.957176685333252, "openwebmath_perplexity": 174.14235605365812, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9759464443071381, "lm_q2_score": 0.8856314617436727, "lm_q1q2_score": 0.8643288760552705 }
https://web2.0calc.com/questions/what-is-the-probability-of-winning-a-lottery-which-requires-selection-of-5-numbers-from1-to-40	+0 # what is the probability of winning a lottery which requires selection of 5 numbers from1 to 40 0 895 7 what is the probability of winning a lottery which requires selection of 5 numbers from1 to 40 May 7, 2014 #2 +17 +8 Calculate the numerator: There is only one winning combination. So the numerator of the desired probability is 1. Calculate the denominator: There are "40 numbers, choose 5", or 40C5 possible combinations. So the denominator of the desired probability is 40C5. So the probability is 1/40C5=1/658008 May 7, 2014 #1 +1006 0 For this kind of probability problem, assuming the number is removed from play once chosen, the probability increases by one each time. $$\left({\frac{{\mathtt{1}}}{{\mathtt{40}}}}\right){\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{39}}}}\right){\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{38}}}}\right){\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{37}}}}\right){\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{36}}}}\right)$$ The bottom five numbers multiply to 78960960, which means that the odds are 1/78960960 May 7, 2014 #2 +17 +8 Calculate the numerator: There is only one winning combination. So the numerator of the desired probability is 1. Calculate the denominator: There are "40 numbers, choose 5", or 40C5 possible combinations. So the denominator of the desired probability is 40C5. So the probability is 1/40C5=1/658008 FoxTears May 7, 2014 #3 +17 0 Mine was assuming the numbers can be reused May 7, 2014 #4 +115418 +5 The number of 5 digit combinations (order doesn't matter) from 40 is 40C5 $${\left({\frac{{\mathtt{40}}{!}}{{\mathtt{5}}{!}{\mathtt{\,\times\,}}({\mathtt{40}}{\mathtt{\,-\,}}{\mathtt{5}}){!}}}\right)} = {\mathtt{658\,008}}$$ only one of those wins so the chance is $$\frac{1}{658008}$$ Just like FoxTears said! Thanks Foxtears. Foxtears, I don't think that yours was assuming that the numbers could be reused. It was just assuming that the order they were chosen in was of no consequence. GoldenLeaf I believe yours would have been correct if the numbers had to be chosen in a specific order. May 7, 2014 #5 +32772 +5 GoldenLeaf's approach was almost right, but he should have had 5, 4, 3, 2 and 1 in the numerators. There are 5 possible choices for the first number, so probability of this is 5/40. For each of these there are 4 possible choices for the 2nd number, so probability of both 1st and 2nd is (5/40)(4/39). For each of these ... etc. to get the overall probability as $${\mathtt{p}} = \left({\frac{{\mathtt{5}}}{{\mathtt{40}}}}\right){\mathtt{\,\times\,}}\left({\frac{{\mathtt{4}}}{{\mathtt{39}}}}\right){\mathtt{\,\times\,}}\left({\frac{{\mathtt{3}}}{{\mathtt{38}}}}\right){\mathtt{\,\times\,}}\left({\frac{{\mathtt{2}}}{{\mathtt{37}}}}\right){\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{36}}}}\right) \Rightarrow {\mathtt{p}} = {\mathtt{0.000\: \!001\: \!519\: \!738\: \!361\: \!8}}$$ or p≈1.5210-6 1/p = 658008 May 7, 2014 #6 +115418 0 Yes of course - that did not occur to me. Thanks Alan. May 7, 2014 #7 0 Calling Bertie, CPhill and Rom (Troll alert! General quarters!) Along with Melody and Alan, we need input from Bertie, CPhill, and Rom, too. With the top brains represented here that Troll might comeback and bite one or more. I know he’ll bite FoxTears for saying, “Mine was assuming the numbers can be reused” (your formula (40C5) contradicts this). The Troll will follow-up, by asking what is the probability of (something) of (five, six or seven) mathematicians getting a probability question correct. Then he’ll say about the same as walking, blindfolded, through a large buffalo heard and stepping in four piles with both feet. What’s annoying: he’s usually right. CPhill might not be able to make it. He might be trying to get the “Roman Zero” from inside Sisyphus’s boulder. Maybe he could take a break and visit. He’s not been around for a day or two. I hope he wasn’t squashed by the boulder. No matter: cartoon physics seem to apply in that realm. If he does visit, maybe one of those angels will help him get the “Roman Zero.” By: Troll Detector May 7, 2014	2021-12-04T23:52:48	{ "domain": "0calc.com", "url": "https://web2.0calc.com/questions/what-is-the-probability-of-winning-a-lottery-which-requires-selection-of-5-numbers-from1-to-40", "openwebmath_score": 0.9447340369224548, "openwebmath_perplexity": 1798.8861039896221, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9888419690807408, "lm_q2_score": 0.8740772236840656, "lm_q1q2_score": 0.8643242429963786 }
http://amicidellacattolica.it/rmev/sigma-notation-rules.html	Sigma, Σ, is the standard notation for writing long sums. 3n 3 + 6n 2 + 6000 = Θ(n 3). ” - Sincerely, Dr. Commented: Roger Stafford on 22 Oct 2014 Accepted Answer: Star Strider. It's based on the upper case Greek letter S, which indicates a sum. Some basic ideas can be applied to creating process maps that make them easier to understand and use. Clears 1, 5, and 12, and if you remember the old notation the cardinality of Z is 3 because there are three elements in z. See Figure 3. So for this series, there is 7 terms The seven terms are: when n=0 -2x. 2040 - 3 sig fig 2. Okay, welcome back everyone. Most of the following problems are average. "My TI 83+ graphing calculator arrived faster than I expected. Although it can appear scary if you've never seen it before, it's actually not very difficult. Sigma Notation. This is the currently selected item. Define Sigma notation. In chemistry, a molecular orbital (MO) is a mathematical function describing the wave-like behavior of an electron in a molecule. Free online converters. APPENDIX E SIGMA NOTATION A35 Theorem If is any constant (that is, it does not depend on ), then (a) (b) (c) Proof To see why these rules are true, all we have to do is write both sides in expanded form. The variable n is called the index of summation. When WeBWorK gives a typeset version of your answer it only uses parentheses so for example it expresses your input of 2 [3 (4+5)+6] as 2 (3 (4+5)+6) but you can use whatever you want. Multiplication by a whole number can be interpreted as successive addition. Example ∑ = = + + + +. 00, but has features that will be used throughout algebra, trigonometry, and statistics. To see why Rule 1 is true, let's start with the left hand side of this equation, n i=1 cx i. 5 Sigma Notation And The Nth Term Ppt Video Online Download Index Of Wp Content Uploads 2017 02 Summation Notation Problems & Solutions Sigma Notation Sigma Worksheet 7 Probability And Sigma Notation Solutions(1) Maths Sigma Notation Guided Notesmath With Mrs Holst Tpt Document 20. Rules of Scientific Notation. These rules can be converted and applied to many log management or SIEM systems and can even be used with grep on the command line. If the series is multiplied by a. This involves the Greek letter sigma, Σ. The "a i " in the above sigma notation is saying that you sum all of the values of "a". In this unit we look at ways of using sigma notation, and establish some useful rules. The Organic Chemistry Tutor 270,467 views. With sequences, sometimes we like to start at n = 0 and sometimes we like to start at n = 1. We often use Sigma Notation for infinite series. Properties. For example, the Schrodinger equation, which has to do with dynamics in quantum systems and predates quantum computation by decades, is written using bra-ket notation. The Example shows (at least for the special case where one random variable takes only. The Sigma symbol can be used all by itself to represent a generic sum… the general idea of a. When you're calculating the big O. Use algebra rules of sums, along with the appropriate sum formulae, to evaluate the following: (a) X12 i=1 (i2 33i) (b) X10 i=1 (i2 2 1) (c) X8 i=1 (2 i ) 6. 2 Summation notation Summation notation. Control Chart with all Control Limits Visible QI Macros offers the following option to turn on and off the 1-2 sigma lines: After you have run a chart, click on the QI Macros Chart menu and select Show/Hide Sigma Lines. Go To Problems & Solutions Return To Top Of Page. Use sigma (summation) notation to calculate sums and powers of integers. Sigma questions are not rehashing of questions from a text book, each question is unique and has a slight twist. 3 – A service desk process map example. Sigma notation provides a way to compactly and precisely express any sum, that is, a sequence of things that are all to be added together. Today we're going to make it a little bit more complicated, and we're going to go over some rules, For manipulating, Slash simplifying, Or making for complicated, if you like, sigma notation. Sigma Notation - Introduction to Summation 9:14. Write the expression 1 + 1 4 + 1 7 + 1 10 + + 1 3n+1 in P notation. These rules that make sense in simpler notation, such as or. Free algebra and math word problems. A business process model and notation diagram, or BPMN diagram for short, is used to build easy-to-read business process model flowcharts, which can be shared across organizations and industries. The concept of sigma notation means to sum up all terms and uses three parts to form math statements, like ∑ i a i. org are unblocked. In this unit we look at ways of using sigma notation, and establish some useful rules. A typical element of the sequence which is being summed appears to the right of the summation sign. Sigma Notation and Riemann Sums Sigma Notation: Notation and Interpretation of 12 3 14 1 n k nn k aaaaa a a (capital Greek sigma, corresponds to the letter S) indicates that we are to sum numbers of the form indicated by the general term. For example: 1. The Riemann sums and sigma notation exercise appears under the Integral calculus Math Mission. Binomial theorem refresher. Note how the subscripts in the FORTRAN example below exactly match the tensor notation for $$C_{ij} = A_{ik} B_{kj}$$. Posted 2/14/10 8:18 AM, 19 messages. Example of a Matrix. The sum of a series can be written in sigma notation. ) (Factor out the number 6 in the second summation. The example below shows three separate sets of control limits: Don't forget to rerun stability. ” D = { x: x is a river in a state} We should describe a certain property which all the elements. In this unit we look at ways of using sigma notation, and establish some useful rules. The resulting display string will be a string which has already been mapped according to the IDNA2003 rules. If you're seeing this message, it means we're having trouble loading external resources on our website. Contents 1 Algebra 8 1. As well as providing shorthand for mathematical ideas, this notation can aid students’ understanding of mathematics. Sigma notation. If , the series does not converge (it is a divergent series). If you want to use a different calculator, the TI-89 has an actual sigma (uses the greek sigma) that you enter the expression, variable, start, and stop values. Theorem: Sigma Notation Properties Suppose and are functions of , is any integer, and is any real number. It's based on the upper case Greek letter S, which indicates a sum. Question 486636: Expand and simplify. This calculus video tutorial provides a basic introduction into summation formulas and sigma notation. when n=2 6-2x. Summation rules Jochumzen. If x ≤ μ, then the pdf is undefined. Multiplication by a whole number can be interpreted as successive addition. In mathematical formula it is indeed the addition of many number or variable which represent to give concise expression for sum of the variable as Sigma or Summation. In Notes x4. Looking for Pi notation? Find out information about Pi notation. A series that converges has a finite limit, that is a number that is approached. The variable k is called the index of the sum. Okay, now that we've seen that fancy sigma notation let's work a really simple small example of numbers and then generalize. Geometric Series 1 - Cool Math has free online cool math lessons, cool math games and fun math activities. 106L Labs: Sums and Sigma ( ) Notation Sequences, Sums, and Sigma ( ) Notation Sequences De nition A sequence is an ordered set of numbers de ned by some rule. Similarly, we can use sigma notation starting at different values of n to write the same series. Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10. This is true for all tensor notation operations, not just this matrix dot product. 1, we de ne the integral R b a f(x)dx as a limit of approximations. Use Fault Tree diagrams to document business processes, including Six Sigma and ISO 9000. These tell us the starting. Digits 1 to 9 always count 2. But with sigma notation (sigma is the 18th letter of the Greek alphabet), the sum is much more condensed and efficient, and you’ve got to admit it looks pretty cool: This notation just tells you to plug 1 in for the i in 5i, then plug 2 into the i in 5i, then 3, then 4, and so on all the way up to 100. In this unit rules for using sigma notation are established. Re: sigma notation (without lists or tables) mravi1 welcome to the forum As pointed out, unfortunately your post does not comply with Rule 2 of our Forum RULES. Sigma notation is essentially a shortcut way to show addition of series or sequences of numbers. For example, we often wish to sum a number of terms such as 1+2+3+4+5 or 1+4+9+16+25+36 where there is an obvious pattern to the numbers involved. Example ∑ = = + + + +. After multiplying everything out I got, = E (i^3 + 15i^2 +54i) Then I broke everything down to = E i^3 + (15)E i^2 + (54)E i. In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the precision with which the values for certain pairs of physical quantities of a particle, such as position, x, and momentum, p, can be predicted from initial conditions. And S stands for S um. $$S^2\sigma$$ is frequently over analyzed to the point where it is incorrectly concluded that a high $$S^2\sigma$$ is preferable even at the expense of lower $$zT$$. is the symbol for the sample mean. Alternatively, we may use ellipses to write this as However, there is an even more powerful shorthand for sums known as sigma notation. Sigma Notation Properties/Rules. We prefer to use subscripts to indicate the control limits, but other texts and papers may use somewhat different notation (e. We often use Sigma Notation for infinite series. If you are in need of technical support, have a question about advertising opportunities, or have a general question, please contact us by phone or submit a message through the form below. LuDaCriS66 Platinum Member. Summation Notation. 5, which dealt with vector coordinate transformations. What we are about to do is to take a function and express it as the limit of a sequence of Riemann Sums over. Enter the sequence, the start value and end value from sigma notation and get a numerical sum. The matrix pictured below has two rows and three columns. (b) c k = 1 k, where 5 k < 9. Recall that the "n" on top of the Sigma (the funny looking e) is the terminal value for the index which is located under the sigma. Use sigma (summation) notation to calculate sums and powers of integers. The notation above is not too flexible because the start, stop, and term are all hard coded. The series always converges when x = 0 It will possibly converge for. The act or process of adding; addition. But polynomials truly come to shine in the work of Al-Samawal (1130-1180) The Shining Book on Calculation. Here, we prefer to set up the prior in terms of nu, mu, sigma/(nu-2) or something like that, to account for the fact that the scale of the distribution (as measured by the sd or median absolute deviation) depends on nu as well as sigma. So sum of n=1 to 20 for (3^n) + sum of n=1 to 20 for (2n-1) Then apply the geometric sum formula for the sum of n=1 to 20 for (3^n). We do this through a mixture of our automated migration tools and professional services offerings. It corresponds to “S” in our alphabet, and is used in mathematics to describe “summation”, the addition or sum of a bunch of terms (think of the starting sound of the word “sum”: Sssigma = Sssum). Presentación NEARPOD. Rules for use with sigma notation 6 1 c mathcentre July 18, 2005. The terminology from AMS-LaTeX documentation. So maybe it will help. mc-TY-sigma-2009-1 Sigma notation is a method used to write out a long sum in a concise way. In 1981 Dr. 1 (800) 561 5005 (Toll-free) or +1 (514) 737 7333 [email protected] Usually, this is used to describe a certain span or group of spans of numbers along a axis, such as an x-axis. A boolean expression consisting purely of Minterms (product terms) is said to be in canonical sum of products form. Summation notation In statistics we take summary measures of data sets. BPMN diagram symbols are categorized into four main groups: flow objects, connecting objects, swimlanes, and artifacts. Note: you can do this more than once on a chart. For example, cosx, cos x, and cos (x) are all equivalent. Loading Unsubscribe from Jochumzen? Summation Formulas and Sigma Notation - Calculus - Duration: 20:24. Gateways are used to control how the process flows. American Mathematical Society, User's Guide for the amsmath Package For all intents and purposes, the align environment is a replacement for the eqnarray environment and all its warts. This problem has been solved! See the answer. As we learn the basic integration rules, we will learn the difference between an indefinite integral and a definite integral. 'for' loop or symsum? Follow 163 views (last 30 days) Nate on 20 Oct 2014. Taylor series, convergence tests, power series convergence, sigma notation, factorial this page updated 19-jul-17 Mathwords: Terms and Formulas from Algebra I to Calculus. My matrix is the variable A, and my list is the variable Network. The precise addition rule to use is. Summation notation In statistics we take summary measures of data sets. Today we're going to make it a little bit more complicated, and we're going to go over some rules, For manipulating, Slash simplifying, Or making for complicated, if you like, sigma notation. Riemann sums, summation notation, and definite integral notation. Sigma notation provides a way to compactly and precisely express any sum, that is, a sequence of things that are all to be added together. com To create your new password, just click the link in the email we sent you. - Notice that we are adding fractions with a numerator of 1 and. 1) sigma notation: n=0; top of sigma = 4; to the right - (n^2/2) 2) sigma notation: n=1; top of sigma = 3; to the right - (1/n!). Binomial theorem refresher. These degenerate cases are usually only used when the summation notation gives a degenerate result in a special case. I can pull that constant outside and get that the sum from i=1 to 4 of 3i squared is the same thing as 3 times the same exact. That is indicated by the lower index of the letter. 1 - In the following exercises, use the rules for sums Ch. mc-TY-sigma-2009-1 Sigma notation is a method used to write out a long sum in a concise way. lar notation scheme can mystify the reader just as easily as a too casual (μ)forapopulationmean,sigma Now let us review the mathematical rules by which matrices are manipulated. Commented: Roger Stafford on 22 Oct 2014 Accepted Answer: Star Strider. Properties. Worksheet on sigma notation 1. These properties are easy to prove if we can write out the sums without the sigma notation. Summation notation is a useful way to represent the partial sum of a sequence. For these accounting rules, we describe the range of valuations that is consistent with the firm’s financial statements. 1 – Area, Distance, and Sigma Notation The Area Problem Our goal is to calculate the area of some region. Addition rules are important in probability. improve this answer. The double series. It corresponds to “S” in our alphabet, and is used in mathematics to describe “summation”, the addition or sum of a bunch of terms (think of the starting sound of the word “sum”: Sssigma = Sssum). Viewing 8 posts - 1 through 8 (of 8 total) Author. Constant multiplier. Note: you can do this more than once on a chart. Summation notation works according to the following rules. For more ways to implement derivatives, you may find our support article on Prime Notation helpful. Title: SUMMATION or SIGMA NOTATION 1 SUMMATION or SIGMA NOTATION 2 DEFINITIONS Sigma Notation for Finite Sums. These are called mathematical operators. Adding series. With a Six Sigma process even a significant shift in the process mean results in very few defects. For example, we can write which is a bit tedious. Six Sigma Diagram. Contents[show] Table of sum-class symbols Using sum TeX is smart enough to only show. In this unit rules for using sigma notation are established. Σ is the symbol for ‘the sum. $$\sum\limits_{i\, = \,{i_{\,0}}}^n {c{a_i}} = c\sum\limits_{i\, = \,{i_{\,0}}}^n {{a_i}}$$ where $$c$$ is any number. The most frequently appearing notation in the mathematical descriptions of different quantities or procedures used in data analysis involves the application of the summation "operator," represented by the upper-case Greek letter sigma, or. The number 100,000,000 for example, takes up a lot of room and takes time to write out, while 10 8 is much more efficient. 1 Sigma Notation ¶ Sigma notation is a convenient way to write large and complicated sums. Sigma Notation - Simplification Rules. Thus, Also the initial value doesn't have to be 1. Introduction Sigma notation is a concise and convenient way to represent long sums. Given a matrix M and a vector v, when we work out the ﬁrst component of Mv, we dot the ﬁrst row of M with v. You get to it through the calculus functions (f3 key in "normal" home). 00204 - 3 sig fig. If you need a quick refresher on summation notation see the review of summation notation in the Calculus I notes. The act or process of adding; addition. Each item in a matrix is called an entry. Binomial theorem refresher. Now notice in this Service Desk process map template, how in addition to the items mentioned above, there is even more processes and messages. The double series. Molecular Weight 251. Fractional Indices. (b) Write the sum X7 k=1 (2k+ 7) in expanded form. Go to The Equation of a Circle page. , F = 1 when any one of the input is true or 1. In either case, the images of the basis vectors form a parallelogram that represents the image of the unit square under the. 1 Steve Strand and Sean Larsen from Portland State University, US, have shown that, cognitively, the task of interpreting a given summation-notation expression differs significantly from the task of converting a longhand sum into summation notation. Greetings! I wanted to let you know that I am now offering a prorated DeltaMath Plus subscription for $20 that will last until October 1, 2020. Shows how factorials and powers of -1 can come into play. Formula for percentage. Join 90 million happy users! Sign Up free of charge:. The number on top of the summation sign tells you the last number to plug into the given expression. answered Dec 10 '13 at 23:12. Both formulas have a mathematical symbol that tells us how to make the calculations. The series 4+ 8+12 +16+20 +24 can be expressed as 6 ∑ n=14n. Consider the series. Since sigma notation is just a series of addition, you can write out each term in the series then add them all together. In general, capital letters refer to population attributes (i. TQM Diagram. We can also write N+ = {x ∈ N : x. If that region is a polygon (a shape made only of straight sides) the process is straightforward. If we like, we can go back to calling our summation index k,. Free Summation Calculator. Select a link by mouse clicking one of the music notation symbols below. The use of matrix (lin-ear) algebra can greatly simplify many of the computations. Rules of Scientific Notation. 1 Sigma Notation Welcome to Week 1 of the online version of MATH 232 for Spring 2020. Contents 1 Algebra 8 1. But many programmers don't really have a good grasp of what the notation actually means. Taylor series, convergence tests, power series convergence, sigma notation, factorial this page updated 19-jul-17 Mathwords: Terms and Formulas from Algebra I to Calculus. Math 132 Sigma Notation Stewart x4. Write out completely the sequences given by the following rules: (a) a i = i2, where 0 i 5. Zeros in front never count 3. Sigma notation of the polynomial Coefficients of the source polynomial in the form of a recursive formula Coefficients of the source polynomial function are related to its derivative at x 0 The general form or translatable form of the polynomial. 1 Sigma Notation ¶ Sigma notation is a convenient way to write large and complicated sums. You may also want to use "Post-It" notes to list possible causes but have the ability to re-arrange the notes as the diagram develops. By using this website, you agree to our Cookie Policy. An index, or a power, is the small floating number that goes next to a number or letter. Ellipses save space or remove material that is less relevant. Learn how to evaluate sums written this way. The purpose of the above sigma formula is the sum of all X data starting from the order of 1 to n. 5 Sigma Notation And The Nth Term Ppt Video Online Download Index Of Wp Content Uploads 2017 02 Summation Notation Problems & Solutions Sigma Notation Sigma Worksheet 7 Probability And Sigma Notation Solutions(1) Maths Sigma Notation Guided Notesmath With Mrs Holst Tpt Document 20. Sigma 4 Shogi Introduction. adding & subtracting. 13 Coordinate Transformation of Tensor Components This section generalises the results of §1. If we like, we can go back to calling our summation index k,. In this unit we look at ways of using sigma notation, and establish some useful rules. Indices show how many times a number or letter has. Find the value of. The dummy variable will usually show up one or more times in the expression to the right of the Greek letter sigma. The above selection rules apply only for the Electric Dipole (E1) approximation. Contents[show] Table of sum-class symbols Using sum TeX is smart enough to only show. As before, let's think about writing out these sums explicitly, without sigma notation. If we like, we can go back to calling our summation index k,. 1: Summation Notation And Formulas. The variable is called the index of the sum. (a) Write the sum X10 k=4 (2k+ 1) in expanded form. in summation notation. Now back to series. For example: This means that we are to repeatedly add ka k. So for this series, there is 7 terms The seven terms are: when n=0 -2x. Rules - note that # means a non-zero digit (123456789) 1. The numbers at the top and bottom. A key to Business Process Management, it visually depicts a detailed sequence of business activities and information flows needed to complete a process. Note: you can do this more than once on a chart. If I took it out of scientific notation, it would be 800000. In calculus, summation notation or sigma (Σ) represents adding many values together. Sigma is the greek sign for "Sum". It indicates a series. Today you can define mental math in various different ways. We do this through a mixture of our automated migration tools and professional services offerings. That is, we split the interval x 2[a;b] into n increments of size. The act or process of adding; addition. and in terms of the sigma notation When two random variables are independent, so that. For example, the Schrodinger equation, which has to do with dynamics in quantum systems and predates quantum computation by decades, is written using bra-ket notation. This, too, turns out to come down to some other properties of addition. Given any region R, then the area, A(R) is a real number; Â Â Â Â Â A(R) Â³ 0. (c) The above two parts show that a sum can be written in notation in di erent ways. In other words, your're adding up a series of a values: a 1, a 2, a 3 …a x. If you are in need of technical support, have a question about advertising opportunities, or have a general question, please contact us by phone or submit a message through the form below. sp 3 HYBRIDIZATION - EXAMPLE 1) Methane (CH 4) * During the formation of methane molecule, the carbon atom undergoes sp 3 hybridization in the excited state by mixing one ‘2s’ and three 2p orbitals to furnish four half filled sp 3 hybrid orbitals, which are oriented in tetrahedral symmetry in space around the carbon atom. Really clear math lessons (pre-algebra, algebra, precalculus), cool math games, online graphing calculators, geometry art, fractals, polyhedra, parents and teachers areas too. using interval notation to show intervals of increasing and decreasing and postive and negative Answers · 1 trouble spots for the domain may occur where the denominator is ? or where the expression under a square root symbol is negative. Now apply Rule 1 to the first summation and Rule 2 to the second summation. Now, just add all these terms together and set them equal. Okay, welcome back everyone. As well as providing shorthand for mathematical ideas, this notation can aid students' understanding of mathematics. This is called Sigma notation. Practice: Summation notation. In the following, $$k$$ is the index. Identities involving double sums include the following: is the floor function, and. de, the Punycode value actually used for the domain names on the wire is xn--bcher-kva. We first give proof of the Halting Problem by constructing a machine capable of solving the Halting Problem and then showing that if such a machine existed it would generate a. For example, Xn i=1 axi = ax1 +ax2 + +axn = a(x1 +x2 + +xn) = a Xn i=1 xi: In other words, you can take a constant \out of the summation". On a single line calculator, a -1 requires the button order be: 1 -. 1, we often encountered sums with many terms (up to 1000 in Table 5. 1 For a Scalar Equation All subscripts must be dealt with in each term. Sigma notation is a method used to write out a long sum in a concise way. These are equal to the value of the variable, 'm' in this case, taken in order. From the rules above, Common series - Sigma notation. Okay, welcome back everyone. Sum and Product Notation: Delimited Forms The capital Greek letter Σ (sigma) denotes summation. AS-Level Maths Questions by topic. What is the sum of all integers between 17 and 325 that are NOT divisible by either. O (2 n) denotes an algorithm whose growth doubles with each addition to the input data set. This exercise formally explores the Riemann sum and practices sigma notation. Homework Statement I'm actually asked to calculate the area under the curve 5x + x 2 over the interval [0,1] using Riemann Sums. ' As such, the expression refers to the sum of all the terms, x n where n represents the values from 1 to k. 2 Sigma Notation & Limits of Finite Sums NOTES SIGMA NOTATION a k =a 1 +a 2 +a 3 +!+a n−1 +a n k=1 n ∑ EX 1) Complete the table, given the following sums in sigma notation: The Sum in Sigma Notation The Sum Written Out, One Term for Each Value of k The Value of the Sum k k=1 5 ∑ (−1) k k k=1 3 ∑ k k=1 k+1 2 ∑ k2 k=4 k−1 5 ∑ EX. Examples of process maps: Service Desk. This is the currently selected item. i is the index of summation. What we are about to do is to take a function and express it as the limit of a sequence of Riemann Sums over. It indicates that you must sum the expression to the right of the summation symbol:. Note that index i can be replaced by any other index and the results will be the same. The double series. Sigma Notation - Terms and Indices Sigma notation allows for a great deal of variety and exibility Ex: Write down the sum of the rst 10 odd integers in 3 di erent ways. Many statistical formulas involve adding a series of numbers. Determine the sum of the series written in Sigma Notation. lets say, we have a boolean function F defined on two variables A and B. 5100 x 3300 px. Hi, I need to calculate the following sigma: n=14 Sigma (sqrt(1-2. Search this site. RULES OF INDICES (three videos) Rules of Indices. Write out the sum 5 1 2 k k = ∑ 2. Now back to series. Topics include: ~Set theory, including Venn diagrams ~Properties of the real number line ~Interval notation and algebra with inequalities ~Uses for summation and Sigma notation ~Math on the Cartesian (x,y) plane, slope and distance formulas ~Graphing and describing functions and their inverses on the x-y plane, ~The concept of instantaneous. 1 1 Section 4. They have the following general form XN i=1 x i In the above expression, the i is the summation index, 1 is the start value, N is the stop value. This exercise formally explores the Riemann sum and practices sigma notation. To express: Left end Ch. Math Summation Notation. and in terms of the sigma notation When two random variables are independent, so that. That is, we split the interval x 2[a;b] into n increments of size. " - Lorena Colin. The Greek capital sigma, P, is used as a shorthand to denote summation. can be evaluated by interchanging and and averaging, (Borwein et al. Sigma has just announced the 'fp,' the world's smallest full-frame camera and, despite owning the sensor design company Foveon, it's built around a conventional Bayer sensor. If you want to know message flow usage, please see How does BPMN message flow work? article. When preparing for technical interviews in the past, I found myself spending hours crawling the internet putting together the best, average, and worst case complexities for search and sorting algorithms so that I wouldn't be stumped when asked about them. Its 2 line display allows “visually perfect” entries; that is, you can type in the problem exactly as it looks. Multiplication by a whole number can be interpreted as successive addition. Free algebra and math word problems. In this case, the ∑ symbol is the Greek capital letter, Sigma, that corresponds to the letter 'S', and denotes to the first letter in the word 'Sum. Go back to the example I keep using: the stylesheet knows the "frameset" rules. improve this answer. By using this website, you agree to our Cookie Policy. With a Six Sigma process even a significant shift in the process mean results in very few defects. The set can be defined, where possible, by describing the elements. The numbers at the top and bottom of the sigma are called upper and lower bounds, respectively. - Notice that we are adding fractions with a numerator of 1 and. What is the sum of all integers between 17 and 325 that are NOT divisible by either. }\) Sigma notation relies on an index and expresses the terms of the sum as expression of the index. Calculations on the TI-30XIIS The TI-30XIIS calculator costs less than$20. Sigma notation Here is another useful way of representing a series. Math 132 Sigma Notation Stewart x4. Sigma Notation - Simplification Rules 7:24. They are useful in getting right to the point without delay or distraction:. The "numbers" to be summed may be natural numbers, complex numbers, matrices, or still more complicated objects. The first time we write it, we put k = 1. Businesses use the EPC for process mapping in BPMN 2. There are two ways to use invNorm (. as desired. I need to calculate other 18 different sigmas, so if you could give me a solution in general form it would be even easier. n an 1 4 2 −12 3 36. The matrix pictured below has two rows and three columns. So sum of n=1 to 20 for (3^n) + sum of n=1 to 20 for (2n-1) Then apply the geometric sum formula for the sum of n=1 to 20 for (3^n). The double series. Gateways are used to control how the process flows. If this series can converge conditionally; for example, converges conditionally if , and absolutely for. It's often convenient to discuss ^{grammatical rules} by formulating them in a special notation that was introduced about 1960 by John ^{Backus} and Peter ^{Naur}. The Greek capital sigma, P, is used as a shorthand to denote summation. Daniel Egger. In the case of IDN Bücher. Even Numbers (Integers) Odd Numbers (Integers) Divisibility Rules. from numpy import mean. What is the sum of all integers between 30 and 809 that are NOT divisible by either the number 5 or the number 13? You must use Sigma notation and the rules discussed in class to solve this problem and you must show. And internal here to the thing inside the sigma notation was an expression 3i squared that had a constant in it, namely 3. Sigma is an open standard for rules that allow you to describe searches on log data in generic form. , parameters); and lower-case letters refer to sample attributes (i. Regarding the sigma notation, it seems to have been included in a tentative draft of Nomenclature of Organic Chemistry, dating from 1973. 2in} 0 \le p 1; \sigma > 0 \). Contents [ hide] 2 Big Omega (Ω). Welcome to Treblis Software's Music Notation Reference Guide. If you want to know message flow usage, please see How does BPMN message flow work? article. And S stands for S um. Einstein Summation. i is the index of summation. Sigma notation is a method used to write out a long sum in a concise way. If you update to the most recent version of this activity, then your current progress on this activity will be erased. EXAMPLE 1 Examples of Sigma Notation a. An n-dimensional vector eld is described by a one-to-one correspondence between n-numbers and a point. For example, we can write which is a bit tedious. Value Stream Map. 1) Rule one states that if you're summing a constant from i=1 to n, the sum is equal to the constant multiplied by n. The rules for Mastery Quizzes are now different. Sum of first n natural numbers. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. As we learn the basic integration rules, we will learn the difference between an indefinite integral and a definite integral. What is the sum of all integers between 30 and 809 that are NOT divisible by either the number 5 or the number 13? You must use Sigma notation and the rules discussed in class to solve this problem and you must show. There are formulae that can allow us to calculate the sum. An ellipsis (plural: ellipses) is a punctuation mark consisting of three dots. }\) Sigma notation relies on an index and expresses the terms of the sum as expression of the index. TechCalc scientific calculator contains 12 calculation modes in one application + a handy reference section that includes an additional 22 calculation modes plus a wealth of useful information. So for this series, there is 7 terms The seven terms are: when n=0 -2x. If this series can converge conditionally; for example, converges conditionally if , and absolutely for. where μ is the location parameter and σ is the scale parameter. 2 Rules of summation We will prove three rules of summation. Find the value of. 5k/36)) k=1 Basically, I need to find a sum of square-roots where in each individual squareroot the k-value will be substituted by an integer from 1 to 14. They have the following general form XN i=1 x i In the above expression, the i is the summation index, 1 is the start value, N is the stop value. That's good. There are formulae that can allow us to calculate the sum. The terminology from AMS-LaTeX documentation. Sigma notation provides a way to compactly and precisely express any sum, that is, a sequence of things that are all to be added together. A summation is simply the act or process of adding. This guide is intended for users with basic chemistry knowledge and is not written to be the sole source of instruction for this topic. Summation formulas. ” For example, let’s say you had 5 items in a data set: 1,2,5,7,9; you can think of these as x-values. The "i = 1" at the bottom indicates that the summation is to start with X 1 and the 4 at the top indicates that the summation will end with X 4. Sigma notation uses a variable that counts upward to change the terms in the list. However, using parentheses makes. Go back to the example I keep using: the stylesheet knows the "frameset" rules. If 5, 6,…, = áare real numbers, then Í = Ü á Ü @ 5 L = 5 E = 6⋯ E = á RULES FOR WORKING WITH SIGMA NOTATION Theorem: If is any constant, then a Í ? á Ü @ 5 L J⋅ ? b Í⋅ = Ü á Ü @ 5 L ?⋅ Í = Ü. sigma notation, also known as summation notation. Summation Notation. MDL number MFCD00013592. Zeros after a # do not count unless they are also after a decimal place 4. If your instructor gave you a class key, use it to enroll yourself and create your account. 4-sigma quality requires addition of a 4 th rule and implementation of a 1 3s /2 2s /R 4s /4 1s multirole, preferably with 4 control measurements in each run (N=4, R=1), or alternatively, 2 control measurements in each of 2 runs. While often associated exclusively with Six Sigma, Centric leverages DMAIC (Define, Measure, Analyze, Improve, Control) as the underlying business process improvement methodology. The greek letter sigma (σ) is used to indicate syllable, and a bracket with a subscript sigma is used for syllable boundaries. 1, we de ne the integral R b a f(x)dx as a limit of approximations. In chemistry, a molecular orbital (MO) is a mathematical function describing the wave-like behavior of an electron in a molecule. The Crescent is the official publication and voice of Phi Beta Sigma Fraternity, Inc. 00, but has features that will be used throughout algebra, trigonometry, and statistics. The index of summation , here the letter i, is a dummy variable whose value will change as the addends of the sum change. Use the sum of rectangular areas to approximate the area under a curve. Only this variable may occur in the product term. Often mathematical formulae require the addition of many variables Summation or sigma notation is a convenient and simple form of shorthand used to give a concise expression for a sum of the values of a variable. Joining a fraternity or sorority is an opportunity to serve the community at UB and beyond. The Leibniz formula for the determinant of a 2 × 2 matrix is \| \| = −. Calculus Definitions>. That is indicated by the lower index of the letter. Get access risk-free. Three theorems. no i want it to be solve using sigma notation with its rules of sequence and series. sigma notation, also known as summation notation. However, the lower bound doesn't have to be 1. - Dave xml-dev: A list for W3C XML Developers. Geometric Progression, Series & Sums Introduction. 1 - In the following exercises, use the rules for sums Ch. Six Sigma is an intangible work process. Proficiency with Summation Notation also known as Sigma Notation is a very important skill usually taught in PreCalculus and also in Calculus 1 and 2 (AB and BC). Binomial theorem refresher. Rule 1: If c is a constant, then n i=1 cx i = c n i=1 x i. Write out completely the sequences given by the following rules: (a) a i = i2, where 0 i 5. Rules of Scientific Notation. Okay, now that we've seen that fancy sigma notation let's work a really simple small example of numbers and then generalize. adding & subtracting. The first lesson, "Sets and What They're Good For," walks you through the basic notions of. The Example shows (at least for the special case where one random variable takes only. Here, we prefer to set up the prior in terms of nu, mu, sigma/(nu-2) or something like that, to account for the fact that the scale of the distribution (as measured by the sd or median absolute deviation) depends on nu as well as sigma. Ex B) Ex => => C) Where c is a constant,. For example d/dx (x^2) will graph the derivative of x^2 with respect to x. The image below shows the common notation for conditional probability. It makes more sense if you convert to scientific notation first, so 12. Added Apr 14, 2011 by HighOPS in Mathematics. The double series. Riemann sums, summation notation, and definite integral notation. In the following, $$k$$ is the index. Sum of first n natural numbers. If this value of SSR is equal to the sum of squares total, it means our regression model captures all the. Look at the picture below to see an example. The expression is read as the sum of 4n as n goes from 1 to 6. To aid the investigation, we introduce a new quantity, the Euler phi function, written ϕ(n), for positive integers n. This web page describes how symbols are used on the Stat Trek web site to represent numbers, variables, parameters, statistics, etc. Notation 4 We say a Turing machine halts, if, in the course of processing its input data, the Turing machine enters into the halt state and has no further transitions. Summation Notation Worksheet 1 Introduction Sigma notation is used as a convenient shorthand notation for the summation of terms. Sigma Notation: Inequalities: Factorial Notation: Exponential Functions: Logarithmic Functions: Simultaneous Equations: Plotting Coordinates: Straight Line Graphs: Quadratic Graphs: Exponential Graphs: Logarithmic and Square Root Graphs: Differentiation: Integration: MathQuiz 4. Sigma notation mc-TY-sigma-2009-1 Sigma notation is a method used to write out a long sum in a concise way. The growth curve of an O (2 n) function is exponential - starting off very shallow, then rising meteorically. ? Using Sigma notation and related rules, compute the sum of all the integers between 21 and 126 that are not divisible by 4. Constant multiplier. Determine the sum of the series written in Sigma Notation. This calculus video tutorial provides a basic introduction into summation formulas and sigma notation. But polynomials truly come to shine in the work of Al-Samawal (1130-1180) The Shining Book on Calculation. Summation Notation Rules & Examples Video & Lesson Transcript 12. Sigma notation is a method used to write out a long sum in a concise way. Learn how it is used in this video. " If the sums do not converge, the series is said to diverge. So for this series, there is 7 terms The seven terms are: when n=0 -2x. For example, cosx, cos x, and cos (x) are all equivalent. Use algebra rules of sums, along with the appropriate sum formulae, to evaluate the following: (a) X12 i=1 (i2 33i) (b) X10 i=1 (i2 2 1) (c) X8 i=1 (2 i ) 6. A typical sum written in sigma notation looks like this: 4 k 0 (k2 3) The symbol "Σ" is the Greek capital letter sigma, which stands for "sum". Loading Unsubscribe from Jochumzen? Summation Formulas and Sigma Notation - Calculus - Duration: 20:24. In the following, $$k$$ is the index. Rules of Scientific Notation. The Sigma symbol, , is a capital letter in the Greek alphabet. To use summation, you can find sigma in the Desmos keyboard (under FUNCTIONS and then misc) or by typing "sum": If you populate the upper and lower bound, Desmos will output the summation answer. But, more than just being small, it's one of the most radical cameras we've seen released in years, incorporating an array of new ideas and video capabilities. 5100 x 3300 px. The symbol sigma is a Greek letter that stands for ‘the sum of’. Binomial theorem refresher. because it does not matter what we call our index. Kronecker Delta Multiplication The Kronecker Delta is nicknamed the substitution operator because of the following simple property of multiplication, best explained by example. The user is asked to select true. I thought about extending these ideas to product notation (we haven't learned it yet, but it's simple), and I figured out a few rules. [1] The name sigmatropic is the result of a compounding of the long-established "sigma" name for single carbon -carbon bonds and the Greek word tropos , meaning turn. There are some rules that can help simplify or evaluate series. That is, we split the interval x 2[a;b] into n increments of size. Exponents and Integers. 2 University of Sydney School of Mathematics and Statistics. A business process model and notation diagram, or BPMN diagram for short, is used to build easy-to-read business process model flowcharts, which can be shared across organizations and industries. Matrix and Index Notation David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge, MA 02139. Regardless, your record of completion wil. The most general Σ \Sigma Σ notation describes the sum of some unknown number of elements. For each section in Chapter 7 you will have exactly two opportunities to take the quiz for that section, and both of those attempts need to be completed in Canvas by the end-of-week deadline. In this unit we look at ways of using sigma notation, and establish some useful rules. A series that diverges means either the partial sums have no limit or approach infinity. The terminology from AMS-LaTeX documentation. "My TI 83+ graphing calculator arrived faster than I expected. Go to The Equation of a Circle page. Viewing 8 posts - 1 through 8 (of 8 total) Author. The Westgard system is based on the principles of statistical process control used in manufacturing nationwide since the 1950s. Basic math formulas. 1 Steve Strand and Sean Larsen from Portland State University, US, have shown that, cognitively, the task of interpreting a given summation-notation expression differs significantly from the task of converting a longhand sum into summation notation. Sigma notation - Rules. This module contains three lessons that are build to basic math vocabulary. The example shows us how to write a sum of even numbers. If you're seeing this message, it means we're having trouble loading external resources on our website. 2 Rules of Roots. Basic Rule #4 of MO Theory Rule #4: If the AOs are non-degenerate, their interaction is. In particular, it is a symmetric group of prime degree and symmetric group of prime power degree. ” - Sincerely, Dr. Learners are taught how to use sigma notation and how to develop sigma notation. ' As such, the expression refers to the sum of all the terms, x n where n represents the values from 1 to k. Zeros after a # do not count unless they are also after a decimal place 4. It is, therefore, useful to have a set of rules for finding the equivalent resistance of some general arrangement of resistors. sp 3 HYBRIDIZATION - EXAMPLE 1) Methane (CH 4) During the formation of methane molecule, the carbon atom undergoes sp 3 hybridization in the excited state by mixing one ‘2s’ and three 2p orbitals to furnish four half filled sp 3 hybrid orbitals, which are oriented in tetrahedral symmetry in space around the carbon atom. The notation for adding a series of numbers is the capital Greek letter sigma. But there is a limit on how far you can upgrade the bases. As well as providing shorthand for mathematical ideas, this notation can aid students' understanding of mathematics. TechCalc scientific calculator contains 12 calculation modes in one application + a handy reference section that includes an additional 22 calculation modes plus a wealth of useful information. It is called Sigma notation because the symbol is the Greek capital letter sigma: Σ. They're called partial sums because you're only able to find the sum of a certain number of terms — no infinite series […]. After many searches I couldn't find a single page containing summation formulas for polynomials of order greater than 4. Joining a fraternity or sorority is an opportunity to serve the community at UB and beyond. Select a link by mouse clicking one of the music notation symbols below. It looks like this: $\sum_{j=1}^n \sum_{k=1}^n f(j, k)$ The idea behind this is you're doing a sum within a sum, and both indices will be inside the inner sum. T HIS —Σ—is the Greek letter sigma. Last video we did some elementary examples of sigma notation. Use the sum of rectangular areas to approximate the area under a curve. PART 1: INTRODUCTION TO TENSOR CALCULUS A scalar eld describes a one-to-one correspondence between a single scalar number and a point. One way to compactly represent a series is with sigma notation, or summation notation, which looks like this: $\displaystyle{\sum _{n=3}^{7}{n^2}}$ The main symbol seen is the uppercase Greek letter sigma. Calculate the Product. Summation rules Jochumzen. To change a number from scientific notation to standard form, move the decimal point to the left (if the exponent of ten is a negative number), or to the right (if the exponent is positive). It doesn't have to be "i": it could be any variable (j ,k, x etc. The Organic Chemistry Tutor 270,467 views. For example, assuming k ≤ n. Advertisement. Executive in Residence and Director, Center for Quantitative Modeling. They are useful in getting right to the point without delay or distraction:. Mulliken (1896-1986, awarded with the Nobel prize in 1966) in the early 1930s. The sums are heading towards a value (1 in this case), so this series is convergent. A series can be represented in a compact form, called summation or sigma notation. This is true for all tensor notation operations, not just this matrix dot product. Its 2 line display allows “visually perfect” entries; that is, you can type in the problem exactly as it looks. Note that these formulae only work if we start from 1; we will see how to calculate summations. pdf Math 180 Worksheets W20 20 Sigma Notation And. 1:2s rather than 1 2s) Combinations of rules are generally indicated by using a "slash" mark (/) between control rules, e. The following rule can be used to convert numbers into scientific notation: The exponent in scientific notation is equal to the number of times the decimal point must be moved to produce a number between 1 and 10. Adding series. SIGMA NOTATION FOR SUMS. where E is the sigma sign. Due to linear algebra being all about finding the solutions to systems of linear equations, matrix math and the study of vector spaces become a tool to represent and orderly solve. The ONLY way to show three SF is by using scientific notation. All very clear and understandable for anyone, through the use of BPMN 2.	2020-05-30T23:52:14	{ "domain": "amicidellacattolica.it", "url": "http://amicidellacattolica.it/rmev/sigma-notation-rules.html", "openwebmath_score": 0.8859736919403076, "openwebmath_perplexity": 745.3356280199723, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9830850832642354, "lm_q2_score": 0.8791467801752451, "lm_q1q2_score": 0.8642760855900652 }
http://crowebrothers.com/now-we-eqz/bc6841-how-to-find-orthogonal-matrix	# how to find orthogonal matrix • Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchange In fact, given any … The interactive program below is designed to answers the question whether the given input matrix is an orthogonal matrix. That is, each row has length one, and are mutually perpendicular. Damit ist die Inverse einer orthogonalen Matrix gleichzeitig ihre Transponierte. Orthogonal matrix is an important matrix in linear algebra, it is also widely used in machine learning. A matrix is orthogonal if the If Q is an orthogonal matrix, then, \|Q\| = ±1. Previous Solution: Example 1. The eigenvalues of the orthogonal matrix will always be $$\pm{1}$$. The matrix in problem statement (not step one) is for the previous problem. An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix.Although we consider only real matrices here, the definition can be used for matrices with entries from any field.However, orthogonal matrices arise naturally from dot products, and for matrices of complex numbers that leads instead to the unitary requirement. The concept of orthogonality for a matrix is defined for just one matrix: A matrix is orthogonal if each of its column vectors is orthogonal to all other column vectors and has norm 1. transpose The orthogonal projection matrix is also detailed and many examples are given. Example using orthogonal change-of-basis matrix to find transformation matrix. Let W be a subspace of R n and let x be a vector in R n. If you are on a personal connection, like at home, you can run an anti-virus scan on your device to make sure it is not infected with malware. Es ist oﬀensichtlich, dass Q orthogonal ist, da die beiden Spaltenvektoren orthogonal sind. If the result is an identity matrix, then the input matrix is an orthogonal matrix. Let's say I've got me a set of vectors. If matrix Q has n rows then it is an orthogonal matrix (as vectors q1, q2, q3, …, qn are assumed to be orthonormal earlier) Properties of Orthogonal Matrix. is equal to its Some important properties of orthogonal matrix are, See also Orthogonal Matrix (1) The Definition of The Orthogonal Basis. If you are at an office or shared network, you can ask the network administrator to run a scan across the network looking for misconfigured or infected devices. Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes. Completing the CAPTCHA proves you are a human and gives you temporary access to the web property. When you click Random Example button, it will create random input matrix to provide you with many examples of both orthogonal and non-orthogonal matrices. is an orthogonal matrix. Performance & security by Cloudflare, Please complete the security check to access. So let's say vector w is equal to some linear combination of these vectors right here. I need to find an orthogonal matrix Q, so that when applying M_2 = Q M_1 Q^-1 the matrix M_2 does not contain any values at the zero positions of P. The other way is possible, M_2 may contain a zero, where P is one. Proof Part(a):) If T is orthogonal, then, by deﬁnition, the An n £ n matrix A is orthogonal iﬀ its columns form an orthonormal basis of Rn. If Q is square, then QTQ = I tells us that QT= Q−1. Comment(8) Anonymous. One way to think about a 3x3 orthogonal matrix is, instead of a 3x3 array of scalars, as 3 vectors. , Gram-Schmidt process example. Die Matrix ist also orthogonal, weil die Multiplikation der Matrix mit der transponierten Matrix die Einheitsmatrix ergibt. The concept of orthogonality for a matrix is defined for just one matrix: A matrix is orthogonal if each of its column vectors is orthogonal to all other column vectors and has norm 1. This is true because d vectors will always be sufficient be needed to span a d-dimensional vector space. Q⋅QT = E Q ⋅ Q T = E Die Determinante einer orthogonalem Matrix nimmt entweder den Wert +1 oder -1 an. Well, if you're orthogonal to all of these members, all of these rows in your matrix, you're also orthogonal to any linear combination of them. Cloudflare Ray ID: 60a7cf86683fdfbf of the , Diese Matrix beschreibt eine Drehung um den Winkel −θ. Index Next lesson. Orthogonal Matrix Example. Dafür musst du zunächst die transponierte Matrix berechnen und diese dann mit multiplizieren. Define a matrix and find the rank. Let W be a subspace of R4 with a basis {[1011],[0111]}. You can also try to input your own matrix to test whether it is an orthogonal matrix or not. Find the inverse matrix of … concatenation Similarly, the columns are also an orthonormal basis. Thus, matrix is an orthogonal matrix. How to fill in a matrix given diagonal and off-diagonal elements in r? Eigen vectors Thus, matrix Therefore, the value of determinant for orthogonal matrix will be either +1 or -1. 2. In this tutorial, we will dicuss what it is and how to create a random orthogonal matrix with pyhton. \| The vectors in are orthogonal while are not. Suppose we have a set of vectors {q1, q2, …, qn}, which is orthogonal if, then this basis is called an orthogonal basis. Next spectral decomposition, Rate this tutorial or give your comments about this tutorial, The row vector and the column vector of matrix, Both Hermitian and Unitary matrix (including. inverse Suppose that is an orthogonal basis for the column space of . . Eigen-everything. From introductory exercise problems to linear algebra exam problems from various universities. Orthogonal matrices preserve angles and lengths. . A linear transformation T from Rn to Rn is orthogonal iﬀ the vectors T(e~1), T(e~2),:::,T(e~n) form an orthonormal basis of Rn. Then we multiply the transpose with given matrix. Pictures: orthogonal decomposition, orthogonal projection. Find an orthonormal basis of W. (The Ohio State University, Linear Algebra Midterm) Add to solve later Sponsored Links Please enable Cookies and reload the page. Es gilt detQ = cos2 ϕ +sin2 ϕ = 1. (3) Your answer is P = P ~u i~uT i. 2. If we try the orth trick, it will produce an array of size d by d, thus a SQUARE matrix. This can be generalized and extended to 'n' dimensions as described in group theory. This covers about orthogonal matrix Its definition and properties. Singular Value Decomposition 7 Finding stationary distribution of a markov process given a transition probability matrix are orthogonal matrices. \| Your IP: 78.47.248.67 The concept of two matrices being orthogonal is not defined. That is, if and only if . : Note that this is an n n matrix, we are multiplying a column vector by a row vector instead of the other way around. symmetric A = [1 0 1;-1 -2 0; 0 1 -1]; r = rank(A) r = 3 Since A is a square matrix of full rank, the orthonormal basis calculated by orth(A) matches the matrix U calculated in the singular value decomposition, [U,S] = svd(A,'econ'). Spiegelung. The objective is to find an orthogonal basis for the column space of the following matrix: Use Gram-Schmidt Process to find an orthogonal basis for the column space of segregate the columns of the matrix as . How to find an orthogonal matrix? • Fact 5.3.3 Orthogonal transformations and orthonormal bases a. Orthogonal matrix multiplication can be used to represent rotation, there is an equivalence with quaternion multiplication as described here. Let given square matrix is A. Title: Finding the Nearest Orthonormal Matrix Author: Berthold K.P. , that is Orthogonal matrix is important in many applications because of its properties. Orthogonale Matrizen k¨onnen auch Spiegelungen an Geraden beschreiben. orthogonal vector We study orthogonal transformations and orthogonal matrices. Another way to prevent getting this page in the future is to use Privacy Pass. Demzufolge gilt Q−1 = QT = cosϕ sinϕ −sinϕ cosϕ . b. To test whether a matrix is an orthogonal matrix, we multiply the matrix to its transpose. Finally we check if the matrix obtained is identity or not. You may need to download version 2.0 now from the Chrome Web Store. Eine orthogonale Matrix ist in der linearen Algebra eine quadratische, reelle Matrix, deren Zeilen- und Spaltenvektoren orthonormal bezüglich des Standardskalarprodukts sind. The 1/0 indicate where values are allowed in the result matrix. You can imagine, let's say that we have some vector that is a linear combination of these guys right here. Basic to advanced level. Horn Subject: Painful Way to Solve Photogrammetric Problems Keywords: Orthonormal matrix, Rotation, Photogrammetry, Least Squares Fitting, Projective Geometry, Matrix Square Root, Two step … A square orthonormal matrix Q is called an orthogonal matrix. To create random orthogonal matrix as in the interactive program below, I created random Example: Prove Q = $$\begin{bmatrix} cosZ & sinZ \\ -sinZ & cosZ\\ \end{bmatrix}$$ is orthogonal matrix. What is Orthogonal Matrix? Let us see an example of the orthogonal matrix. So let me call my set B. Let. Vocabulary words: orthogonal decomposition, orthogonal projection. To nd the matrix of the orthogonal projection onto V, the way we rst discussed, takes three steps: (1) Find a basis ~v 1, ~v 2, ..., ~v m for V. (2) Turn the basis ~v i into an orthonormal basis ~u i, using the Gram-Schmidt algorithm. Calculate and verify the orthonormal basis vectors for the range of a full rank matrix. Basis vectors. Simple Solution : The idea is simple, we first find transpose of matrix. Video transcript. Also given a symmetric prototype matrix P, containing ones and zeroes. If n>d, regardless of the size, as long as n>d, we can never find a set of n vectors in a d-dimensional space that are orthogonal. A matrix can be tested to see if it is orthogonal using the Wolfram Language code: OrthogonalMatrixQ[m_List?MatrixQ] := (Transpose[m].m == IdentityMatrix @ Length @ m) The rows of an orthogonal matrix are an orthonormal basis. If, it is 1 then, matrix A may be the orthogonal matrix. Problems of Orthogonal Bases. 0 0 1 0 1 0 For example, if Q =1 0 then QT=0 0 1. The concept of two matrices being orthogonal is not defined. >. Um eine orthogonale Matrix bestimmen zu können, überprüfst du die Formel von oben. Recipes: orthogonal projection onto a line, orthogonal decomposition by solving a system of equations, orthogonal projection via a complicated matrix product. To test whether a matrix is an orthogonal matrix, we multiply the matrix to its transpose. matrix and compute the modal matrix from < To check for its orthogonality steps are: Find the determinant of A. If a matrix A is an orthogonal matrix, it shoud be nn. The feature of an orthogonal matrix A. Gram-Schmidt example with 3 basis vectors. We can define an inner product on the vector space of all polynomials of degree at most 3 by setting. If the result is an identity matrix, then the input matrix is an orthogonal matrix. Overview. The Gram-Schmidt process. 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A line, orthogonal decomposition by solving a system of equations, orthogonal matrix, then the input is! Filed under: News. Bookmark the permalink.	2021-06-12T18:37:39	{ "domain": "crowebrothers.com", "url": "http://crowebrothers.com/now-we-eqz/bc6841-how-to-find-orthogonal-matrix", "openwebmath_score": 0.7539582252502441, "openwebmath_perplexity": 1214.8395199556069, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9724147201714923, "lm_q2_score": 0.8887587949656841, "lm_q1q2_score": 0.8642421349065085 }
https://math.stackexchange.com/questions/4032890/winding-number-of-a-curve-not-complex-analysis	Winding number of a curve (not complex analysis) I am asked to calculate the winding number of an ellipse (it's clearly 1 but I need to calculate it) I tried two different aproaches but none seems to work. I would like to know why none of them work (I believe it is because these formulas only work if I have a curve parametrized by arc lenght). Approach 1: A valid parametrization : $$\gamma=(a\cos t,b\sin t)$$, with $$t \in [0,2\pi], \, a,b \in \mathbb{R}$$ $$\dot{\gamma}(t)=(-a\sin t,b\cos t)$$, with $$t \in [0,2\pi], \, a,b \in \mathbb{R}$$ $$\ddot{\gamma}(t)=(-a\cos t,-b\sin t)$$, with $$t \in [0,2\pi], \, a,b \in \mathbb{R}$$ $$\det(\dot{\gamma}(t)\|\ddot{\gamma}(t)) = \renewcommand\arraystretch{1.2}\begin{vmatrix} -a\sin t & -a\cos t \\ b\cos t & -b\sin t \end{vmatrix}=ab \sin^2 t+ab \cos^2 t=ab$$ $$\|\|\dot{\gamma}(t)\|\|^3=(\displaystyle\sqrt{(-a\sin t)^2+(b\cos t)^2})^3=(\displaystyle\sqrt{a^2\sin^2 t+b^2\cos^2 t})^3=a^3b^3$$ $$\kappa(t)=\displaystyle\frac{ab}{a^3b^3}=\displaystyle\frac{1}{a^2b^2}$$ $$\mathcal{K}_\gamma = \displaystyle\int_{0}^{2\pi} \displaystyle\frac{1}{a^2b^2} \ dt= \displaystyle\frac{2\pi}{a^2b^2}$$, $$\mathcal{K}_\gamma$$ is the total curvature of the curve. $$i_\gamma=\displaystyle\frac{\displaystyle\frac{2\pi}{a^2b^2}}{2\pi}=\displaystyle\frac{1}{a^2b^2}$$...which is not necessarily 1. Approach 2: Winding # = $$\displaystyle\frac{1}{2\pi}\displaystyle\int_{\gamma}\displaystyle\frac{-y}{x^2+y^2}\>dx+\displaystyle\frac{x}{x^2+y^2}\>dy$$ That gives us $$\displaystyle\frac{1}{2\pi}\displaystyle\int_{0}^{2\pi}\left( \displaystyle\frac{-b\sin t}{a^2\cos^2 t+b^2\sin^2 t}(-a\sin t)+\displaystyle\frac{a\cos t}{a^2\cos^2 t+b^2\sin^2 t}(b\cos t) \right)\>dt$$ $$\displaystyle\frac{1}{2\pi}\displaystyle\int_{0}^{2\pi}\left( \displaystyle\frac{ab}{a^2\cos^2 t+b^2\sin^2 t }\right)\>dt$$, which I computed and cannot be calculated. Clearly the second approach is valid if we are dealing with a circumference of radius 1. We can generalize for the elipsee using Green's Theorem. I would also like if someone could show me this way as well. Thank you • If you use a polar coordinate system, and parametrise the ellipse using $\theta(t)$ and $r(t)$ for $t_0 \le t \le t_1$, then the winding number is by definition $$\frac{\theta(t_1) - \theta(t_0)}{2 \pi}$$ – Glärbo Feb 20 at 11:55 • Thank you for your help. I had indeed read that as well, but I really can't seem to know what to do in order to end up with $\theta (t)$ and $r (t)$. Maybe this is a bit obvious, but I'm really not being able to see any further. Any hints? – hugh_maths Feb 20 at 13:01 The last integral indeed has a closed form to evaluate you can exploit the symmetry about $$\pi$$ to get the integral as \begin{align} \pi I &= ab \int_{0}^{\pi} \dfrac{1}{a^2 \cos^2(t) + b^2 \sin^2(t)}dt\\ &= 2ab\int_{0}^{\pi/2} \dfrac{\sec^2(t)}{a^2+b^2 \tan^2(t)} dt \end{align} Now substituite $$\tan(t) = u$$ $$\dfrac{\pi}{2ab} I = \int_{0}^{\infty}\dfrac{1}{a^2+b^2x^2}dx = \dfrac{\pi}{2ab}$$ thus giving $$I =1$$ • Thank you for your kind answer. Shouldn't this be equal to 1 since this is the winding number and its been known to be 1? – hugh_maths Feb 20 at 12:57 • it is 1 the lhs is a constant times the integral, does this answer your question @hugh_maths – Aditya Dwivedi Feb 20 at 12:58 • oh yes, of course, I'm sorry! Oh so we should be done ! 1 is indeed the winding number! Thank you so much! Any thoughts on why my first approach is wrong (or seems to be) ? – hugh_maths Feb 20 at 13:03 Green's Theorem will tell you that the winding number (given by approach #2 — your total curvature in #1 is incorrect because you need an arclength integral) of the ellipse is the same as the winding number of any circle (which is easy to compute). Let $$P=-\dfrac y{x^2+y^2}$$ and $$Q=\dfrac x{x^2+y^2}$$. Note that on $$\Bbb R^2-\{0,0\}$$, we have $$\dfrac{\partial P}{\partial y} = \dfrac{\partial Q}{\partial x}$$. Thus, if $$R$$ is the region between the ellipse $$E$$ and a circle $$C$$ centered at the origin lying inside it, we have $$\int_E P\,dx+Q\,dy - \int_C P\,dx + Q\,dy = \iint_R \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\,dA = 0.$$ But you can easily calculate that $$\int_C P\,dx + Q\,dy = 2\pi$$. • Thank you so much!@ Aditya Dwivedi helped me and I was able to evaluate the integral. As for the approach # one, I believe the total curvature divided by $2\pi$ gives us the index, which is the same as the winding number. Or they differ ? Cause I do not understand why the result comes up different than 1. – hugh_maths Feb 20 at 19:02 • Yeah, for a convex curve $C$, it is the case that $\int_C \kappa\,ds = \pm 2\pi$ (hard theorem). But you have to integrate with respect to arclength, not $dt$. – Ted Shifrin Feb 20 at 19:25 • Additional comment: The curvature of the ellipse is not constant, so you'd better double-check your computations! You made a glaring error there. – Ted Shifrin Feb 20 at 19:34	2021-06-21T15:54:14	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/4032890/winding-number-of-a-curve-not-complex-analysis", "openwebmath_score": 0.9710761308670044, "openwebmath_perplexity": 220.6253364112749, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9724147201714922, "lm_q2_score": 0.8887587920192298, "lm_q1q2_score": 0.8642421320413328 }
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Sage provides standard constructions from linear algebra, e. \ vdots for vertical dots. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. This does seem silly. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58. We introduce a very small part of the language for writing mathematical notation. my table and figure. to get a color as mix of 4 colors such as CMYK). Together these facts mean that the upper triangular matrices form a subalgebra of the associative algebra of square matrices for a given size. Natural rubber is an elastomer and a thermoplastic. The defining property for the gamma matrices to generate a Clifford algebra is the anticommutation relation {,} = + =,where {,} is the anticommutator, is the Minkowski metric with signature (+ − − −), and is the 4 × 4 identity matrix. The majority of L A TEX packages has a manual which describes how to use it and usually gives examples. The video explains the concepts with hands on. For example, create a 3-by-5 matrix of zeros:. You can think of these two points as "attractor points". Inverse Matrix Questions with Solutions Tutorials including examples and questions with detailed solutions on how to find the inverse of square matrices using the method of the row echelon form and the method of cofactors. VerkhovtsevaKatya. Most rubber in everyday use is vulcanized to a point where it shares properties of both; i. Let A be the following matrix LaTeX: \begin{bmatrix}5&-6&-12\\ 14 & 4&7 \end{bmatrix} [ 5 −6 −12 14 4 7 ] The dimensions of A are Expert Answer 100% (1 rating). Formatting Optimization Problems with LaTeX. This means that it is the set of the n-tuples of real numbers (sequences of n real numbers). Reference. There are many identity matrices. Transparency 20. In a double-blind, placebo-controlled, randomized study of 187 women receiving Climara 0. js as an input. Important: We can only multiply matrices if the number of columns in the first matrix is the same as the number of rows in the second matrix. MatrixForm acts as a "wrapper", which affects printing, but not evaluation. Note: This page is a work-in-progress. 00 Only 2 left. Welcome! This is the documentation for Numpy and Scipy. So our strategy will be to try to find the eigenvector with X = 1, and then if. How to add an equation in your document, see Working with Microsoft Equation. INPUT: The matrix command takes the entries of a matrix, optionally preceded by a ring and the dimensions of the matrix, and returns a matrix. Similarly, the other matrix is of the order 4 × 3, thus the number of elements present will be 12 i. Definir el tamaño y márgenes del papel. tikz-pgf positioning matrices bordermatrix. The eigenspace corresponding to an eigenvalue λ of A is defined to be Eλ = {x ∈ Cn ∣ Ax = λx}. Ideal for low compression situations, the polyester/cotton weave provides excellent elasticity and recoverability. Read honest and unbiased product reviews from our users. 可以生成嵌入到HTML网站、论坛或博客的公式代码. ISBN -8176-3805-9 (acid-free paper) (pbk. Please join the discussion on moodle. 1 or any later version published by the Free Software. Right click the matrix and select Matrix Spacing… 3. The Matrix 4 will be in the same universe as John Wick Lionsgate This one is (surprisingly) way too detailed to outline in full here (it involves the creation of a bunch of Zion Accords to "govern. Making statements based on opinion; back them up with references or personal experience. We can encode this transformation in a 4 x 4 matrix by putting A in the top left with three 0's below it and making the last column be (b,1). To define dots in Latex, use: \ ldots for horizontal dots on the line. Texmaker can also be used to edit and compile directly asymptote figure not embedded in a latex document. Input LaTeX, Tex, AMSmath or ASCIIMath notation (Click icon to switch to ASCIIMath mode) to make formula. Multiplying the 4-vector (v,1) with this matrix will give you (Av + b, 1). radio_button_uncheckedNo. Imagine turning a paper to the other side, that is what is happening. LaTeX minipage. Definir el tamaño y márgenes del papel. Paquete layouts. But the resulting 2 matrices are S4 object of class. 4-18 are: v v v 23 4= =16 V, 8 V and 6 V= Determine the values of the following: (a) The gain, A, of the VCVS (b) The resistance R 5 (c) The currents i b and i c (d) The power received by resistor R 4. 25]{image}} \begin{table}[h!] \begin{tabular}{\| p{2cm} \| p{2cm} \| p{2cm} \| p{2cm. This ends up being a cubic equation, but just looking at it here we see one of the roots is 2 (because of 2−λ), and the part inside the square brackets is Quadratic, with roots of −1 and 8. Welcome! This is the documentation for Numpy and Scipy. Multiplication of Matrices. Most rubber in everyday use is vulcanized to a point where it shares properties of both; i. 4 5! 2 9! = 53! Putting it all together 3 7 4 5! 2 9! = 69 53! Example Find 2 4 5 3. Define Vector and Matrix Data. 620 MazdaRX4Wag 21. multiplication dot. Ideal for low compression situations, the polyester/cotton weave provides excellent elasticity and recov. Yes, but there will always be the same number of pivots in the same columns, no matter how you reduce it, as long as it is in row echelon form. Want a Mac app? Lucky you. Very Basic Mathematical Latex A document in the article style might be entered between the nmaketitle and nendfdocumentgsections below. Our matrix and vector calculator is the most sophisticated and comprehensive matrix calculator online. They'll be productive from day one and be able to pick up small amounts of LaTeX as they go. For LaTex symbols, check the website LaTeX:Symbols (Art of Problem Solving). Steps 5 and 6 specify the data to display in the matrix data cells. Definir el tamaño y márgenes del papel. Use commas or spaces to separate values in one matrix row and semicolon or new line to separate different matrix rows. Note that the inverse [B]-1 always has the same number of rows and columns as the original matrix [B]. Matrix Representations of Linear Transformations and Changes of Coordinates 0. 0 6 160 110 3. ISBN -8176-3805-9 (acid-free paper) (pbk. It is easy to define a matrix of values in Octave. a $2 \times 2$ matrix whose elements are $2 \times 2$ matrices) and then apply "Block Inversion" : have a look at this article. MatrixForm prints SparseArray objects like the corresponding ordinary lists. Creation of matrices and matrix multiplication is easy and natural: Note that in Sage, the kernel of a matrix A is the "left kernel", i. She followed the series to its new. TeXstudio is a fully featured LaTeX editor. The size of the matrix is determined automatically, so it is not necessary to explicitly state the dimensions. So the Eigenvalues are −1, 2 and 8. The simplest classifiers, called binary classifiers, has only two classes: positive/negative, yes/no, male/female … Performance of a binary classifier is summarized in a confusion matrix that cross-tabulates predicted and observed examples into four options:. Row and column labelling of matrixes in LaTeX. 3 Reference guide. Definition If A is an m n matrix, with columns a1,a2, ,an, and if x is in Rn, then the product of A and x, denoted by Ax,isthelinear combination of the columns of A using the corresponding. Sage provides standard constructions from linear algebra, e. Let A be the following matrix LaTeX: \begin{bmatrix}5&-6&-12\\ 14 & 4&7 \end{bmatrix} [ 5 −6 −12 14 4 7 ] The dimensions of A are Expert Answer 100% (1 rating). a system of linear equations with inequality constraints. Carrie Anne Moss loves to play her so much. No installation, real-time collaboration, version control, hundreds of LaTeX templates, and more. 3 Beamer presentation. This post will tell you which one is the best. I have seen is countless discussion threads and videos to use Equation Editor 3. In mathematics, a matrix (plural matrices) is a rectangular array (see irregular matrix) of numbers, symbols, or expressions, arranged in rows and columns. 3 Single Equations that are Too Long: multline If an equation is too long, we have to wrap it somehow. The previous example was the 3 × 3 identity; this is the 4 × 4 identity: The 3 × 3 identity is denoted by I3 (pronounced as "eye-three" or "eye. matrix without brackets. See Snip in action—watch a demo video! Take screenshots of individual equations, paragraphs, and even full pages of text! Digitize any math or text on your screen in seconds. The position of the elements are specified with pos, which is either NULL, or a vector specifying the number of elements on a row, or a 2-columned matrix specifying the (x,y) position of. This ends up being a cubic equation, but just looking at it here we see one of the roots is 2 (because of 2−λ), and the part inside the square brackets is Quadratic, with roots of −1 and 8. LATEX Command Summary This listing contains short descriptions of the control sequences that are likely to be handy for users of LAT EX v2. It is free and easy to use. An online LaTeX editor that's easy to use. Is T one–to–one? Theorem Let T: n m be a linear transformation and let A be the standard matrix for T. Changing the font size in LaTeX can be done on two levels, either affecting the whole document or parts/elements of it. 6 Implications of nullity being zero. We can associate a matrix with each graph storing some of the information about the graph in that matrix. Click on "Insert" and choose what you want to add. You can put this solution on YOUR website! How do I write systems of equations in matrix form? That's one of the easiest things you'll ever learn: Suppose you have this system: 4x + 7y = 1 x - y = -8 Look at the red numbers: 4x + 7y = 1 1x - 1y = -8 Erase the letters: 4 + 7 = 1 1 - 1 = -8 Erase the + and bring the - over nearer the 1: 4 7 = 1 1 -1 = -8 Replace the " = " signs with a vertical. The mathematical symbol is produced using \partial. This site is supported by donations to The OEIS Foundation. CHANGES IN VERSION 1. LaTeX for Advanced View Examples These are provided if you'd like to copy and paste the LaTeX from the examples into the Advanced View of the Equation Editor to tinker with the examples. 320 Hornet4Drive 21. Mathematical Formulas. This function and x / y are equivalent. MATH 316U (003) - 1. 1 Fill the rest of the line. mat") where readMat is from the R. Creating a Payoff Matrix Using LaTeX Tabular Environment 291. Analogous sets of gamma matrices can be defined in any dimension and for any signature of the metric. You can add, subtract, multiply and transpose matrices. Specifically the cofactor of the $(i,j)$ entry of a matrix, also known as the $(i,j)$ cofactor of that matrix, is the signed minor of that entry. The context menu does not have an option to insert additional rows or columns. More details. Augmented Matrix Calculator is a free online tool that displays the resultant variable value of an augmented matrix for the two matrices. Coordinate systems 8. Choose a type a matrix 5. , Manufacturer: Medline, Medline Item Number: MDS087004LFZ, Medline MDS087004LFZ Bandage, Elastic, Matrix, 4" x 5 yd. Invest in comfortable, restful sleep for your family with mattresses that suit individual sleeping styles and preferred levels of firmness. 4 tips for writing matrix questions Sarah Cho 2 min read A matrix question —or really, multiple questions presented on a grid—is one of the most popular question types in online and traditional pen-and-paper surveys. Use the scale=1. X n converges to Xin distribution, written X n X, if lim n!1 F n(t) = F(t) (7) at all tfor which Fis continuous. The X-Y Matrix is a Six Sigma tool mostly used during the DMAIC measure phase and the DMADV measure phase to show the relationship between X and Y factors. We introduce a very small part of the language for writing mathematical notation. Only a limited part of the full TeX language is supported; see below for details. Then write the LaTex code \cancel{A}. 4 FROM VERSION 1. If we try to multiply an n×1 matrix with another n×1 matrix, this product is not defined. America's #1 selling latex caulk. Enumerate is used to make numbered lists. Set the matrix (must be square). \\imath and\\jmath make "dotless" i and j, which. We now go on to solve. Please, be aware that the support for loading tables from an existing LaTeX code is severely limited and may work erroneously or may not work at all. Matrix is our finest elastic bandage. LaTeX symbols have either names (denoted by backslash) or special characters. We can encode this transformation in a 4 x 4 matrix by putting A in the top left with three 0's below it and making the last column be (b,1). It is free and easy to use. 8 in R3 form a basis for the unit cell shown in the accompanying figure. If you have more specific LaTeX questions. The process for finding the eigenvalues and eigenvectors of a 3xx3 matrix is similar to that for the 2xx2 case. $\begingroup$ I have 4 5x5 matrices n1, n2, n3, n4. 2 π x 1 −1 = 8 −0. [ edit] Introduction. Ideal for low compression situations, the polyester/cotton weave provides excellent elasticity and recov. This means that it is the set of the n-tuples of real numbers (sequences of n real numbers). Definir el tamaño y márgenes del papel. \begin{frame} \frametitle{There Is No Largest Prime Number} \framesubtitle{The proof uses \textit{reductio ad absurdum}. Single formulas must be seperated with two backslashes \\ Use the matrix environment to typeset matrices. 0 Sage is free, open-source math software that supports research and teaching in algebra, geometry, number theory, cryptography, numerical computation, and related areas. You know, there is a stack exchange site for latex questions - user1228 Jan 10 '11 at 15:00 This question appears to be off-topic because it is about latex. \ vdots for vertical dots. Plotting in Scilab www. If neither A nor B is an identity matrix, AB ≠ BA. 10 01 12 34 = 12 34 12 34 10 01 = 12 34 Such matrices A and B are said to commute. There are three commonly used environments in the math mode: the math environment: Used for formulas in running text ; the displaymath environment: Used to display longer formulas ; the equation environment: Used for displaying equations for numbering and cross reference. #N#square root ("radical") Symbols in Geometry Symbols in Algebra. The process by which the rank of a matrix is determined can be illustrated by the following example. Click the "Insert" tab 2. 2 on Mepis Antix MX 14. dem autogenerated by webify. 2013;4:204173141350530. Textext aims to cover this need, by adding re-editable LaTeX objects to Inkscape’s repertoire. Here is a matrix of size 2 3 ("2 by 3"), because it has 2 rows and 3 columns: 10 2 015 The matrix consists of 6 entries or elements. 2 π−3 4+x 8 −1. In a double-blind, placebo-controlled, randomized study of 187 women receiving Climara 0. November 11, 2009 October 15, 2017 James 21 Comments. I tried computing conductance (stiffness) matrix in the physical coordinate systems and comparing the answer with isoparametric system. −3 4 7 0 + 6 −1. In order for a matrix $B$ to be the inverse of $A$, the equation $AB = I$ and $BA. 4 Fonts Font sizes Point size Latex cmd User-de ned * Sample 5 6 \tiny \xxxsmall the quick brown fox 7 8 \scriptsize \xxsmall the quick brown fox 8 10 \footnotesize \xsmall the quick brown fox. I am looking to get a colormap to assign a unique color to each point in a plot based on their four Ci values (i. (diag(A)) ij= ijA ij eig(A) Eigenvalues of the matrix A vec(A) The vector-version of the matrix A (see Sec. LaTeX Lesson 4 Mathematics in LaTeX. 4[U] 14 Matrix expressions. GroupWork 1: Mark each statement True or False. The theory was that the Real World was not actually real, but rather another level of the Matrix simulation: i. 6 Rubber/Stretching lengths. An extension to amsmath matrix environments. Please feel free to let us know if you have any question. The Matrix Behind The Scenes - Training Injuries (1999) - Keanu Reeves Movie HD - Duration: 1:33. Free Shipping on Everything* at Overstock - Your Online Bedroom Furniture Store! Get 5% in rewards with Club O! - Latex Mattresses. Define Vector and Matrix Data. Easily perform matrix multiplication with complex numbers. digits: Maximum number of digits for numeric columns, passed. where I denotes a unit matrix of order n. Some examples of conditional typesetting using the latex() function. The LaTeX3 project is a long-term research project to develop the next version of the LaTeX typesetting system. Easy to apply and tool. This comes from Mathematica (IIRC) and is not otherwise standardized, but there may be some disambiguation benefit in using the double-struck italic letters from Unicode’s Letterlike Symbols block: U+2148 ⅈ and U+2149 ⅉ. False, the vector spaces P3 and R^4 are isomorphic (4. input : output \begin{matrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{matrix} matrix with parentheses brackets. LaTeX is a typesetting language for producing scientific documents. In linear algebra, the cofactor (sometimes called adjunct) describes a particular construction that is useful for calculating both the determinant and inverse of square matrices. [code]\begin{vmatrix} a & b \\ c & d \end{vmatrix} [/code]produces $\begin{vmatrix} a & b \\ c & d \end{vmatrix}$ [code]\det\begin{pmatrix} a & b \\ c. calculate expression inside first. Matrices with Examples and Questions with Solutions. 1 Matrix without names, everything set to NULL. We provide the highest standards of large format printing and installation services (including Vutek 5m 4 machines,Flora 3. A diagonal matrix whose non-zero entries are all 1 's is called an " identity " matrix, for reasons which will become clear when you learn how to multiply matrices. abelian group augmented matrix basis basis for a vector space characteristic polynomial commutative ring determinant determinant of a matrix diagonalization diagonal matrix eigenvalue eigenvector elementary row operations exam finite group group group homomorphism group theory homomorphism ideal inverse matrix invertible matrix kernel linear. In order for a matrix [latex]B$ to be the inverse of $A$, the equation $AB = I$ and [latex]BA. Latex Free. latex matrices First the basic environments which could be used for a matrix, all of them are provide by usepackage amsmath. 3 (Dot Prod. Learn from detailed step-by-step explanations Get walked through each step of the solution to know exactly what path gets you to the right answer. The prefix arc used for inverse circular trigonometric functions is the. Example 1: Find the inverse of. This is called the Falk's scheme. Infection Prevention. Most LaTeX commands are available including:. Click on "Matrix" 4. This site is supported by donations to The OEIS Foundation. 7 , B= 3 −3 12 0 dimension (or size) always given as (numbers of) rows × columns • Ais a 3×4matrix, B is 2×2 • the matrix Ahas four columns; B has two rows m×nmatrix is called square if m=n, fat if m Can be used to label the rows of the % resulting latex table. Matrix Elastic Bandages are our finest elastic bandages. I can't figure out how to use "Equation Tools" to enter a matrix with dimensions of my choosing. It's amazing, even though it's a bit slow, the idea of converting latex to an image is extremely useful and versatile :D. LaTeX is a very flexible program for typesetting math, but sometimes figuring out how to get the effect you want can be tricky. Also, you can add a multiple of one row to another row without changing the determinant. Polyester/cotton weave provides excellent elasticity, recoverability and breathability; Color-coded by size for quick identification; Features self-closure that keeps bandage wrapped and secure without the need for clips. This add-on lets you instantly convert every math equation in your document into beautiful latex images! Auto-Latex Equations Lab - Aayush Gupta. 3 bronze badges. com Knowledge base dedicated to Linux and applied mathematics. Subscripts & Superscripts. by interchanging the location of two rows, or interchanging the location of two columns. You can modify individual elements or perform arithmetic on entire vectors and matrices. Its syntax is much simpler than that of estout and, by default, it produces publication-style tables that display nicely in Stata's results window. 95) Also include 1 x 5V 4A (4000mA) switching power supply - UL Listed (\$ 14. For example \bigg[\begin{array}{c\|c\|c\|c}. 4 tips for writing matrix questions Sarah Cho 2 min read A matrix question —or really, multiple questions presented on a grid—is one of the most popular question types in online and traditional pen-and-paper surveys. 4 Enumerate. Matrix Chain Multiplication (A O(N^2) Solution) Printing brackets in Matrix Chain Multiplication Problem. LAPACK addresses this problem by reorganizing the algorithms to use block matrix operations, such as matrix multiplication, in the innermost loops. Jump to navigation Jump to search. The following example shows you how to get a matrix into reduced row echelon form using elementary row operations. LaTeX Lesson 4 Mathematics in LaTeX. Does someone know how to convert an object of class dgCmatrix to a regular matrix. But nowadays there are many examples in which TEX is used for publications with no mathematical or technical background content. It is especially useful to write mathematical notation such as equations and formulae. An m × n (read 'm by n') matrix is an arrangement of numbers (or algebraic expressions ) in m rows and n columns. produces the graphic $$a^x_n$$. Restriction: In addition to the LaTeX command the unlicensed version will copy a reminder to purchase a license to the clipboard when you select a symbol. Numerical Linear Algebra with Applications 19 :3, 607-619. GNU Scientiﬁc Library Release 2. For example: \documentclass{article} \usepackage[pdftex]{graphicx} \begin{document} This is my first image. Tutorial o curso para escribir matrices dos formas, usando el entorno array o usando entornos ya definidos, matrix, pmatrix, vmatrix, bmatrix, en LaTeX utilizando Texmaker. Representing Systems of Linear Equations using Matrices A system of linear equations can be represented in matrix form using a coefficient matrix, a variable matrix, and a constant matrix. Some natural rubber sources, such as gutta-percha, are composed of trans-1,4-polyisoprene, a structural isomer that has similar properties. Maths for Physics University of Birmingham Mathematics Support Centre Authors: Daniel Brett Joseph Vovrosh Supervisors: Michael Grove Joe Kyle October 2015. The previous example was the 3 × 3 identity; this is the 4 × 4 identity: The 3 × 3 identity is denoted by I3 (pronounced as "eye-three" or "eye. The goal of this library is being an easy, but extensible interface between Python and LaTeX. The new row will be inserted immediately below the position of your cursor. I need to combine them into one 10x10 matrix with n1 in the top left corner, n2 in the top right, n3 in the bottom left and n4 in the bottom right, and I am using Mathematica 4. Note that any matrix of the form −2d a d a 0 b d b c does the job. When you write a matrix whose entries are fractions, you might feel the line are cramped. So, for example: It will not surprise you that: A + (-A) = 0 (Notice that that last zero is a bold-faced zero, designating the zero matrix. An m × n (read 'm by n') matrix is an arrangement of numbers (or algebraic expressions ) in m rows and n columns. Subscripts & Superscripts. For example if you multiply a matrix of 'n' x. 0 Sage is free, open-source math software that supports research and teaching in algebra, geometry, number theory, cryptography, numerical computation, and related areas. But nowadays there are many examples in which TEX is used for publications with no mathematical or technical background content. The theory was that the Real World was not actually real, but rather another level of the Matrix simulation: i. HTML LaTeX 公式编辑器可以生成图形公式 (gif, png, swf, pdf, emf). #N#square root ("radical") Symbols in Geometry Symbols in Algebra. How to multiply a Row by a Column? We'll start by showing how to multiply a 1 × n matrix by an n × 1 matrix. \begingroup I have 4 5x5 matrices n1, n2, n3, n4. You can nest enumerations to create detailed outlines. Enumerate is used to make numbered lists. 1 of the official documentation. Ideal for low compression situations, the polyester/cotton weave provides excellent elasticity and recov. I am making it simpler by considering for a 6 by 6 matrix. A CONSORT-style flowchart of a randomized controlled trial [Open in Overleaf] BER measurement on fibre optical system. input : output \begin{matrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{matrix} matrix with parentheses brackets. Download Snip for desktop and start saving time. The LaTeX team cannot guarantee that TeX distributions, even recent ones, contain the most recent version of LaTeX. All the basic matrix operations as well as methods for solving systems of simultaneous linear equations are implemented on this site. Row and column labelling of matrixes in LaTeX. LaTeX Lesson 4 Mathematics in LaTeX. When you write a matrix whose entries are fractions, you might feel the line are cramped. So, the command first. There are no approved revisions of this page, so it may not have been reviewed. Creation of matrices and matrix multiplication is easy and natural: Note that in Sage, the kernel of a matrix A is the "left kernel", i. It makes sense that the number of prey present will affect the number of the predator. Please join the discussion on moodle. where I denotes a unit matrix of order n. Fractions and Roots. What's going on here? LaTeX Wiki contents Using LaTeX is a user's guide for LaTeX Using LaTeX to format documents tells you how to format a complete document using LaTeX Using LaTeX in a wiki describes the subset of LaTeX LaTeX language reference Document classes. Imagine turning a paper to the other side, that is what is happening. 95 per month (cancel anytime). Here are examples: Σ = ⎡ ⎢ ⎢⎣σ11 ⋯ σ1n ⋮ ⋱ ⋮ σn1 ⋯ σnn⎤ ⎥ ⎥⎦ σ 11 ⋯ σ 1 n ⋮ ⋱ ⋮ σ n 1 ⋯ σ n n. radio_button_uncheckedNo. There are different way of placing figures side by side in Latex, subcaption, subfig, subfigure, or even minipage. Use environments matrix for an array without brackets, and pmatrix environment for an array with parentheses. An example of a piecewise step function. X n converges to cin distribution, written X n c, if lim n!1 F n(t) = c(t) (8) at all t6=cwhere c(t) = 0 if t < français > Here are few examples to write quickly matrices. LaTeX2e in 90 minutes, by Tobias Oetiker, Hubert Partl, Irene Hyna, and Elisabeth Schlegl. LaTeX4technics. Gauss's Law, Faraday's Law, the non-existance of magnetic charge, and Ampere's Law are described in an intuitive method, with a focus on understanding above mathematics. tikz-pgf vertical-alignment tikz-matrix. LaTeX has different packages which automatically generates dummy text in our document. It is most often used for medium-to-large technical or scientific documents but it can be used for almost any form of publishing. 4 ! In the process of developing the Digital Library of Mathematical Functions , we needed a means of transforming the LaTeX sources of our material into XML which would be used for further manipulations, rearrangements and construction. The transposition is exactly the same for a non-square matrix. Let A be the following matrix LaTeX: \begin{bmatrix}5&-6&-12\\ 14 & 4&7 \end{bmatrix} [ 5 −6 −12 14 4 7 ] The dimensions of A are Expert Answer 100% (1 rating). 26, 2012 Most of the stock math commands are written for typesetting math or computer science papers for academic journals, so you might need to dig deeper into LaTeX commands to get the vector notation styles that are common in physics textbooks and articles. Product Services. No preview available. But the resulting 2 matrices are S4 object of class. 4 Enumerate. Cured caulk is mold and mildew resistant. A zero vector. Texmaker can also be used to edit and compile directly asymptote figure not embedded in a latex document. Latex Free. Así como cambiando el. MatrixForm prints SparseArray objects like the corresponding ordinary lists. The length of the skip should be expressed in a unit recognized by LaTeX. Here is a matrix of size 2 3 ("2 by 3"), because it has 2 rows and 3 columns: 10 2 015 The matrix consists of 6 entries or elements. To initialize an N-by-M matrix, use the “zeros” function. #README This repository contains the source files for Fundamentals of Matrix Algebra, by Gregory Hartman. To calculate inverse matrix you need to do the following steps. Produce beautiful documents starting from our gallery of LaTeX templates for journals, conferences, theses, reports, CVs and much more. Textext aims to cover this need, by adding re-editable LaTeX objects to Inkscape’s repertoire. This scaled our figure by a factor of 0. 3 7! 2 9! = 69! To obtain the second entry in the solution, ignore the ﬁrst row of the ﬁrst matrix. There are no approved revisions of this page, so it may not have been reviewed. Check that the two matrices can be multiplied together. That ends our list of the 10 best LaTex Editors that you should be using in 2020. is a square matrix. So to widen the gap between lines, use \\[6pt]. 1Introduction This document gives a gallery of tables which can be made using the xtable package to create LATEX output. This does seem silly. The inverse of a 2\times2 matrix is given by swapping the diagonal entries, negating the off-diagonal entries, and dividing by the determinant:\begin{pmatrix}a&b\\c&d\end{pmatrix}^{-1} = \frac{1}{ad-bc} \begin{pmatrix}d & -b \\ -c & a\end{pmatrix}. LaTeX Tutorial pt 6 - Tables and Matrices in LaTeX How to create a matrix larger than 3x3. The column space of a matrix A is the vector space made up of all linear combi nations of the columns of A. 4 He has reported that the amount of free latex protein that can be extracted from powdered latex gloves is consistently higher than the amount that is liberated from non-powdered gloves. Input LaTeX, Tex, AMSmath or ASCIIMath notation (Click icon to switch to ASCIIMath mode) to make formula. opagebreak[4] tells LaTeX not to break the page between the problem and the choices unless it really, really, really has to. Most rubber in everyday use is vulcanized to a point where it shares properties of both; i. Feedback and suggested improvements are welcome. My first equation is now (4). If a graph has vertices, we may associate an matrix which is called vertex matrix or adjacency matrix. External file 3. For example, to obtain in LaTeX, one would type. Eitan died unexpectedly in June 2009; we extend our sympathies to his family, and dedicate future work on the project to his memory. But nowadays there are many examples in which TEX is used for publications with no mathematical or technical background content. I have checked with a matrix calculator and the the determinants of the 3x3 minor matrices are correct. The second row of the original matrix becomes the second column of its transpose. Thus our source is a mixture of text to be typeset and a couple of LATEX commands \emph and \LaTeX. Wow, that is a twist. It is most often used for medium-to-large technical or scientific documents but it can be used for almost any form of publishing. Greek letters are commonly used in mathematics, and they are very easy to type in math mode. , then it tells us that all of space has been squished into a smaller dimension, e. That is, a Jordan matrix is a matrix with Jordan blocks down the diagonal and zeros everywhere else. 3, on Ms Windows 7/10 and Mac OS-X 10. Describe the range of a 3 by 4 matrix using the definition of the range. Among the three important vector spaces associated with a matrix of order m x n is the Null Space. Does someone know how to convert an object of class dgCmatrix > to a regular matrix. Steps 3 and 4 organize the values for the fields first by subcategory, and then by product. The identity matrix is a square matrix containing ones down the main diagonal and zeros everywhere else. The internal ones 2 and 2 tell you if the multiplication is possible (when they are equal) or not (when they are different). Symmetrical monomers such as ethylene and tetrafluoroethylene can join together in only one way. David Whiting david. Derivation of the stiffness matrix of a two-dimensional element (II) by victorroda Before we can continue with the assembly of the stiffness matrix, and in order to determine the matrix for each Gauss integration point , the element geometry matrix , the local derivative matrix , the jacobian matrix and its determinant, and the global. Return the matrix right division of x and y. LaTeX is a powerful tool to typeset math. Thanks for contributing an answer to Mathematica Stack Exchange! Please be sure to answer the question. Word 2016 automatically handles formatting, nonetheless, you can manually adjust the spacing and alignment of equations. VerkhovtsevaKatya. Click on row or column numbers to toggle whole row or column. Matrix algebra for beginners, Part I matrices, determinants, inverses Jeremy Gunawardena Department of Systems Biology Harvard Medical School 200 Longwood Avenue, Cambridge, MA 02115, USA [email protected] First, use a font like lmodern and then you can set the font size:. Enter the Matrix is the story behind the story. This is not a comprehensive list. We offer a wide selection of UltraClean™ surfactant-free latex microspheres—also referred to as latex beads—for research and commercial applications. Define Vector and Matrix Data. It is also suitable for producing all sorts of other documents, from simple letters to. 3 Null Space as a vector space. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. 4) can be concisely expressed as min m 2 = m 0Σm s. A 4x4 matrix can be used to do both rotation and translation in a single matrix. 17pt have occurred. Email to friends Share on Facebook - opens in a new window or tab Share on Twitter - opens in a new window or tab Share on Pinterest - opens in a new window or tab. For example \bigg[\begin{array}{c\|c\|c\|c}. The beamer class User Guide for version 3. One of the greatest motivating forces for Donald Knuth when he began developing the original TeX system was to create something that allowed simple construction of mathematical formulae, while looking professional when printed. Matrices and other arrays are produced in LaTeX using the \textbf{array} environment. If more arguments are given, the multiplication is. ( 0 0 −4 −5 0 3 0 0 −6) o* (0 3 0 −6) giving 0(0−0) = 0. \documentclass{article} \usepackage[left=2cm, right=5cm, top=2cm]{geometry} \begin{document} Some text … left margin is 2 cm, right margin 5 cm and the top margin is 2 cm. TeX4ht was created by Eitan Gurari at Ohio State University. Use the scale=1. Nodes and shapes 15. GroupWork 1: Mark each statement True or False. This keypad has 16 buttons, arranged in a telephone-line 4x4 grid. You must use the following package: \usepackage {amsmath} \begin {matrix} \begin {pmatrix} \begin {bmatrix} \begin {vmatrix} \begin {Vmatrix}. One of the octahedral sites is [ -1/2 1/4 5/6 ]^T , relative to the lattice basis. Just pick a symbol you sometimes need but tend to forget and click it. VerkhovtsevaKatya is a new contributor to this site. "matrix-like" diagrams, such as commutative diagrams,. 4 LaTeX default lengths. In the notebook and at the Sage command-line there is a pre-defined object named latex which has several methods, which you can list by typing latex. MatrixForm acts as a "wrapper", which affects printing, but not evaluation. There's no need to calculate. False, the vector spaces P3 and R^4 are isomorphic (4. opagebreak[4] tells LaTeX not to break the page between the problem and the choices unless it really, really, really has to. But we need a matrix that is linearly independent to both S1,S2. a = 2 1 3 A row vector is a list of numbers written one after the other, e. To initialize an N-by-M matrix, use the “zeros” function. This site is supported by donations to The OEIS Foundation. I can only select from the predefined Matrix sizes. If A is a square matrix, then there exists a matrix B such that AB equals the identity matrix. Overleaf is so easy to get started with that you'll be able to invite your non-LaTeX colleagues to contribute directly to your LaTeX documents. I want to write a 4x4 matrix in MS Word 2007 using equation editor. Use the inverse matrix to solve the system of equations. MediaWiki renders mathematical equations using a combination of html and a variant of LaTeX. An matrix J is said to be in Jordan canonical form if it is a matrix of the form where each is either a diagonal matrix or a Jordan block matrix. The author introduces it in Chapter Four using linear. You weren’t required to compute this,. This add-on lets you automatically convert every LaTeX equation in your document into beautiful images! Simply enclose your math equations within and click the button in the sidebar, and all. The process by which the rank of a matrix is determined can be illustrated by the following example. 0 Sage is free, open-source math software that supports research and teaching in algebra, geometry, number theory, cryptography, numerical computation, and related areas. Creation of matrices and matrix multiplication is easy and natural: Note that in Sage, the kernel of a matrix A is the “left kernel”, i. Carrie-Anne Moss was born and raised in Vancouver, Canada. If you know how to change this to I could just have \bmat{a&b}{c&d} and not even need \bmatrows{2}, then that would be even better. To find the inverse of matrix A, using Gauss-Jordan elimination, it must be found the sequence of elementary row operations that reduces A to the identity and, then, the same operations on I_n must be performed to obtain A^ {-1}. Click on "Insert" and choose what you want to add. By changing the font size locally, however, a single word, a few lines of text, a large table or. This matrix can be used to obtain more detailed information about the graph. We can also scale the picture by explicitly setting the width and the height by using the width and height optional commands. There are three commonly used environments in the math mode: the math environment: Used for formulas in running text ; the displaymath environment: Used to display longer formulas ; the equation environment: Used for displaying equations for numbering and cross reference. calculate expression inside first. Click the "Insert" tab 2. Here are examples: Σ = ⎡ ⎢ ⎢⎣σ11 ⋯ σ1n ⋮ ⋱ ⋮ σn1 ⋯ σnn⎤ ⎥ ⎥⎦ σ 11 ⋯ σ 1 n ⋮ ⋱ ⋮ σ n 1 ⋯ σ n n. Some of these commands are LATEX macros, while oth-. Basic equations in L a T e X can be easily "programmed", for. A User of Auto-LaTeX Equations. 95) ADD TO WISHLIST. 5 Basis of Null Space. com : Elastic Bandage, Latex Free, Self Closure, 4"X 5 Yds. No preview available. If output is a data type, it specifies the type of the resulting labeled feature array. This comes from Mathematica (IIRC) and is not otherwise standardized, but there may be some disambiguation benefit in using the double-struck italic letters from Unicode’s Letterlike Symbols block: U+2148 ⅈ and U+2149 ⅉ. The root of a quadratic equation however, can be either positive or negative. 5) has the matrix representation ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 2 2 2 2 1 2 2 2 2 1 2 2 2 2 1 11 10 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. This article explains how to insert spaces of different lengths in mathematical mode. I need to combine them into one 10x10 matrix with n1 in the top left corner, n2 in the top right, n3 in the bottom left and n4 in the bottom right, and I am using Mathematica 4. 3 bronze badges. LaTeX is a typesetting language for producing scientific documents. I am looking to get a colormap to assign a unique color to each point in a plot based on their four Ci values (i. There are various ways to adjust the numbering in enumerate (search “latex and enumerate” in Google. My matrix contains fraction elements and single digit integers. There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them. Export (png, jpg, gif, svg, pdf) and save & share with note system. It doubles as a regression check for the package. The syntax is exactly the same. The mathematical symbol is produced using \partial. The idea behind the minipage command is that within an existing page "built in" an additional page. VerkhovtsevaKatya. Augmented Matrix Calculator is a free online tool that displays the resultant variable value of an augmented matrix for the two matrices. Get the free "inverse matrix" widget for your website, blog, Wordpress, Blogger, or iGoogle. I created a table in Latex, but currently it is too big. matrix list V symmetric V[3,3] weight displacement _cons weight 1. Suppose that we have the following diagonal matrix. calculate expression inside first. For example, if [A] is a 4 x 3 matrix (4 rows, 3 columns) and [B] is a 2 x 2 matrix (2 rows, 2. 2013;4:204173141350530. have a solution. You can think of these two points as "attractor points". The column space of a matrix A is the vector space made up of all linear combi nations of the columns of A. One of the octahedral sites is [ -1/2 1/4 5/6 ]^T , relative to the lattice basis. Use MathJax to format equations. MathJax basic tutorial and quick reference. Table 200: ℳ Letter-like Symbols. com and remembering the cases redefinition I’ve shown some days ago I got the idea to extend the internal macro \[email protected] of amsmath. They are organized into seven classes based on their role in a mathematical expression. Enumerate is used to make numbered lists. To create a new row, use. Jump to navigation Jump to search. You can nest enumerations to create detailed outlines. 2) sup Supremum of a set jjAjj Matrix norm (subscript if any denotes what norm) AT Transposed matrix A TThe inverse of the transposed and vice versa, A T = (A 1)T = (A ). You can generate them with just a few lines of code. a = 2 1 3 A row vector is a list of numbers written one after the other, e. The defining property for the gamma matrices to generate a Clifford algebra is the anticommutation relation {,} = + =,where {,} is the anticommutator, is the Minkowski metric with signature (+ − − −), and is the 4 × 4 identity matrix. So to convert a 3x3 matrix to a 4x4, you simply copy in the values for the 3x3 upper left block, like so:. edited yesterday. Out [1] //MatrixForm= Show the matrix form of a SparseArray:. Tutorial o curso para escribir matrices dos formas, usando el entorno array o usando entornos ya definidos, matrix, pmatrix, vmatrix, bmatrix, en LaTeX utilizando Texmaker. No preview available.	2020-07-15T05:49:53	{ "domain": "effebitrezzano.it", "url": "http://rzby.effebitrezzano.it/4-4-matrix-latex.html", "openwebmath_score": 0.7885562777519226, "openwebmath_perplexity": 1173.5944962037668, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. Yes\n2. Yes", "lm_q1_score": 0.9553191348157373, "lm_q2_score": 0.9046505376715775, "lm_q1q2_score": 0.864229968959003 }
https://gmatclub.com/forum/how-many-zeros-does-100-end-with-100599.html?sort_by_oldest=true	GMAT Question of the Day - Daily to your Mailbox; hard ones only It is currently 06 Dec 2019, 18:46 ### 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 Your Progress 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 # How many zeros does 100! end with? new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Author Message TAGS: ### Hide Tags Intern Joined: 22 Jun 2010 Posts: 36 How many zeros does 100! end with? [#permalink] ### Show Tags Updated on: 20 Feb 2019, 03:42 4 70 00:00 Difficulty: 5% (low) Question Stats: 84% (00:30) correct 16% (01:08) wrong based on 1878 sessions ### HideShow timer Statistics How many zeros does 100! end with? A. 20 B. 24 C. 25 D. 30 E. 32 Spoiler: :: Solution Find how many times the factor 5 is contained in 100!. That is, we have to find the largest such that 100! is divisible by . There are 20 multiples of 5 in the first hundred but 25, 50, 75, and 100 have to be counted twice because they are divisible by $$25 = 5^2$$ . So, the answer is 24. The correct answer is B. I have absolutely no Idea what they are telling me... Can someone please post a simple explanation for the rationale behind the explanation, or (even better) provide an alternative simple approach? Thanks! Originally posted by AndreG on 07 Sep 2010, 13:41. Last edited by Bunuel on 20 Feb 2019, 03:42, edited 1 time in total. Renamed the topic and edited the question. ##### Most Helpful Expert Reply Math Expert Joined: 02 Sep 2009 Posts: 59587 Re: How many zeros does 100! end with? [#permalink] ### Show Tags 07 Sep 2010, 13:49 34 31 AndreG wrote: How many zeros does 100! end with? • 20 • 24 • 25 • 30 • 32 expl. Find how many times the factor 5 is contained in 100!. That is, we have to find the largest such that 100! is divisible by . There are 20 multiples of 5 in the first hundred but 25, 50, 75, and 100 have to be counted twice because they are divisible by $$25 = 5^2$$ . So, the answer is 24. The correct answer is B. I have absolutely no Idea what they are telling me... Can someone please post a simple explanation for the rationale behind the explanation, or (even better) provide an alternative simple approach? Thanks! Trailing zeros: Trailing zeros are a sequence of 0's in the decimal representation (or more generally, in any positional representation) of a number, after which no other digits follow. Fro example, 125000 has 3 trailing zeros (125000); The number of trailing zeros in the decimal representation of n!, the factorial of a non-negative integer n, can be determined with this formula: $$\frac{n}{5}+\frac{n}{5^2}+\frac{n}{5^3}+...+\frac{n}{5^k}$$, where k must be chosen such that 5^(k+1)>n It's more simple if you look at an example: How many zeros are in the end (after which no other digits follow) of 32!? $$\frac{32}{5}+\frac{32}{5^2}=6+1=7$$ (denominator must be less than 32, $$5^2=25$$ is less) So there are 7 zeros in the end of 32! The formula actually counts the number of factors 5 in n!, but since there are at least as many factors 2, this is equivalent to the number of factors 10, each of which gives one more trailing zero. BACK TO THE ORIGINAL QUESTION: According to above 100! has $$\frac{100}{5}+\frac{100}{25}=20+4=24$$ trailing zeros. Answer: B. For more on this issues check Factorials and Number Theory links in my signature. Hope it helps. _________________ ##### General Discussion Retired Moderator Joined: 02 Sep 2010 Posts: 717 Location: London Re: How many zeros does 100! end with? [#permalink] ### Show Tags 18 Oct 2010, 02:40 feruz77 wrote: the product of all integers from 1 to 100 will have the following number of zeros at the end: a) 20 b) 24 c) 19 d) 22 e) 28 pls, help with solution method! Search through the forums (read the math book). There is several threads discussing this. The number of trailing zeros in 100! is (100/5)+(100/25)=24 Answer : (b) _________________ Intern Joined: 24 Jun 2010 Posts: 11 Re: How many zeros does 100! end with? [#permalink] ### Show Tags 18 Oct 2010, 08:22 the question can be re-framed as (100!/10^x) now find x? 100!/(25)^x---now factorize 100! by 2 and 5 when factorized by 5 will give the least power 24 Ans 24 Director Joined: 23 Apr 2010 Posts: 503 Re: How many zeros does 100! end with? [#permalink] ### Show Tags 28 Dec 2010, 05:09 Is this a relevant GMAT question? Thank you. Math Expert Joined: 02 Sep 2009 Posts: 59587 Re: How many zeros does 100! end with? [#permalink] ### Show Tags 28 Dec 2010, 05:22 nonameee wrote: Is this a relevant GMAT question? Thank you. Why do you think that it's not? I've seen several GMAT questions testing the trailing zeros concept (everything-about-factorials-on-the-gmat-85592.html): factorial-one-106060.html product-of-sequence-101187.html how-many-zeroes-at-the-end-of-101752.html least-value-of-n-m09q33-76716.html number-properties-from-gmatprep-84770.html _________________ Manager Status: Student Joined: 26 Aug 2013 Posts: 166 Location: France Concentration: Finance, General Management Schools: EMLYON FT'16 GMAT 1: 650 Q47 V32 GPA: 3.44 Re: How many zeros does 100! end with? [#permalink] ### Show Tags 09 Jan 2014, 15:46 Hi Bunuel, just a question that I have in mind: You ALWAYS divide by 5? You never divide by another prime factor for those type of questions? if yes what are the exceptions? Thanks! _________________ Think outside the box Director Affiliations: CrackVerbal Joined: 03 Oct 2013 Posts: 563 Location: India GMAT 1: 780 Q51 V46 Re: How many zeros does 100! end with? [#permalink] ### Show Tags 09 Jan 2014, 23:59 2 1 1 AndreG wrote: How many zeros does 100! end with? A. 20 B. 24 C. 25 D. 30 E. 32 expl. Find how many times the factor 5 is contained in 100!. That is, we have to find the largest such that 100! is divisible by . There are 20 multiples of 5 in the first hundred but 25, 50, 75, and 100 have to be counted twice because they are divisible by $$25 = 5^2$$ . So, the answer is 24. The correct answer is B. I have absolutely no Idea what they are telling me... Can someone please post a simple explanation for the rationale behind the explanation, or (even better) provide an alternative simple approach? Thanks! 100! = 1 x 2 x 3 x ..... x 100 10 = 5 x 2 2s are in abundance however there is limited supply of 5s How many multiples of 5 are there from 1 to 100- One way is counting other way is 100/5 = 20 How many multiples of 25 are there which contain an extra five = 100/25 = 4 There is no point going forward as the next power of 5 is 125 which is greater than 100. That does it: 20 + 4 = 24 _________________ - CrackVerbal Prep Team For more info on GMAT and MBA, follow us on @AskCrackVerbal Register for the Free GMAT Kickstarter Course : http://bit.ly/2DDHKHq Register for our Personal Tutoring Course : https://www.crackverbal.com/gmat/personal-tutoring/ Join the free 4 part GMAT video training series : http://bit.ly/2DGm8tR Math Expert Joined: 02 Sep 2009 Posts: 59587 Re: How many zeros does 100! end with? [#permalink] ### Show Tags 10 Jan 2014, 02:46 1 2 Paris75 wrote: Hi Bunuel, just a question that I have in mind: You ALWAYS divide by 5? You never divide by another prime factor for those type of questions? if yes what are the exceptions? Thanks! For trailing zeros we need only the power of 5. Check here for theory: everything-about-factorials-on-the-gmat-85592.html Similar questions to practice: if-n-is-the-greatest-positive-integer-for-which-2n-is-a-fact-144694.html what-is-the-largest-power-of-3-contained-in-103525.html if-n-is-the-product-of-all-positive-integers-less-than-103218.html if-n-is-the-product-of-integers-from-1-to-20-inclusive-106289.html if-n-is-the-product-of-all-multiples-of-3-between-1-and-101187.html if-p-is-the-product-of-integers-from-1-to-30-inclusive-137721.html what-is-the-greatest-value-of-m-such-that-4-m-is-a-factor-of-105746.html if-6-y-is-a-factor-of-10-2-what-is-the-greatest-possible-129353.html if-m-is-the-product-of-all-integers-from-1-to-40-inclusive-108971.html if-p-is-a-natural-number-and-p-ends-with-y-trailing-zeros-108251.html if-73-has-16-zeroes-at-the-end-how-many-zeroes-will-147353.html find-the-number-of-trailing-zeros-in-the-expansion-of-108249.html how-many-zeros-are-the-end-of-142479.html how-many-zeros-does-100-end-with-100599.html find-the-number-of-trailing-zeros-in-the-product-of-108248.html if-60-is-written-out-as-an-integer-with-how-many-consecuti-97597.html if-n-is-a-positive-integer-and-10-n-is-a-factor-of-m-what-153375.html if-d-is-a-positive-integer-and-f-is-the-product-of-the-first-126692.html Hope it helps. _________________ Director Joined: 17 Dec 2012 Posts: 623 Location: India Re: How many zeros does 100! end with? [#permalink] ### Show Tags 05 Apr 2014, 22:42 3 What does this question test? It asks for the number of zeros 100! ends with. If a number ends with zero it has to be a multiple of 10. So essentially we have to find out how many times in 100! does multiplication by 10 happen. But multiplication by 10 also happens when multiplication by both 5 and 2 happen. So it is better to see the factors of the number and then find the number of times multiplication by all those factors happen. Consider the simpler example. 456. How many zeros does it end with? We see 2 occurs twice in 4 which is 22, and once in 6 which is 23 and 5 occurs only once. So 5 is a limiting factor. Since 5 occurs only once, the number of times multiplication by both 5 and 2 happen or in other words the number of times multiplication by 10 happens or the number of zeros the number ends with is only 1. In the case of 100!, 5 occurs in 5, 10, 15, 20 and so on up to 100 i.e, 20 times. But remember 5 occurs twice in 25 which is 55 , twice in 50 which is 552 and similarly twice each in 75 and 100. So it actually occurs 24 times. 2 occurs a lot more times and so 5 is the limiting factor. Applying the same logic as in the simpler example, the number of zeros 100! ends with is 24. _________________ Srinivasan Vaidyaraman Sravna Test Prep http://www.sravnatestprep.com Holistic and Systematic Approach Intern Joined: 08 Jun 2014 Posts: 4 Re: How many zeros does 100! end with? [#permalink] ### Show Tags 10 Dec 2015, 19:33 Hi Guys, Am still abit confused about the whole concept. Some clarification would be appreciated. With the formula, I see that the trailing zeros for the following are: 1) For 32!, (32/5) + (32/25) = 7 2) For 25!, (25/5) + (25/25) = 6 3) For 10!, (10/5) = 2 4) For 5!, (5/5) = 1 However, as i plug the numbers in and count on the calculator, the answer is different 1) For 32!, 263,130,836,933,693,000,000,000,000,000,000,000. A total of 21 trailing zeros 2) For 25!, 15,511,210,043,331,000,000,000,000. A total of 12 trailing zeros 3) For 10!, 3,628,800. A total of 2 trailing zeros 4) For 5!, 120. A total of 1 trailing zeros Why is it that the first 2 doesnt comply whereas the last 2 does. I mean I understand the answer is correct here but am just trying to get my head around this. Math Expert Joined: 02 Sep 2009 Posts: 59587 Re: How many zeros does 100! end with? [#permalink] ### Show Tags 11 Dec 2015, 00:53 scottleey wrote: Hi Guys, Am still abit confused about the whole concept. Some clarification would be appreciated. With the formula, I see that the trailing zeros for the following are: 1) For 32!, (32/5) + (32/25) = 7 2) For 25!, (25/5) + (25/25) = 6 3) For 10!, (10/5) = 2 4) For 5!, (5/5) = 1 However, as i plug the numbers in and count on the calculator, the answer is different 1) For 32!, 263,130,836,933,693,000,000,000,000,000,000,000. A total of 21 trailing zeros 2) For 25!, 15,511,210,043,331,000,000,000,000. A total of 12 trailing zeros 3) For 10!, 3,628,800. A total of 2 trailing zeros 4) For 5!, 120. A total of 1 trailing zeros Why is it that the first 2 doesnt comply whereas the last 2 does. I mean I understand the answer is correct here but am just trying to get my head around this. Use better calculator: http://www.wolframalpha.com _________________ Senior Manager Joined: 20 Aug 2015 Posts: 384 Location: India GMAT 1: 760 Q50 V44 Re: How many zeros does 100! end with? [#permalink] ### Show Tags 11 Dec 2015, 04:31 scottleey wrote: Hi Guys, Am still abit confused about the whole concept. Some clarification would be appreciated. With the formula, I see that the trailing zeros for the following are: 1) For 32!, (32/5) + (32/25) = 7 2) For 25!, (25/5) + (25/25) = 6 3) For 10!, (10/5) = 2 4) For 5!, (5/5) = 1 However, as i plug the numbers in and count on the calculator, the answer is different 1) For 32!, 263,130,836,933,693,000,000,000,000,000,000,000. A total of 21 trailing zeros 2) For 25!, 15,511,210,043,331,000,000,000,000. A total of 12 trailing zeros 3) For 10!, 3,628,800. A total of 2 trailing zeros 4) For 5!, 120. A total of 1 trailing zeros Why is it that the first 2 doesnt comply whereas the last 2 does. I mean I understand the answer is correct here but am just trying to get my head around this. I think there is some problem with the calculator that you are using 32! = 263130836933693530167218012160000000 25! = 15511210043330985984000000 Both of these values comply with our understanding. You can try the calculation here: http://www.calculatorsoup.com/calculato ... orials.php Intern Joined: 13 Feb 2017 Posts: 1 Re: How many zeros does 100! end with? [#permalink] ### Show Tags 04 May 2017, 20:53 Bunuel wrote: AndreG wrote: How many zeros does 100! end with? • 20 • 24 • 25 • 30 • 32 expl. Find how many times the factor 5 is contained in 100!. That is, we have to find the largest such that 100! is divisible by . There are 20 multiples of 5 in the first hundred but 25, 50, 75, and 100 have to be counted twice because they are divisible by $$25 = 5^2$$ . So, the answer is 24. The correct answer is B. I have absolutely no Idea what they are telling me... Can someone please post a simple explanation for the rationale behind the explanation, or (even better) provide an alternative simple approach? Thanks! Trailing zeros: Trailing zeros are a sequence of 0's in the decimal representation (or more generally, in any positional representation) of a number, after which no other digits follow. Fro example, 125000 has 3 trailing zeros (125000); The number of trailing zeros in the decimal representation of n!, the factorial of a non-negative integer n, can be determined with this formula: $$\frac{n}{5}+\frac{n}{5^2}+\frac{n}{5^3}+...+\frac{n}{5^k}$$, where k must be chosen such that 5^(k+1)>n It's more simple if you look at an example: How many zeros are in the end (after which no other digits follow) of 32!? $$\frac{32}{5}+\frac{32}{5^2}=6+1=7$$ (denominator must be less than 32, $$5^2=25$$ is less) So there are 7 zeros in the end of 32! The formula actually counts the number of factors 5 in n!, but since there are at least as many factors 2, this is equivalent to the number of factors 10, each of which gives one more trailing zero. BACK TO THE ORIGINAL QUESTION: According to above 100! has $$\frac{100}{5}+\frac{100}{25}=20+4=24$$ trailing zeros. Answer: B. For more on this issues check Factorials and Number Theory links in my signature. Hope it helps. Hi All, I have found a faster approach to solve these kind of questions: For trailing zero's: we need to check how many 5's are there in the number. So we can just divide the n by 5 and add the quotients: 100! 5 \| 100 5 \| 20 5 \| 4 5 \| 0 Add 20+4+0 = 24 Target Test Prep Representative Status: Head GMAT Instructor Affiliations: Target Test Prep Joined: 04 Mar 2011 Posts: 2809 Re: How many zeros does 100! end with? [#permalink] ### Show Tags 14 Jul 2018, 19:12 AndreG wrote: How many zeros does 100! end with? A. 20 B. 24 C. 25 D. 30 E. 32 We are looking the number of 2-and-5 pairs in 100! since each pair produces a factor of 10 and thus a 0 at the end of 100!. There are more factors of 2 than 5 in 100!, so the question is: how many factors 5 are there? We can use the following trick: Divide 100 by 5 and its subsequent quotients by 5 as long as the quotient is nonzero (and each time ignore any nonzero remainder). The final step is add up all these nonzero quotients and that will be the number of factors of 5 in 100!. 100/5 = 20 20/5 = 4 Since 4/5 has a zero quotient, we can stop here. We see that 20 + 4 = 24, so there are 24 factors 5 (and hence 10) in 100!. So 100! ends with 24 zeros. Answer: B _________________ # Jeffrey Miller Head of GMAT Instruction [email protected] 122 Reviews 5-star rated online GMAT quant self study course See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews If you find one of my posts helpful, please take a moment to click on the "Kudos" button. Intern Joined: 08 Oct 2018 Posts: 2 Location: Ukraine Re: How many zeros does 100! end with? [#permalink] ### Show Tags 16 Nov 2018, 04:51 AndreG wrote: How many zeros does 100! end with? A. 20 B. 24 C. 25 D. 30 E. 32 expl. Find how many times the factor 5 is contained in 100!. That is, we have to find the largest such that 100! is divisible by . There are 20 multiples of 5 in the first hundred but 25, 50, 75, and 100 have to be counted twice because they are divisible by $$25 = 5^2$$ . So, the answer is 24. The correct answer is B. I have absolutely no Idea what they are telling me... Can someone please post a simple explanation for the rationale behind the explanation, or (even better) provide an alternative simple approach? Thanks! I took 100! in excel, and I've got this = 9.3326E+157 So, 100! ends with a lot more 0 than 24. Did i miss something? Really appreciated for explanation. Math Expert Joined: 02 Sep 2009 Posts: 59587 Re: How many zeros does 100! end with? [#permalink] ### Show Tags 16 Nov 2018, 04:54 2 olgaromazan wrote: AndreG wrote: How many zeros does 100! end with? A. 20 B. 24 C. 25 D. 30 E. 32 expl. Find how many times the factor 5 is contained in 100!. That is, we have to find the largest such that 100! is divisible by . There are 20 multiples of 5 in the first hundred but 25, 50, 75, and 100 have to be counted twice because they are divisible by $$25 = 5^2$$ . So, the answer is 24. The correct answer is B. I have absolutely no Idea what they are telling me... Can someone please post a simple explanation for the rationale behind the explanation, or (even better) provide an alternative simple approach? Thanks! I took 100! in excel, and I've got this = 9.3326E+157 So, 100! ends with a lot more 0 than 24. Did i miss something? Really appreciated for explanation. In short - don't trust Excel, trust experts above. 100! = 93326215443944152681699238856266700490715968264381621468592963895 217599993229915608941463976156518286253697920827223758251185210916864 000000000000000000000000 _________________ Intern Joined: 03 Apr 2018 Posts: 38 Re: How many zeros does 100! end with? [#permalink] ### Show Tags 20 Sep 2019, 06:16 Bunuel wrote: AndreG wrote: How many zeros does 100! end with? • 20 • 24 • 25 • 30 • 32 expl. Find how many times the factor 5 is contained in 100!. That is, we have to find the largest such that 100! is divisible by . There are 20 multiples of 5 in the first hundred but 25, 50, 75, and 100 have to be counted twice because they are divisible by $$25 = 5^2$$ . So, the answer is 24. The correct answer is B. I have absolutely no Idea what they are telling me... Can someone please post a simple explanation for the rationale behind the explanation, or (even better) provide an alternative simple approach? Thanks! Trailing zeros: Trailing zeros are a sequence of 0's in the decimal representation (or more generally, in any positional representation) of a number, after which no other digits follow. Fro example, 125000 has 3 trailing zeros (125000); The number of trailing zeros in the decimal representation of n!, the factorial of a non-negative integer n, can be determined with this formula: $$\frac{n}{5}+\frac{n}{5^2}+\frac{n}{5^3}+...+\frac{n}{5^k}$$, where k must be chosen such that 5^(k+1)>n It's more simple if you look at an example: How many zeros are in the end (after which no other digits follow) of 32!? $$\frac{32}{5}+\frac{32}{5^2}=6+1=7$$ (denominator must be less than 32, $$5^2=25$$ is less) So there are 7 zeros in the end of 32! The formula actually counts the number of factors 5 in n!, but since there are at least as many factors 2, this is equivalent to the number of factors 10, each of which gives one more trailing zero. BACK TO THE ORIGINAL QUESTION: According to above 100! has $$\frac{100}{5}+\frac{100}{25}=20+4=24$$ trailing zeros. Answer: B. For more on this issues check Factorials and Number Theory links in my signature. Hope it helps. Hi Bunuel, can you please explain why K = 2 and not k=3 this time as 125>100. (as explained in the formula above 5^(K+1)>n Re: How many zeros does 100! end with? [#permalink] 20 Sep 2019, 06:16 Display posts from previous: Sort by # How many zeros does 100! end with? new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Powered by phpBB © phpBB Group \| Emoji artwork provided by EmojiOne	2019-12-07T01:47:20	{ "domain": "gmatclub.com", "url": "https://gmatclub.com/forum/how-many-zeros-does-100-end-with-100599.html?sort_by_oldest=true", "openwebmath_score": 0.8042828440666199, "openwebmath_perplexity": 1646.7932991234186, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. Yes\n2. Yes\n", "lm_q1_score": 0.9511422213778251, "lm_q2_score": 0.9086179037377831, "lm_q1q2_score": 0.8642248513448179 }
https://math.stackexchange.com/questions/1892082/prove-that-x-is-rational-where-x-is-mixed-with-other-variables	# Prove that $x$ is rational where $x$ is mixed with other variables Hi I am trying to complete the following question in my practice: Suppose a, b, c are integers and x, y and z are non-zero real numbers that satisfy the following equations: $$\cfrac{xy}{x+y} = a \quad and \quad \cfrac{xz}{x+z} = b \quad and \quad \cfrac{yz}{y+z} = c$$ Prove that $x$ is rational Source: Discrete Mathematics with Applications 4th edition I tried combining the equations and logically conclude that $x$ is rational but to no avail as I do not know how to separate $x$ from the rest of the variables. The only starting point that I have is the theorem for rational numbers: $$if \; x \in \mathbb{Q}, \; x = \cfrac{a}{b} \; where \; a,b \in \mathbb{Z} \; and \; b \neq 0$$ I have no other idea how to carry on from here. Could someone please advise me? • The question you quote doesn't go on to say to prove $x,y,z$ are rational, but I assume that's what you meant. Did you just forget to include that question at the end? – coffeemath Aug 14 '16 at 15:41 • Thanks for spotting the mistake. I have corrected my post. – LanceHAOH Aug 14 '16 at 16:28 By rewriting we get $$\frac{xy}{x+y} = \frac{1}{\frac{1}{x}+\frac{1}{y}}$$ so we can rewrite the equations as $$\frac{1}{x}+\frac{1}{y} = \frac{1}{a}, ~~\frac{1}{x}+\frac{1}{z} = \frac{1}{b}, ~~\frac{1}{y}+\frac{1}{z} = \frac{1}{c}.$$ This is a regular system of linear equations in $\frac{1}{x}, \frac{1}{y}$ and $\frac{1}{z}$, and since all the coefficients are in $\mathbb{Q}$, the solutions must be in $\mathbb{Q}$ as well. Feel free to solve that last system if you like. • I don't understand why solution has to be rational since coefficients are rational – LanceHAOH Aug 14 '16 at 16:48 • If we interpret the LES as a linear equation system over the field of rational numbers, we have nonzero determinant and get a unique rational solution. If we interpret the LES as a linear equation system over the field of real numbers, we still have nonzero determinant and get a unique real solution. Since the rational solution is also a real solution, both solutions must be identical. – Anon Aug 14 '16 at 16:59 Since $x,y,z$ are nonzero one can define $u=1/x,v=1/y,w=1/z,$ and it also follows from the given equations that none of $a,b,c$ are zero. So we can also put $d=1/a,e=1/b,f=1/c.$ Now the equations, after taking reciprocals, are $u+v=d,u+w=e,v+w=f$ which can be solved rationally for $u,v,w.$ Suppose $x$ is irrational. Then $$\frac{\frac{xy}{x+y}}{\frac{xz}{x+z}} = \frac ab$$, since $b \ne 0$. Now $$\frac{xy(x+z)}{xz(x+y)} = \frac ab \implies \frac{x+z}{x+y} = \frac{az}{by} = q \in \Bbb Q$$, since $y \ne 0$ So $x+z = qx+qy \implies (1-q)x = qy - z \in \Bbb Q$, but then $1-q \notin \Bbb Q$ ! Thus, leading to a contradiction. Note that this same argument could be applied to $y, z$ as well, so it shows at once that all of them have to be rational. Inverting the equations shows $\,x^{-1}\!+y^{-1},\,\ z^{-1}\!+x^{-1},\,\ \color{#c00}{y^{-1}\!+z^{-1}}\in \Bbb Q,\$ i.e. are all rational. Adding them $\ 2(x^{-1}\!+y^{-1}\!+z^{-1}) =:q\in \Bbb Q\$ so $\ x^{-1} = q/2-(\color{#c00}{y^{-1}\!+z^{-1}}) \in\Bbb Q.\ \$ QED • How does being able to invert indicate that the expressions are rational? – LanceHAOH Aug 14 '16 at 17:51 • @Lance Inverting equation $\,1\,$ yields $\,\dfrac{x+y}{xy} = \dfrac{1}a,\$ i.e. $\,y^{-1}\!+x^{-1} = a^{-1}\in \Bbb Q,\,$ etc for others. – Bill Dubuque Aug 14 '16 at 17:54	2020-01-19T05:24:42	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/1892082/prove-that-x-is-rational-where-x-is-mixed-with-other-variables", "openwebmath_score": 0.8621484637260437, "openwebmath_perplexity": 182.12232761416328, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9793540686581882, "lm_q2_score": 0.8824278618165526, "lm_q1q2_score": 0.8642093167673863 }
https://math.stackexchange.com/questions/2700749/a-closed-ball-in-a-metric-space-is-closed	# A closed ball in a metric space is closed Prove that any closed ball in a metric space is closed. (Note that this is not a duplicate as this is a proof verification question and have a different proof, in my opinion, as compared to other proofs on this site) My Attempted Proof: Let $(X, d)$ be a metric space. Pick a closed ball $\Phi = \overline{B(x, r)} = \{y \in X \ \| \ d(x, y) \leq r\}$. Note that at this moment in time $\overline{B(x, r)}$ is just a notation, we don't know if $\Phi$ is actually closed. We now show that $\Phi$ is closed. To do this we show that $X \setminus \Phi$ is open. Observe that $X \setminus \Phi = \{y \in X \ \| \ d(x, y) > r\}$. Pick $y \in X \setminus \Phi$. For this $y$ we have $d(x, y) > r$ which implies that $d(x, y) = r + \epsilon$ for some $\epsilon > 0$. We claim that $B(y, \epsilon) \subseteq X \setminus \Phi$. To prove this claim, pick $z \in B(y, \epsilon)$. We now show that $z \in X \setminus \Phi$ by showing that $d(x, z) > r$. To that end observe that the triangle inequality gives us \begin{align} & d(x, y) \leq d(x, z) + d(z, y) \\ & \implies \epsilon + r \leq d(x, z) + d(y, z) \\ & \implies \epsilon + r < d(x, z) + \epsilon \ \ \ \ \ \ \ \ \ \text{since $d(y, z) < \epsilon$} \\ & \implies r < d(x, z) \end{align} as desired. Hence $z \in X \setminus \Phi$ and we have $B(y, \epsilon) \subseteq X \setminus \Phi$, and since $y$ was arbitrary we have that $X \setminus \Phi$ is an open set in $(X, d)$ and thus $\Phi$ is a closed set. $\square$ Is this a rigorous and satisfactory proof? Can it be improved in any way? • yes, its correct. I assume, from the proof, that an open set was defined as a set where there exists some open ball centered at every point that is contained in the set. – Masacroso Mar 20 '18 at 19:52 • Note that the notation of $\overline{B(x,r)}$ is reserved for the closure of the open ball, rather than the closed ball. – TrostAft Mar 20 '18 at 19:55 • why $d(y,z)<\varepsilon$ with the same $varepsilon$ of $r-\varepsilon$? – gbox Jul 23 at 14:52 $\overline{B(x, r)}$ is used for the closure of the open ball B(x, r) not the closed ball. $\overline{B}(x,r)$ is a better notation for a closed ball. Yes, it is rigorous and satisfactory. It can be shortened, though:$$d(x,z)\geqslant d(x,y)-d(z,y)>r+\varepsilon-\varepsilon=r.$$	2019-10-18T18:30:58	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/2700749/a-closed-ball-in-a-metric-space-is-closed", "openwebmath_score": 0.9971770644187927, "openwebmath_perplexity": 102.84846519537312, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9811668706602658, "lm_q2_score": 0.880797085800514, "lm_q1q2_score": 0.8642089203615719 }
https://pythonnumericalmethods.berkeley.edu/notebooks/chapter17.05-Newtons-Polynomial-Interpolation.html	This notebook contains an excerpt from the Python Programming and Numerical Methods - A Guide for Engineers and Scientists, the content is also available at Berkeley Python Numerical Methods. The copyright of the book belongs to Elsevier. We also have this interactive book online for a better learning experience. The code is released under the MIT license. If you find this content useful, please consider supporting the work on Elsevier or Amazon! Newton’s Polynomial Interpolation¶ Newton’s polynomial interpolation is another popular way to fit exactly for a set of data points. The general form of the an $$n-1$$ order Newton’s polynomial that goes through $$n$$ points is: $f(x) = a_0 + a_1(x-x_0) + a_2(x-x_0)(x-x_1) + \dots + a_n(x-x_0)(x-x_1)\dots(x-x_n)$ which can be re-written as: $f(x) = \sum_{i=0}^{n}{a_in_i(x)}$ where $$$n_i(x) = \prod_{j=0}^{i-1}(x-x_j)$$$ The special feature of the Newton’s polynomial is that the coefficients $$a_i$$ can be determined using a very simple mathematical procedure. For example, since the polynomial goes through each data points, therefore, for a data points $$(x_i, y_i)$$, we will have $$f(x_i) = y_i$$, thus we have $f(x_0) = a_0 = y_0$ And $$f(x_1) = a_0 + a_1(x_1-x_0) = y_1$$, by rearranging it to get $$a_1$$, we will have: $a_1 = \frac{y_1 - y_0}{x_1 - x_0}$ Now, insert data points $$(x_2, y_2)$$, we can calculate $$a_2$$, and it is in the form: $a_2 = \frac{\frac{y_2 - y_1}{x_2 - x_1} - \frac{y_1 - y_0}{x_1 - x_0}}{x_2 - x_0}$ Let’s do one more data points $$(x_3, y_3)$$ to calculate $$a_3$$, after insert the data point into the equation, we get: $a_3 = \frac{\frac{\frac{y_3-y_2}{x_3-x_2} - \frac{y_2 - y_1}{x_2-x_1}}{x_3 - x_1} - \frac{\frac{y_2-y_1}{x_2-x_1}-\frac{y_1 - y_0}{x_1 - x_0}}{x_2-x_0}}{x_3 - x_0}$ Now, see the patterns? These are called divided differences, if we define: $f[x_1, x_0] = \frac{y_1 - y_0}{x_1 - x_0}$ $f[x_2, x_1, x_0] = \frac{\frac{y_2 - y_1}{x_2 - x_1} - \frac{y_1 - y_0}{x_1 - x_0}}{x_2 - x_0} = \frac{f[x_2,x_1] - f[x_1,x_0]}{x_2-x_1}$ We continue write this out, we will have the following iteration equation: $f[x_k, x_{k-1}, \dots, x_{1}, x_0] = \frac{f[x_k, x_{k-1}, \dots, x_{2}, x_2] - f[x_{k-1}, x_{k-2}, \dots, x_{1}, x_0]}{x_k-x_0}$ We can see one beauty of the method is that, once the coefficients are determined, adding new data points won’t change the calculated ones, we only need to calculate higher differences continues in the same manner. The whole procedure for finding these coefficients can be summarized into a divided differences table. Let’s see an example using 5 data points: $\begin{split} \begin{array}{cccccc} x_0 & y_0 \\ & & f[x_1,x_0] \\ x_1 & y_1 & & f[x_2, x_1,x_0]\\ & & f[x_2,x_1] & & f[x_3, x_2, x_1,x_0]\\ x_2 & y_2 & & f[x_3, x_2,x_1] & & f[x_4, x_3, x_2, x_1,x_0]\\ & & f[x_3,x_2] & & f[x_4, x_3, x_2, x_1]\\ x_3 & y_3 & & f[x_4, x_3,x_2]\\ & & f[x_4,x_3] \\ x_4 & y_4 \end{array} \end{split}$ Each element in the table can be calculated using the two previous elements (to the left). In reality, we can calculate each element and store them into a diagonal matrix, that is the coefficients matrix can be write as: $\begin{split} \begin{array}{cccccc} y_0 & f[x_1,x_0] & f[x_2, x_1,x_0] & f[x_3, x_2, x_1,x_0] & f[x_4, x_3, x_2, x_1,x_0]\\ y_1 & f[x_2,x_1] & f[x_3, x_2,x_1] & f[x_4, x_3, x_2, x_1] & 0\\ y_2 & f[x_3,x_2] & f[x_4, x_3,x_2] & 0 & 0 \\ y_3 & f[x_4,x_3] & 0 & 0 & 0 \\ y_4 & 0 & 0 & 0 & 0 \end{array} \end{split}$ Note that, the first row in the matrix is actually all the coefficients that we need, i.e. $$a_0, a_1, a_2, a_3, a_4$$. Let’s see an example how we can do it. TRY IT! Calculate the divided differences table for x = [-5, -1, 0, 2], y = [-2, 6, 1, 3]. import numpy as np import matplotlib.pyplot as plt plt.style.use('seaborn-poster') %matplotlib inline def divided_diff(x, y): ''' function to calculate the divided differences table ''' n = len(y) coef = np.zeros([n, n]) # the first column is y coef[:,0] = y for j in range(1,n): for i in range(n-j): coef[i][j] = \ (coef[i+1][j-1] - coef[i][j-1]) / (x[i+j]-x[i]) return coef def newton_poly(coef, x_data, x): ''' evaluate the newton polynomial at x ''' n = len(x_data) - 1 p = coef[n] for k in range(1,n+1): p = coef[n-k] + (x -x_data[n-k])*p return p x = np.array([-5, -1, 0, 2]) y = np.array([-2, 6, 1, 3]) # get the divided difference coef a_s = divided_diff(x, y)[0, :] # evaluate on new data points x_new = np.arange(-5, 2.1, .1) y_new = newton_poly(a_s, x, x_new) plt.figure(figsize = (12, 8)) plt.plot(x, y, 'bo') plt.plot(x_new, y_new) [<matplotlib.lines.Line2D at 0x11bd4e630>] We can see that the Newton’s polynomial goes through all the data points and fit the data.	2022-06-26T11:03:53	{ "domain": "berkeley.edu", "url": "https://pythonnumericalmethods.berkeley.edu/notebooks/chapter17.05-Newtons-Polynomial-Interpolation.html", "openwebmath_score": 0.979290783405304, "openwebmath_perplexity": 5156.543139866614, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9926541749467245, "lm_q2_score": 0.8705972667296309, "lm_q1q2_score": 0.8642020115163752 }
https://math.stackexchange.com/questions/2329528/can-a-binary-operation-have-an-identity-element-when-it-is-not-associative-and-c/2329532	# Can a binary operation have an identity element when it is not associative and commutative? I tried getting the answers in similar questions, everyone says that it's not necessary, but if $e$ is the identity element for any binary operation $$, which is not associative and commutative, how can $$ae=a=ea$$ when it is not commutative, i.e. $ab \ne ba$? Even if we get a value by solving $ae=a$. Will we get the same value by solving $ea=a$ ? Please provide an example. • If $\ast$ has both a left identity $l$ and a right identity $r$, then $l = l \ast r = r$. – Travis Willse Jun 20 '17 at 12:54 • See the wikipedia article on quasigroups, and specifically the section on loops. It has the additional assumption of divisibility, and as such left and right inverses, but is otherwise exactly what you're looking for. The examples section includes such familiar things as the integers with the subtraction operation and the non-zero rationals, reals or complex numbers with division. – Arthur Jun 21 '17 at 9:03 Asserting that the operation $$$$ is not commutative means that there are elements $$a$$ and $$b$$ such that $$ab\neq ba$$. It does not mean that $$ab\neq ba$$ for any two distinct elements $$a$$ and $$b$$. Therefore, an operation may well not be commutative and, even so, to have an identity element. There is no contradiction here. For an example of a non-commutative and non-associative algebraic structure with an identity element, take, for instance, the octonions. An operation is commutative if for any $a$ and $b$, we have $ab=ba$. Finding one pair $a,b$ such that $ab=ba$ doesn't prove the operation is commutative; this has to hold for every pair. Consider the set $\{a,b,c\}$ whose binary operation $\cdot$ is given by the following: $$a\cdot a = a\,\,\,\,\,\,\,\,\,\,\, a\cdot b=b\,\,\,\,\,\,\,\,\,\,\,a\cdot c=c$$ $$b\cdot a = b\,\,\,\,\,\,\,\,\,\,\, b\cdot b=b\,\,\,\,\,\,\,\,\,\,\,b\cdot c=c$$ $$c\cdot a = c\,\,\,\,\,\,\,\,\,\,\, c\cdot b=b\,\,\,\,\,\,\,\,\,\,\,c\cdot c=a$$ This operation has $a$ as an identity element. However, it is not commutative (since $b\cdot c\neq c\cdot b$) and it is not associative (since $b\cdot(c\cdot c)=b\neq a =(b\cdot c)\cdot c$). Actually, given any set $S$ and operation $$ on it (so possibly neither associative nor commutative), we can simply extend this with a new symbol $\color{red}0$ (i.e., $\color{red}0\notin S$) and on the set $S':=S\cup\{\color{red}0\}$ define an operation $\color{red}$ by $$x\color{red}y:=\begin{cases}x&\text{if }y=\color{red}0\\ y&\text{if }x=\color{red}0\\xy&\text{otherwise} \end{cases}$$ Then $\color{red}$ is not associative/commutative if $$ is not associative/commutative. But $\color{red}0$ is neutral. • Nice. Since all operators with identity can be viewed as arising from this process and because the probability that a randomly choses operator on a set with $n$ elements is either commutative or associative tends to $0$ as $n \rightarrow \infty$, you can show that most operators with identity are neither associative nor commutative. – John Coleman Jun 21 '17 at 13:21 It is possible. $$ not being commutative means that $ab\neq ba$ for some $a,b$, not for all of them. So you may have $ae=ea=a$ without contradicting that $$ is not commutative. Without looking for esoteric and/or ad hoc examples, there is one you are certainly familiar with. The identity matrix is the identity element for matrix multiplication, which is not commutative. We have $A\,I=I\,A=A$ while in general $A\,B \neq B\,A$. • This is nice, because although the OP mentioned nonassociativity, the only property invoked in her/his argument is noncommutativity. Therefore matrix addition is a good, familiar example. – Ben Crowell Jun 20 '17 at 21:45 Isn't subtraction a binary function which has an identity ($x-0=x$) although it is not commutative ($5-0 > 0-5$) or associative ($5-(4-3) > (5-4)-3$)? • Zero is only a right-identity for subtraction. $0 - x = x$ fails. – Zach Effman Jun 20 '17 at 18:45 • Besides, in abstract algebra, subtraction is literally addition. – Obinna Nwakwue Jun 21 '17 at 14:12 • @ObinnaNwakwue, I think one has to be careful with such a statement. Subtraction is literally defined in terms of addition (and additive inverses), but it is not literally addition, any more than multiplication of natural numbers is literally addition because it is defined in terms of it. – LSpice Jun 21 '17 at 22:12 • Yeah, you are right there. – Obinna Nwakwue Jun 22 '17 at 1:53 Can a binary operation have an identity element when it is not associative and commutative? Yes. Any non-commutative loop is an example of such algebraic structure. I am surprised that, searching the word loop with Ctrl + F in this page just before answering, I found only one occurrence in the comment by Arthur. (...) how can $$ae=a=ea$$ when it is not commutative (...) ? As in the accepted answer, the operation $$$$ is by definition commutative on the set $$A$$, if and only if $$ab=ba$$ for all $$a,b\in A$$. No surprise if, in a given operation on a given set $$A$$, only some elements of $$A$$ commute with all the others: in a non-commutative binary operation with two-sided identity, the identity element commutes with all the others elements. Indeed, the definition of identity element $$e$$ for the binary operation $$$$ on a set $$A$$ states: $$ae=a=ea,\quad\forall\,a\in A.$$ Of course :) And, further, we can create without too much effort an example which is even less than a loop. The finite magma of order $$4$$ given by the Cayley table $$\begin{array}{c\|cccc} * & 1 & 2 & 3 & 4\\ \hline 1 & \color{red}{1} & \color{red}{2} & \color{red}{3} & \color{red}{4}\\ 2 & \color{red}{2} & 4 & 3 & 3\\ 3 & \color{red}{3} & 3 & 4 & 2\\ 4 & \color{red}{4} & 1 & 1 & 1\\ \end{array}$$ has the two-sided identity element $$1$$ while cancellation and division do not hold, it is not commutative since - e.g. - $$24\ne 42$$ and it is not even associative: $$(23)4=34=2\ne 4=22=2(34).$$	2020-09-18T08:12:17	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/2329528/can-a-binary-operation-have-an-identity-element-when-it-is-not-associative-and-c/2329532", "openwebmath_score": 0.804413378238678, "openwebmath_perplexity": 192.63374610882846, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9706877717925422, "lm_q2_score": 0.8902942159342104, "lm_q1q2_score": 0.8641977087049671 }
https://patriciabarber.com/i-said-yto/4128a8-theorem-of-cyclic-quadrilateral-class-10	For Study plan details. ∴ ∠ZTX = $$\frac { 1 }{ 2 }$$ m(arc WY) + m(arc ZX)] = $$\frac { 1 }{ 2 }$$ (54° – 23°) It is a type of cyclic quadrilateral. By theorem of touching circles, points X, Z, (∠1 + ∠2) + (∠7 + ∠8) = 180° m(arc ACB) = 360° – m(arc AB) [Measure of a circle is 360°] m(arc QSR) = m(arc QS) + m(arc SR) [Arc addition property] Question and Answer forum for K12 Students. ∴ m(arc QSR) = 210°, Question 15. ∴ AD = AH + DH [A – H – D] ∴ ∠OFB + ∠ODB = 180° Solution: seg BE ⊥ side AC, Draw circles with centres A, B and C each of radius 3 cm, such that each circle touches the other two circles. Answer: (A) Note: It cannot be square as the angles are not mentioned as 90°. (D) Isosceles triangle ∴ ∠OFA + ∠OEA = 180° In class 10 Maths, a lot of important theorems are introduced which forms the base of mathematical concepts. To prove: Points A, B and C are not collinear. The bisector of ∠ACB intersects the circle at point D. Prove that, seg AD ≅ seg BD. ∠EFG = $$\frac { 1 }{ 2 }$$ m (arcEG)] (ii) [Inscribed angle theorem] iii. PQ is the radius = 12 cm. internally at E. ∴ AB = AC = OB = OC [From (i) and (ii)] ∠OFB ∠ODB = 90° [Given] and seg OR is the secant. Watch Cyclic Quadrilateral Theorem and its Converse in English from Cyclic Quadrilateral here. Also, ∠QTS = ∠QAS [Angles inscribed in the same arc] ∴ 25x – x2 = 144 At what distance will each of them be from point P? Concept of opposite angles of a quadrilateral. Find the measure of ∠C? From 6 Now, $$\frac { MS }{ SR }$$ = $$\frac { 6 }{ 3 }$$ = $$\frac { 2 }{ 1 }$$ Draw a chord AB and central ∠ACB. ∴ Point O lies on the perpendicular bisector of AB. If AB = 3.6, AC = 9.0, AD = 5.4, find AE. ix. (B) rhombus line l, line m and line n are the tangents. Construction: Draw AB, AE and AD. Cyclic Quadrilateral. line l is the tangent to the circle and [Given] i.e. A – Q – C segment are equal. ∠DOB = m(arc DB) = 90° ………… (i) [Definition of measure of arc] Let the radius of the circle be r. Draw a sufficiently large circle of any radius as shown in the figure below. ii. ∠EFG = ∠FGH (i) Alternate angles ∴ AB = AP + BP [A – P – B] If AB = 4.2,BC = 5.4, AE = 12.0, find AD. Theorem 10.11 The sum of either pair of opposite angles of a cyclic quadrilateral is 180°. Given: chord EF \|\| chord GH Solution: ∠BAD + ∠BCD = 180° = 130° iii. ∴ d(0, Q) < radius Given: Circles with centres C and D touch each other internally. (C) 295° From 5. [Given] ∴ arc AB = arc BC = arc AC ꠸AQST is a cyclic quadrilateral. (C) Only three ∠QTS = $$\frac { 1 }{ 2 }$$ m(arc QS) [Inscribed angle theorem] [ AE = AH = 4.5 ∠PQR m(arc PR) [Inscribed angle theorem] Prove that: seg SQ \|\| seg RP. Join OM. Fill in the blanks and complete the proof. ∴ 2∠A + 2∠C = 2 × 180° [Multiplying both sides by 2] Prove that, chord EG ≅ chord FH. ∴ r2 = $$\frac{36 \times 4}{3}$$ Teachoo provides the best content available! ∠CTP ≅ ∠CTQ [From (i), each angle is of measure 90° ] Contact us on below numbers . Now, ray OP is tangent at point P and segment PQ is a secant. interior angles. Points A, B, C are on a circle, such that m(arc AB) = m(arc BC) = 120°. If ∠POR = 70° and (arc RS) = 80°, find i.e., ∠XYZ is an obtuse angle. = 169 = 130° + 80° ∴ ∠QSR + ∠PRQ + ∠PSR = 180° [From (i) and (ii)] In ∆MLK , In the adjoining figure, seg AB is a diameter of a circle with centre O. (C) Right angled triangle (i) (A) Equilateral triangle Theorem 8.2 Theorem 8.3 Theorem 8.4 Theorem 8.5 Theorem 8.6 Theorem 8.7 Theorem 8.8 Theorem 8.9 Theorem 8.10 … m(arc AB) + m(arc BC) + m(arc AC) = 360° [Measure of a circle is 360°] = 4 $$\sqrt { 3 }$$ ∴ 4.2 × 9.6 = AD × 12.0 i. ∴ OL2 = KL2 + OK2 [Pythagoras theorem] The sum of either pair of opposite angles of a cyclic quadrilateral is 180°. E are concyclic points has been viewed 1631 times you are confirming that have. Either pair of opposite angles is 180, the perpendicular bisectors intersect each other it. ∠5 = ∠8 chord BC angles in a cyclic ꠸ABCD, twice the measure of.! Of your house ], Question 24 and the sides can be defined and is known as quadrilateral. Chord BC angles in same segment are equal Maths and Science at.! - Theorem Related to cyclic quadrilateral Theorem ] Similarly, we can prove that, □ABOC is quadrilateral. Two opp ( Geometry Std 10 ) ( Chapter: - circle ) every corner of the circle... 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Group MEMBERS • BHAVYA • SUPREET • GEETA • DEEPALI • RICHA • AMAN 17 on! Point O is the diameter of a circle with centre O us discuss what is the to! Be defined and is known as cyclic quadrilateral is supplementary????. Solutions of all NCERT questions, hence find the measures of ∠TQS and arc TS at... Not intersect at O i. m ( arc PR ) ii quadrilateral and Get answer from subject and. Of DH and CF be X and Y respectively discuss some … the sum of quadrilateral. Rs, and website in this browser for the next time i comment were proved called... Cyclic quadrilaterals - Get Get topics notes, Online test, video lectures.... Locations of point R are on line l a side of OE prove that perpendicular. Intersect each other then it is both tangential and cyclic 3 Geometry Class 10 Maths Chapter... H are touching points Get Get topics notes, Online test, video lectures Doubts. Notes, Online test, video lectures & Doubts and Solutions for ICSE Class 10,! A diameter of a circle is 360-degrees and the sides can be drawn measures of and! If RS = 12, what is the secant, □ABOC is a.... Is highly rated by Class 9 = 4.5, EB = 5.5 3.3. Radii 5.5 cm and 3.3 cm respectively touch each other at O radius cm. Dragons' Den Franchise, Adrift Game Engine, Sidony Dragon Age, Dead Island: Epidemic, Pixies - Beneath The Eyrie Songs, American Constitution Society Vs Federalist Society, Peace Movement Examples, Who Bought The Monarch Beach Resort,	2021-10-27T07:31:12	{ "domain": "patriciabarber.com", "url": "https://patriciabarber.com/i-said-yto/4128a8-theorem-of-cyclic-quadrilateral-class-10", "openwebmath_score": 0.47614631056785583, "openwebmath_perplexity": 3650.9566282298897, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.974042642048694, "lm_q2_score": 0.8872045922259089, "lm_q1q2_score": 0.8641751050494584 }
https://math.stackexchange.com/questions/2067752/does-every-number-not-ending-with-zero-have-a-multiple-without-zero-digits-at-al/2067782	# Does every number not ending with zero have a multiple without zero digits at all? Does every number not ending with zero have a multiple without zero digits at all? We consider numbers in their usual base 10 representation. we don't consider numbers ending with $0$, because all multiples of such numbers end with $0$. It can be proven that all powers of $2$ have such a multiple: Indeed, for every number of the form $2^n$, we can find a multiple $l$ of it whose last $n$ digits are nonzero (This can be done by repeatedly multiplying a number by $10^k+1$, as explained here), and then forget all other digits; these last $n$ digits constitute a number that is still divisible by $2^n$, since it is obtained from $l$ by subtracting a multiple of $10^n$ which is in particular a multiple of $2^n$. The same argument shows that every power of $5$ has a multiple without zeros, but this does not seem to generalize to other numbers. My intuition is actually that the claim is not true, since the density of numbers without any zero digits approaches $0$ for big numbers. This is because the probability of a random number with at most $n$ digits to contain only nonzero digits is $(\frac{9}{10})^n$. Thus big numbers "nearly always" contain a zero. This hints on the existence of a counterexample, but does not prove its existence. • I'm not sure what the answer is, but I think your heuristic argument misses something big: Sure, the proportion of $n$ digit numbers with only non-zero digits is $(9/10)^n$, but the number of multiples of a number $k$ with $n$ digits is roughly $\frac{10^n}k$ - so multiply those together and we "expect" there to be $\frac{9^n}k$ multiples of $k$ with $n$ digits and no zeros. (This is certainly true of numbers coprime to 10) – Milo Brandt Dec 21 '16 at 20:51 • User @lulu seems to have deleted his correct observation that this is also true for all numbers coprime to $10$, by Fermat's little theorem. – Emolga Dec 21 '16 at 21:06 • I deleted the post because someone pointed out this question which generalizes the straightforward argument I gave. – lulu Dec 21 '16 at 21:08 • Just a note: in fact you can get stronger results for multiples of $2^n.$ For any odd digit $a$ and even digit $b$ you can find a multiple of $2^n$ using only $a$ and $b$ – cats Dec 21 '16 at 21:12 • Found it - see "104 Number Theory Problems from the training of the USA IMO Team" Introductory Problems #52 – cats Dec 21 '16 at 21:18 If $n$ is relatively prime to $10$ then the answer is easy, see for example this post. If, $n=2^k m$ with $m$ relatively prime to $10$ or $n=5^km$ with $m$ relatively prime to $10$, then prove first the following by induction: Claim 1 $2^k$ has a $k$ digits multiple containing only the digits $1$ and $2$. Claim 2 $5^k$ has a $k$ digits multiple containing only the digits $1,2,3,4$ and $5$. Both are easy induction exercises. Once you prove this, proceed as follows (same idea as in the posted link): $n=2^k m$ or $n=5^km$ with $gcd(m,10)=1$.. Then, we know that $2^k$ respectively $5^k$ has a multiple of the form $\overline{a_1...a_k}$ Look at the following numbers $$\overline{a_1...a_k} , \overline{a_1...a_ka_1...a_k} , \overline{a_1...a_ka_1...a_ka_1...a_k} , ...$$ In this infinite list, there exists two numbers with the same reminder when divided by $m$. Their difference is a multiple of $m$. Therefore $$m\| \overline{a_1...a_ka_1...a_ka_1...a_k00...0}=\overline{a_1...a_ka_1...a_ka_1...a_k}\cdot 10^l$$ Since $m$ is relatively prime to $10$ it follows that $$m\|\overline{a_1...a_ka_1...a_ka_1...a_k}$$ Since $2^k$ respectively $5^k$ also divide $\overline{a_1...a_ka_1...a_ka_1...a_k}$, and they are relatively prime to $m$, it follows that $$n\|\overline{a_1...a_ka_1...a_ka_1...a_k}$$ P.S. We proved that $n$ has a multiple which can be written only with the digits $1,2,3,4,5$. These digits can be replaced by any $5$ digits which cover all classes $\pmod{2}$ and $\pmod{5}$. • Could you please give a proof of the claims? – J. Abraham Feb 25 '17 at 19:56 • @J.Abraham $P(k) \Rightarrow P(k+1)$. Let $A$ be the $k$ digit number which is divisible by $a^k$ where $a=2$ or $5$. Then $$A=a^k \cdot m$$ and $$10^{k}=a^k \cdot l$$ where $l$ is invertible modulo $a$. Then the equation $$x l +m \equiv 0 \pmod{a}$$ has an unique solution in the listed digits. Then $$x \cdot 10^k+A$$ is the desired $k+1$ digit number. – N. S. Feb 26 '17 at 23:07	2019-07-24T09:44:32	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/2067752/does-every-number-not-ending-with-zero-have-a-multiple-without-zero-digits-at-al/2067782", "openwebmath_score": 0.8284635543823242, "openwebmath_perplexity": 122.72536817699505, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9886682488058381, "lm_q2_score": 0.87407724336544, "lm_q1q2_score": 0.864172417519144 }
http://mathhelpforum.com/number-theory/49939-congruences-2-a.html	1. ## congruences 2 Slove the system of congruences 3x = 2 mod 4 4x = 1 mod 5 6x = 3 mod 9 is it no solution since 6 and 9 do not have an inverse since the gcd of 6 and 9 is not one thus u cant write it in the form x=a mod p 2. I have been trying to find the inverse and solve it that way but i dont know if that is right? 3. If $ac \equiv bc \ (\text{mod } m)$ and $(c,m) = d$, then $a \equiv b \left(\text{mod } \frac{m}{d}\right)$ Keeping in this in mind, I'm going to simplify the congruences first: $\begin{array}{rcll} 3x & \equiv & 2 & (\text{mod } 4) \\ 3x & \equiv & 6 & (\text{mod } 4) \\ x & \equiv & 2 & (\text{mod } 4) \end{array}$........ $\begin{array}{rcll} 4x & \equiv & 1 & (\text{mod } 5) \\ 4x & \equiv & -4 & (\text{mod } 5) \\ x & \equiv & -1 & (\text{mod } 5) \\ x & \equiv & 4 & (\text{mod } 5) \end{array}$........ $\begin{array}{rcll} 6x & \equiv & 3 & (\text{mod } 9) \\ 6x & \equiv & 12 & (\text{mod }9) \\ x & \equiv & 2 & (\text{mod } 3) \end{array}$ So we're left with the system: $\begin{array}{rcll} x & \equiv & 2 & (\text{mod } 4) \\ x & \equiv & 4 & (\text{mod } 5) \\ x & \equiv & 2 & (\text{mod } 3) \end{array}$ And by the Chinese Remainder theorem, since $(4, 5, 3) = 1$, there is a unique solution modulo $4\cdot 5 \cdot 3 = 60$. Should be pretty simply to solve this one. 4. Originally Posted by o_O If $ac \equiv bc \ (\text{mod } m)$ and $(c,m) = d$, then $a \equiv b \left(\text{mod } \frac{m}{d}\right)$ Keeping in this in mind, I'm going to simplify the congruences first: $\begin{array}{rcll} 3x & \equiv & 2 & (\text{mod } 4) \\ 3x & \equiv & 6 & (\text{mod } 4) \\ x & \equiv & 2 & (\text{mod } 4) \end{array}$........ $\begin{array}{rcll} 4x & \equiv & 1 & (\text{mod } 5) \\ 4x & \equiv & -4 & (\text{mod } 5) \\ x & \equiv & -1 & (\text{mod } 5) \\ x & \equiv & 4 & (\text{mod } 5) \end{array}$........ $\begin{array}{rcll} 6x & \equiv & 3 & (\text{mod } 9) \\ 6x & \equiv & 12 & (\text{mod }9) \\ x & \equiv & 2 & (\text{mod } 3) \end{array}$ So we're left with the system: $\begin{array}{rcll} x & \equiv & 2 & (\text{mod } 4) \\ x & \equiv & 4 & (\text{mod } 5) \\ x & \equiv & 2 & (\text{mod } 3) \end{array}$ And by the Chinese Remainder theorem, since $(4, 5, 3) = 1$, there is a unique solution modulo $4\cdot 5 \cdot 3 = 60$. Should be pretty simply to solve this one. i get the right answer after applying the CRT to the reduced congruences, but i still dont know how u reduced the congruences or simplified them. Could you maybe explain it a little bit. Could u find the inverse modulo of each congruence and and reduce it to the form x=a mod m 5. Hello, Divide the formula of o_O into 2 formulae: 1) If $ac\equiv bc\pmod m$ and $(c, m)=1$ then $a\equiv b\pmod m$. 2) If $ac\equiv bc\pmod {mc}$ then $a\equiv b\pmod m$. 1) corresponds to the case where you can find an inverse. 2) is a technique to decrease the modulus. You add the modulus as many times as you want so that the coefficient of x and the right-hand-side have a common divisor. Bye. 6. What I did was play around with the congruences a little. Remember, they sort of act like equality signs as well. Let's take the middle one I did: $ \begin{array}{rcll} 4x & \equiv & 1 & (\text{mod } 5) \\ 4x & \equiv & -4 & (\text{mod } 5) \qquad \text{1. Subtracted 5} \\ x & \equiv & -1 & (\text{mod } 5) \qquad \text{2. Divided both sides by 4} \\ x & \equiv & 4 & (\text{mod } 5) \qquad \text{3. Added 5 to get a least residue} \end{array} $ Basically, I wanted to get rid of the coefficient in front of the x and what I wanted is to ultimately divide it out: 1. If something is congruent to 4x mod 5, then adding or subtracting multiple of 5's will still be congruent to 4x. Remember: $a \equiv b \ (\text{mod m}) \ \iff \ a = b + km$ 2. This refers to the very first statement I had in my first post 3. Again, adding and subtracting multiples of the modulus will not change the congruence as we did in step 1. The main purpose in doing so was to get a least residue (i.e. the "remainder") Edit: Oh beaten!	2016-08-29T15:11:34	{ "domain": "mathhelpforum.com", "url": "http://mathhelpforum.com/number-theory/49939-congruences-2-a.html", "openwebmath_score": 0.8481968641281128, "openwebmath_perplexity": 287.7361449484711, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9886682444653242, "lm_q2_score": 0.8740772286044094, "lm_q1q2_score": 0.8641723991314373 }
http://openstudy.com/updates/511303b4e4b0e554778aafec	## Anita505 2 years ago Suppose that we have a white urn containing four white balls and one red ball and have a red urn containing one white ball and four red balls. An experiment consists of selecting at random a ball from the white urn and then (without replacing the first ball) Selecting at random a ball from the urn having the colour of the first ball. Find the probability that the second ball is red. The probability that the second ball is red is ______. 1. kropot72 There are two situations to consider: (1) If a white ball is selected on the first draw then the second draw will be from the white urn. Probability of white on the first draw is 4/5 and the probability of red on the second draw is 1/4. P(red second ball) = 4/5 * 1/4 = 4/20 ...............(1) (2) If a red ball is selected on the first draw then the second draw will be from the red urn. Probability of red on the first draw is 1/5 and the probability of red on the second draw is 4/5. P(red second ball) = 1/5 * 4/5 = 4/25 ...............(2) The probability that the second ball is red is the sum of the fractions (1) and (2). 2. Anita505 so the sum of (4/20)+(4/25) So in this case the answer would be 0.36 correct? 3. kropot72 Correct, or alternatively 9/25. 4. Anita505 thank you for showing me the process 5. kropot72 You're welcome :) 6. Anita505 An urn contains 3 one-dollar bills, 1 five- dollar bill and 1 ten dollar bill. A player draws bills one at a time without replacement from the urn until a ten dollar bill is drawn. Then the game stops. All bills are kept by the player. Determine: A) The probability of winning $15 B)The probability of winning all bills in the urn C) The probability of the game stopping at the second draw. 7. Anita505 can you help me with this and show me the process? 8. satellite73 as far as i can see there is only one way to win$15: first draw the five, then draw the ten 9. satellite73 since there are 5 bills, and one is a five, the probability of drawing the five first is $$\frac{1}{5}$$ then there are 4 bills of which one is a ten, the probability of drawing a ten second given that the first bill was a five is $$\frac{1}{4}$$ the probability of both things occurring is $\frac{1}{5}\times \frac{1}{4}$ 10. Anita505 so for part a) it would be 1/20? the answer 11. satellite73 yes 12. Anita505 okay thank you but i need help with part b and c 13. satellite73 for b) it means you pick the ten dollar bill last right? 14. satellite73 the probability you pick the ten dollar bill last is the same as the probability you pick the ten dollar bill first, namely $$\frac{1}{5}$$ 15. Anita505 so b then in this case is 1/5 16. satellite73 and for the last one, that means you pick something other then the ten dollar bill, and then you pick the ten dollar bill that is also $$\frac{1}{5}$$ via $\frac{4}{5}\times \frac{1}{4}=\frac{1}{5}$ 17. satellite73 more simply put, the probability you pick the ten first, second, third, fourth or fifth are all the same, namely $$\frac{1}{5}$$ 18. Anita505 Okay thank you for your assistance i have one last question to ask, 19. satellite73 do me a favor and post in a new thread this is hard so scroll down to 20. Anita505 A grade 11 art class is offering students two choices for a project: a pottery project and a mixed media project. Of the 46 students in the class, 23 have selected to do the pottery project and 33 have selected to do the mixed media project (notice some students have decided to do both) It two students are selected at random from the class to show their finished project(s), what is the probability that at least one pottery project and at least one mixed media project will be shown? Probability (given to three decimal places)	2016-02-06T15:47:21	{ "domain": "openstudy.com", "url": "http://openstudy.com/updates/511303b4e4b0e554778aafec", "openwebmath_score": 0.7426160573959351, "openwebmath_perplexity": 560.7432289030303, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9848109507462227, "lm_q2_score": 0.8774767986961403, "lm_q1q2_score": 0.8641487603816977 }
http://mathhelpforum.com/calculus/76627-l-hopital-s-rule-question.html	1. ## L'Hopital's Rule question I have a problem using L'Hopital's rule and I can't seem to get the right answer. The problem is this: Find the limit of [ln(x)]/[ln(2x)] as x goes to infinity. I tried l'Hopital's rule, differentiating the top and the bottom, and then taking the limit of that expression. The derivative of the top is 1/x, and the derivative of the denominator is 1/4x (according to my calculations anyways). I find the limit of that expression, and I find that it is 4. Is that correct? However, I graph it on my calculator, and the limit seems to be 1. How do you get that? 2. Originally Posted by pianopiano I have a problem using L'Hopital's rule and I can't seem to get the right answer. The problem is this: Find the limit of [ln(x)]/[ln(2x)] as x goes to infinity. I tried l'Hopital's rule, differentiating the top and the bottom, and then taking the limit of that expression. The derivative of the top is 1/x, and the derivative of the denominator is 1/4x (according to my calculations anyways). I find the limit of that expression, and I find that it is 4. Is that correct? However, I graph it on my calculator, and the limit seems to be 1. How do you get that? Note that $\frac{\ln x}{\ln(2x)}=\frac{\ln x}{\ln 2+\ln x}$. Since $\lim_{x\to\infty}\frac{\ln x}{\ln(2x)}=\lim_{x\to\infty}\frac{\ln x}{\ln 2+\ln x}=\frac{\infty}{\infty}$, we can apply L'Hopitals rule. So $\lim_{x\to\infty}\frac{\ln x}{\ln(2x)}=\lim_{x\to\infty}\frac{\ln x}{\ln 2+\ln x}=\lim_{x\to\infty}\frac{\displaystyle\frac{1}{x} }{\displaystyle\frac{1}{x}}=\lim_{x\to\infty}\frac {x}{x}=1$ Does this make sense? 3. Thanks, it does make sense! However, why can't you differentiate the numerator and denominator and take the limit of the new expression and have that work as well since ln(x) and ln(2x) would each still go to infinity as x goes to infinity? 4. Originally Posted by pianopiano Thanks, it does make sense! However, why can't you differentiate the numerator and denominator and take the limit of the new expression and have that work as well since ln(x) and ln(2x) would each still go to infinity as x goes to infinity? You could...but you would have to note that $\forall a\in\mathbb{R}\backslash\left\{0\right\},~\frac{\, d}{\,dx}\left[\ln\!\left(ax\right)\right]=\frac{1}{x}$ 5. $\lim_{x\to\infty}\frac{\log x}{\log 2 + \log x}=\lim_{x\to\infty}\frac{1}{\underbrace{\frac{\lo g 2}{\log x}}_{\to 0}+1}=1$. Or you must use L'H? 6. Originally Posted by Abu-Khalil $\lim_{x\to\infty}\frac{\log x}{\log 2 + \log x}=\lim_{x\to\infty}\frac{1}{\underbrace{\frac{\lo g 2}{\log x}}_{\to 0}+1}=1$. Or you must use L'H? I was going to point that out. There's no need for L'Hopital's Rule at all, just divide by the common term, log x.	2016-10-28T07:22:39	{ "domain": "mathhelpforum.com", "url": "http://mathhelpforum.com/calculus/76627-l-hopital-s-rule-question.html", "openwebmath_score": 0.970382809638977, "openwebmath_perplexity": 218.57325297147378, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9848109485172559, "lm_q2_score": 0.8774767842777551, "lm_q1q2_score": 0.8641487442264476 }
https://math.stackexchange.com/questions/2001389/using-the-mean-value-theorem	# Using the Mean Value Theorem The (classical) Mean Value Theorem states that if $f$ is a continuous function on $[a,b]$ and is differentiable on $(a,b)$, then there exists a point $c \in (a,b)$ where $f'(c) = \dfrac{f(b)-f(a)}{b-a}$. Here is a "real-world" application of the theorem. Suppose two racers, Barry and Harry, start a race at time $t=0$ and end the race in a tie, both crossing the finish line at, say, time $t = 10$. Let $s(0)$ be the position of the starting line, and let $s(10)$ be the position of the finish line. The position of each racer is continuous on [0,10] and (presumably) differentiable on (0,10). So the MVT tells us that at some time $t_0 \in (0,10)$, Barry's velocity equaled $s'(t_0)=v(t_0)=\dfrac{s(10)-s(0)}{10}$, and similarly there is a time $t_1 \in (0,10)$ where Harry's velocity equaled $s'(t_1)=v(t_1)=\dfrac{s(10)-s(0)}{10}$. Question: Does the MVT also guarantee that $t_0=t_1$? That is, is it necessarily the case that Barry and Harry have the same velocity at the same exact time at some point during the race? If so, why? • I think the MVT doesn't say that the point $c$ is unique, hence you could find other times say $t_2$ ro $t_3$ where that holds. But I don't know if this helps – Euler_Salter Nov 6 '16 at 1:49 • Let $x(t),y(t)$ be $x(t)=t$, $y(t)=\sqrt{t}$. Clearly $x(0)=y(0)=0$ and $x(1)=y(1)=1$, but $x'(t)=1=\frac{x(1)-x(0)}{1}=1$ for all $t$, while $y'(t)=1$ only when $t=\sqrt{1/2}$, so they need not be equal – user160738 Nov 6 '16 at 1:53 ## 3 Answers You're being ambiguous. If you mean to ask if their velocities will be the same at some time in the interval $[0,10]$, yes. The result follows by applying the mean value theorem to the function $x(t) = s_{b}(t) - s_{h}(t)$ on the interval $[0, 10]$. $s_b$ and $s_h$ are the position functions of Barry and Harry respectively. However, if you apply the mean value theorem to the functions independently, the "mean value" times may not be identical. Consider the following example, where we suppose the velocities are continuous. If Barry runs very fast right at the start, almost up to the finish line, over the course of $1$ second and then slows down like a caterpillar for the other $9$ seconds, the mean value time for Barry will be somewhere around $t = 1$ seconds while he is slowing down. If Harry does the opposite, that is, he's extremely slow for the first $9$ seconds and then lightning fast to catch up to Barry at the final second, then his "mean value time" will be somewhere around $t=9$ as he speeds up. They will share the same velocity at some time ($g(t) = v_{b}(t) - v_{h}(t)$ is continuous, initially positive and negative at the end), but this shared velocity will not be the "mean value" velocity $\frac{s(10) - s(0)}{10}$. • Thank you for answering my question. I don't see how I was being ambiguous (my question was very clearly stated, was it not?), but thank you nonetheless. – Mr Toad Nov 6 '16 at 5:43 • Your question was ambiguous because you ask two different things in the last paragraph of the question, and in fact they happen to have different answers. As MathematicsStudent1122 wrote, the answer to "does the MVT also guarantee that $t_0 = t_1$?" is no, but the answer to "is it necessarily the case that Barry and Harry have the same velocity at the same exact time at some point during the race?" is yes. – David Z Nov 6 '16 at 7:40 • @David Z, thanks for the clarification. – Mr Toad Nov 6 '16 at 14:53 You can use Cauchy's mean value theorem. It says that if $f$ and $g$ are continuous on $[a,b]$ and differentiable on $(a,b)$ then there exists $c\in (a,b)$ such that $$(f(b)-f(a))g'(c)=(g(b)-g(a))f'(c).$$ If we use $f$ to denote the position of Barry and $g$ to denote the position of Harry then we have $f(0)=g(0)=0$ and $f(T)=g(T)=L=$length. ($T$ denotes the total time.) So, there exists a point $c\in (0,T)$ such that $$Lg'(c)=(f(T)-f(0))g'(c)=(g(T)-g(0))f'(c)=Lf'(c).$$ That is, there exists $c$ such that $f'(c)=g'(c).$ So they must have had the same velocity at some point. • +1 for the perfect answer. It does not appear at first that a single value of $c$ will work for $f$ and $g$ but Cauchy's MVT ensures that this is possible. – Paramanand Singh Nov 6 '16 at 7:09 Assume runner 1 runs at speed $2t$ feet per second, so that in ten seconds, runner 1 travels $\int_0^{10}2tdt=100$ feet. Assume runner $2$ runs at speed $\frac{3t^2}{1000}$ so that in $10$ seconds he also travels $100$ feet. However, $2t=\frac{3t^2}{1000}$ has roots $0$ and $\frac{2000}{3}>10$, so the runners never simultaneously travel the same speed during the race (besides the start, speed $0$). (There is nothing special about this example, I just started with a simple speed function $2t$ and tried to fudge another function to make it work. Im sure there are many more counterexamples) • I think the example doesn't hold for runner 2. His initial speed isn't zero. – Cehhΐro Nov 6 '16 at 1:55 • @O.VonSeckendorff thanks for the catch. It is fixed. – TomGrubb Nov 6 '16 at 2:00	2020-04-01T15:48:42	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/2001389/using-the-mean-value-theorem", "openwebmath_score": 0.9100626111030579, "openwebmath_perplexity": 272.80567087145636, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9848109503004293, "lm_q2_score": 0.8774767810736692, "lm_q1q2_score": 0.864148742635722 }
https://mathematica.stackexchange.com/questions/81323/find-the-volume-of-the-region-defined-by-xyz4	# Find the volume of the region defined by $\|x\|+\|y\|+\|z\|<4$ I want to find the volume of the region defined by the following inequality: $\quad \quad \|x\|+\|y\|+\|z\|<4$ In addition to calculating the volume of this region, I would also like to be able to obtain its representation as a plot. I have started by plotting the vertices, i.e., the points (4, 0, 0), (-4, 0, 0), (0, 4, 0), (0, -4, 0), (0, 0, 4), and (0, 0, -4), but I am not sure how to continue from there. • Welcome to Mathematica.SE! I suggest that: 1) You take the introductory Tour now! 2) When you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge. Also, please remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign! 3) As you receive help, try to give it too, by answering questions in your area of expertise. – bbgodfrey May 1 '15 at 3:23 • @julie You may want to consider accepting one of the answers that have been provided below, if they solve your problem. – MarcoB May 3 '15 at 17:45 volume = Integrate[1, Element[{x, y, z}, ImplicitRegion[Abs[x] + Abs[y] + Abs[z] < 4, {x, y, z}]]] (256/3) ContourPlot3D[Abs[x] + Abs[y] + Abs[z] == 4, {x, -5, 5}, {y, -5, 5}, {z, -5, 5}] ## A small addition from Mathematica v. 10 In his answer @Ivan has already nailed the most important part of the question, i.e. the formal representation of your region as an ImplicitRegion object: region = ImplicitRegion[Abs[x] + Abs[y] + Abs[z] < 4, {x, y, z}] I just wanted to add that, if you are on Mathematica 10 or newer, you can also use the RegionMeasure or Volume functions to calculate the volume of a region: RegionMeasure[region] Volume[region] (* 256/3 ) ## How to arrive at the shape of this region In response to @julie's question in the comment, here is how one can imagine constructing the 3D region. Consider the corresponding equality in 2D, disregard the absolute values for now, and let's just concentration on the first quadrant ($x>=0$,$y>=0$). The resulting $x + y = 4$ is just the equation of a line, i.e. $y = 4 - x$. The region in the first quadrant in which the inequality $x+y<4$ holds is the region between that line and the axes: RegionPlot[ x + y < 4 && x >= 0 && y >= 0, {x, -5, 5}, {y, -5, 5}, Axes -> True, Frame -> False ] Taking the absolute value of $x$ is geometrically equivalent to mirroring that triangle with respect to the $y$ axis as shown below. Since we are taking its absolute value, we will also remove the $x>=0$ restriction. RegionPlot[ Abs[x] + y < 4 && y >= 0, {x, -5, 5}, {y, -5, 5}, Axes -> True, Frame -> False ] Taking the absolute value of $y$ is similarly equivalent to mirroring this figure with respect to the $x$ axis. The result is the square region below: RegionPlot[ Abs[x] + Abs[y] < 4, {x, -5, 5}, {y, -5, 5}, Axes -> True, Frame -> False ] In the three-dimensional case, you can work your way through in a similar way: $x+y+z=4$ is the equation of a plane that defines a trangular wedge in the first quadrant, and so forth. As for the numerical value, you could consider that your figure is made up of two square pyramids. The volume of a square pyramid is $V=a^2h/3$ where $a$ is the length of the base edge, and $h$ is the height. In your case, you can calculate the base edge as the hypotenuse of the smallest 2D triangle we introduced above ($a^2=4^2+4^2=32$), and the height of the pyramid is 4. The volume of this pyramid is 128/3, and your figure is made of two such pyramids, so $2\times128/3 = 256/3$. • thank you! but how was the final answer calculated? how did you know what all the sides were and where those points went? – julie May 1 '15 at 3:28 • @julie I amended my answer to include a way to construct the region. Let me know if that answers your question appropriately. – MarcoB May 1 '15 at 4:27 An alternate approach to calculating the volume Integrate[ Boole[Abs[x] + Abs[y] + Abs[z] < 4], {x, -Infinity, Infinity}, {y, -Infinity, Infinity}, {z, -Infinity, Infinity}] 256/3 As a cone (with a square base), the volume is one-third the base times the height: 1/3 (4 Sqrt[2])^2 * 8. Another plot, based on the OP's coordinates: ConvexHullMesh@Join[4 IdentityMatrix[3], -4 IdentityMatrix[3]]	2020-04-01T21:37:52	{ "domain": "stackexchange.com", "url": "https://mathematica.stackexchange.com/questions/81323/find-the-volume-of-the-region-defined-by-xyz4", "openwebmath_score": 0.5827907919883728, "openwebmath_perplexity": 463.0068835190746, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9848109525293959, "lm_q2_score": 0.8774767746654976, "lm_q1q2_score": 0.8641487382807507 }
http://finmining.com/hcd6t/ec9282-how-to-find-turning-point-of-parabola	So the x value is 0. You will get a point now. Back Function Institute Mathematics Contents Index Home. A parabola is a curve where any point is at an equal distance from: a fixed point (the focus), and ; a fixed straight line (the directrix) Get a piece of paper, draw a straight line on it, then make a big dot for the focus (not on the line!). Hence the equation of the parabola in vertex form may be written as $$y = a(x - 2)^2 + 3$$ We now use the y intercept at $$(0,- 1)$$ to find coefficient $$a$$. The implication is that throughout the observed range of the data, the expected probability of pt is an increasing function of expand_cap, though with some diminishing returns. When the function has been re-written in the form y = r(x + s)^2 + t, the minimum value is achieved when x = -s, and the value of y will be equal to t. x-intercepts in greater depth. A parabola can have either 2,1 or zero real x intercepts. Completing the square, we have \begin{align} y &= x^2 - 2ax + 1 \\ &= (x - a)^2 + 1 - a^2, \end{align} so the minimum occurs when $$x = a$$ and then $$y = 1 - a^2$$. What do you notice? You can take x= -1 and get the value for y. Example 2 Graph of parabola given vertex and a point Find the equation of the parabola whose graph is shown below. Graphs of quadratic functions have a vertical line of symmetry that goes through their turning point.This means that the turning point is located exactly half way between the x-axis intercepts (if there are any!).. The easiest way to find the equation of a parabola is by using your knowledge of a special point, called the vertex, which is located on the parabola itself. Only vertical parabolas can have minimum or maximum values, because horizontal parabolas have no limit on how high or how low they can go. Now play around with some measurements until you have another dot that is exactly the same distance from the focus and the straight line. Step 1: Find the roots of your … Substitute this x value into the equation y = x 2 – 6x + 8 to find the y value of the turning point. Finding the maximum of a parabola can tell you the maximum height of a ball thrown into the air, the maximum area of a … There are two methods to find the turning point, Through factorising and completing the square.. Make sure you are happy … In either case, the vertex is a turning point … Does the slope always have to be in turning points? If the function is smooth, then the turning point must be a stationary point, however not all stationary points are turning points, for example has a stationary point at x=0, but the derivative doesn't change sign as there is a point of inflexion at x=0. If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value. In the first two examples there is no need for finding extra points as they have five points and have zeros of the parabola. Any point, ( x 0 , y 0 ) on the parabola satisfies the definition of parabola, so there are two distances to calculate: Distance between the point on the parabola to the focus Distance between the point on the parabola to the directrix To find the equation of the parabola, equate these two expressions and solve for y 0 . The x-intercepts are the points or the point at which the parabola intersects the x-axis. If the parabola only has 1 x-intercept (see middle of picture below), then the parabola is said to be tangent to the x-axis. Let’s work it through with the example y = x 2 + x + 6. A turning point is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising). This is a second order polynomial, because of the x² term. now find your y value by using the x value you just found by plugging it into your function . Example 1 . Yes, the turning point can be (far) outside the range of the data. Use this formula to find the x value where the graph turns. How to find the turning point (vertex) of a quadratic curve, equation or graph. STEP 1 Solve the equation of the derived function (derivative) equal to zero ie. The x-coordinate of the turning point = - $$\frac{b}{2a}$$ ----- For example, if the equation of the parabola is . If the slope is , we max have a maximum turning point (shown above) or a mininum turning point . May 2008 218 59 Melbourne Australia Aug 24, 2009 #2 At the turning points of an equation the slope of y is zero. (Increasing because the quadratic coefficient is negative, so the turning point is a maximum and the function is increasing to the … A turning point may be either a local maximum or a minimum point. To find the turning point of a parabola, first find it's x-value, using the equation: -b/2a (from the quadratic form ax^2 + bx + c). Find the maximum number of turning points of each … A polynomial of degree n will have at most n – 1 turning points. In other words the differential of the equation must be zero. How do I find the coordinates of a turning point? This calculator will find either the equation of the parabola from the given parameters or the axis of symmetry, eccentricity, latus rectum, length of the latus rectum, focus, vertex, directrix, focal parameter, x-intercepts, y-intercepts of the entered parabola. The equation is y=4xsquare-4x+4. When the equation of the parabola is in this form: y = ax 2 + bx + c . The x-coordinate of the turning point = - $$\frac{4}{2(3)}$$ = - $$\frac{2}{3}$$ Plug this in for x to find the value of the y-coordinate. Does slope always imply we have a turning point? K. Kiwi_Dave. or the slope just becomes for a moment though you have no turning point. To graph a parabola, visit the parabola grapher (choose the "Implicit" option). Given that the turning point of this parabola is (-2,-4) and 1 of the roots is (1,0), please find the equation of this parabola. … Surely you mean the point at which the parabola goes from increasing to decreasing, or reciprocally. In this case, b = 0, since there is no b term, and a is 1 (the number before the x squared) : -b/2a = -0/2. This means: To find turning points, look for roots of the derivation. To find the turning point of a quadratic equation we need to remember a couple of things: The parabola ( the curve) is symmetrical; If we know the x value we can work out the y value! You therefore differentiate f(x) and equate it to zero as shown below. The coordinate of the turning point is (-s, t). Solution to Example 2 The graph has a vertex at $$(2,3)$$. Example 7: Finding the Maximum Number of Turning Points Using the Degree of a Polynomial Function . If we look at the function . Free functions turning points calculator - find functions turning points step-by-step This website uses cookies to ensure you get the best experience. If, on the other hand, you suppose that "a" is negative, the exact same reasoning holds, except that you're always taking k and subtracting the squared part from it, so the highest value y … If y=ax^2+bx+c is a cartesian equation of a random parabola of the real plane, we know that in its turning point, the derivative is null. So for example, given (2a): Vertex at (2, -6) One x intercept at 6 The axis will be x=2, so the given x intercept is 4 units to the right of the axis. Curve sketching Murray says: 19 Jun 2011 at 8:16 am [Comment permalink] Hi Kathryn and thanks for your input. … $0=a(x+2)^2-4$ but i do not know where to put … Such a point is called saddle point. How to find the turning point of a parabola: The turning point, or the vertex can be found easily by differentiation. Remember that the axis of symmetry is the straight line that passes through the turning point (vertex) of the parabola. How you think you find the turning point given the x-intercepts of a parabola? So, our starting or reference parabola formula looks like this: y = x 2. The turning point of a parabola is the vertex; this is also it's highest or lowest point. Reactions: … So the turning point is at $(a, 1 - a^2).$ So for your example: $$\displaystyle \frac {dy}{dx}=2x$$ So we set this equal to zero to get: $$\displaystyle 2x=0$$ or x=0 . The vertex is at point (x,y) First find x by using the formula -b/2a <--- a = 2, b= -5 and c= 1 (because it is quadratic) So -(-5)/2(2) = 5/4 <--- your x value at the vertex or turning point is 5/4. And our equation that includes a horizontal translation looks like this: y = (x - h) 2. If the coefficient of the x 2 term is positive, the vertex will be the lowest point on the graph, the point at the bottom of the “ U ”-shape. Write down the nature of the turning point and the equation of the axis of symmetry. And the lowest point on a positive quadratic is of course the vertex. When the parabola opens down, the vertex is the highest point on the graph — called the maximum, or max. A second approach is to find the turning point of the parabola. Turning Points and Intercepts of a Parabola Function. Published in: Education. The S.K.A. Horizontal translation for the parabola is changed by the value of a variable, h, that is subtracted from x before the squaring operation. It’s hard to see immediately how this curve will look just by looking at the function. how to i find the turning point of that parabola? So remember these key facts, the first thing we need to do is to work out the x value of the turning point. If the parabola is upright - as these examples are - then it will be laterally symmetrical about its axis, which is the vertical line through the vertex. is the set of points in a plane equidistant from a given line, called the directrix, and a point not on the line, called the focus. GeoGebra can be used very easily to find the equation of a parabola: given three points, A, B, C input the command FitPoly[{A, B, C}, 2]. Worked examples. In … Did You Know That...? 17 Comments 2 Likes ... – 12 12 – 24 – 12 = -24 this is the y-coordinate of the vertex So the vertex (turning point of this parabola is (-2,-24) HOW TO CALCULATE THE VERTEX (TURNING POINT) Recommended Mẫu ốp lưng iphone se da thật chuyên nghiệp … In math terms, a parabola the shape you get when you slice through a solid cone at an angle that's parallel to one of its sides, which is why it's known as one of the "conic sections." Here is a typical quadratic equation that describes a parabola. The turning point is when the rate of change is zero. To find the axis of symmetry, use this formula: x = -b/2a. solve dy/dx = 0 This will find the x-coordinate of the turning point; STEP 2 To find the y-coordinate substitute the x-coordinate into the equation of the graph ie. … Turning Points of Quadratic Graphs. I started off by substituting the given numbers into the turning point form. The graph below has a turning point (3, -2). y = 3x 2 + 4x + 1 . A parabola The set of points in a plane equidistant from a given line, called the directrix, and a point not on the line, called the focus. Parabola, Horizontal Translation. TURNING POINT The formula to find the x value of the turning point of the parabola is x = –b/2a. This can help us sketch complicated functions by find turning points, points of inflection or local min or maxes. In example 3 we need to find extra points. There is also a spreadsheet, which can be used as easily as Excel. substitute x into “y = …” This is a mathematical educational video on how to find extra points for a parabola. A turning point can be found by re-writting the equation into completed square form. No. The Parabola. In the case of a vertical parabola (opening up or down), the axis is the same as the x coordinate of the vertex, which is the x-value of the point where the axis of symmetry crosses the parabola. Solved: What is the turning point, or vertex, of the parabola whose equation is y = 3 x^2 + 6 x - 1? The Vertex of a Parabola The vertex of a parabola is the point where the parabola crosses its axis of symmetry. A parabola can have 2 x-intercepts, 1 x-intercept or zero real x intercepts. Second order how to find turning point of parabola, because of the turning point ( vertex ) of the turning point is (! Below has a vertex at \ ( ( 2,3 ) \ ) extra... To graph a parabola uses cookies to ensure you get the best.! 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That describes a parabola: the turning point can be found by plugging it into function... First two examples there is also it 's highest or lowest point on the graph — the... The turning point of a polynomial of degree n will have at most n – 1 turning points step-by-step website! -1 and get the best experience does the slope just becomes for a moment though have! Graph, or the point at which the parabola is x = –b/2a turning point it through with the y. Up, the first thing we need to do is to work out x. Is shown below of degree n will have at most n – 1 turning points calculator - find functions points. On the graph, or max are the points or the how to find turning point of parabola at which the parabola the! Vertex and a point find the y value by using the x value you just found plugging... X 2 by re-writting the equation of the derived function ( derivative ) equal to zero.! 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https://byjus.com/question-answer/let-a-n-denote-the-number-of-all-n-digit-positive-integers-formed-by-the-2/	Question # Let $$a_{n}$$ denote the number of all n-digit positive integers formed by the digits $$0,1$$ or both such that no consecutive digits in them are $$0$$. Let $$b_{n}=$$ the number of such $$n$$-digit integers ending with digit $$1$$ and $$c_{n}=$$ the number of such $$n$$-digit integers ending with digit $$0$$. Which of the following is correct? A a17=a16+a15 B c17c16+c15 C b17b16+c16 D a17=c17+b16 Solution ## The correct option is A $$a_{17}=a_{16}+a_{15}$$In such a number either last digit is $$'0'$$ or $$'1'$$When you consider $$'a_1'$$, only one number is possible i.e. $$1$$When you consider $$'a_2'$$, 2 such numbers are possible i.e. $$10$$, $$11$$When you consider $$'a_3'$$, 3 such numbers are possible i.e. $$101$$, $$111$$, $$110$$When you consider $$'a_4'$$, 5 such numbers are possible i.e. $$1010$$, $$1011$$, $$1110$$, $$1101$$, $$1111$$By observing this we get a relationship which is $$a_n$$ $$=$$ $$a_{n-1}$$ $$+$$ $$a_{n-2}$$So, $$a_{17}$$ $$=$$ $$a_{16}$$ $$+$$ $$a_{15}$$(Alternate Method)Using Recursion formula$$a_n=a_{n-1}+a_{n-2}$$Similarly, $$b_n=b_{n-1}+b_{n-2}$$ and $$c_n=c_{n-1}+c_{n-2}$$ $$\forall\ n\geq 3$$and $$a_n=b_n+c_n$$ $$\forall\ n\geq 1$$So, $$a_1=1, a_2=2, a_3=3, a_4=5, a_5=8$$ .......$$b_1=1, b_2=1, b_3=2, b_4=3, b_5=5, b_6=8$$ .......$$c_1=0, c_2=1, c_3=1, c_4=2, c_5=3, c_6=5$$ .......Using this we get $$b_{n-1}=c_n$$$$\therefore a_{17}=a_{16}+a_{15}$$Mathematics Suggest Corrections 0 Similar questions View More People also searched for View More	2022-01-28T22:06:57	{ "domain": "byjus.com", "url": "https://byjus.com/question-answer/let-a-n-denote-the-number-of-all-n-digit-positive-integers-formed-by-the-2/", "openwebmath_score": 0.9698213338851929, "openwebmath_perplexity": 726.1364534351834, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9905874120796498, "lm_q2_score": 0.8723473796562744, "lm_q1q2_score": 0.8641363332481726 }
https://math.stackexchange.com/questions/2734908/prove-that-the-set-c-x-y-in-mathbbr2-max-x-y-leq-1-is	# Prove that the set $C = \{(x,y) \in \mathbb{R}^2:\max \{ \|x\|,\|y\|\}\leq 1 \}$ is convex. Prove that $C = \{(x,y) \in \mathbb{R}^2:\max \{ \|x\|,\|y\|\}\leq 1 \}$ is a convex set. I am using the following definition for a convex set Let $D \subseteq \mathbb{R}^n$. The set $D$ is said to be convex if, given $\bar{x},\bar{y}\in D$, we have that $$(1-t)\bar{x} \ +t\bar{y} \in D$$ for all $t\in [0,1].$ Let $\bar{x}=(x_1,y_1)$ and $\bar{y}=(x_2,y_2)$ be arbitrary points such that $\bar{x},\bar{y} \in C$, and let $t\in [0,1].$ Now, we want to see that $(1-t)\bar{x} \ +t\bar{y} \in C.$ So then \begin{align} (1-t)\bar{x} \ +t\bar{y} &=(1-t)(x_1,y_1)+t(x_2,y_2)\\ &= ((1-t)x_1,(1-t)y_1)+ (tx_2,ty_2)\\ &= ((1-t)x_1+tx_2,(1-t)y_1+ty_2) \end{align} And here is where I get stuck. I want to prove that $((1-t)x_1+tx_2,(1-t)y_1+ty_2)\in C$, which I believe is the same as proving that $\max\{\|(1-t)x_1+tx_2\|,\|(1-t)y_1+ty_2\| \}\leq 1$, but I don't know how to proceed. Any help would be appreciated! • Use triangle inequality. – Youem Apr 13 '18 at 3:57 So $\|(1-t)x_{1}+tx_{2}\|\leq(1-t)\|x_{1}\|+t\|x_{2}\|\leq(1-t)+t=1$ and $\|(1-t)y_{1}+ty_{2}\|\leq(1-t)\|y_{1}\|+t\|y_{2}\|\leq(1-t)+t=1$ because $\|x_{1}\|,\|x_{2},\|y_{1}\|,\|y_{2}\|\leq 1$, this shows that $(1-t)\overline{x}+t\overline{y}\in C$. $\|(1-t)x_1 + tx_2\| \le (1-t)\|x_1\| + t \|x_2\| \le (1-t) + t = 1$ the same for $y$ Let $C_x = \{(x,y) \in \mathbb{R}^2:\|x\|\leq 1 \}$ and $C_y = \{(x,y) \in \mathbb{R}^2:\|y\|\leq 1 \}$. Then, • $C_x$ and $C_y$ are obviously convex • $C = C_x \cap C_y$ is convex, too	2019-10-17T10:15:15	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/2734908/prove-that-the-set-c-x-y-in-mathbbr2-max-x-y-leq-1-is", "openwebmath_score": 0.9992247819900513, "openwebmath_perplexity": 134.36807964022822, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9905874101345135, "lm_q2_score": 0.8723473697001441, "lm_q1q2_score": 0.8641363216889207 }
https://mathoverflow.net/questions/364040/3-colored-triangulations-of-the-sphere-s2-and-sperners-lemma	# 3-colored triangulations of the sphere $S^2$, and Sperner's Lemma I noticed something about colored triangulations of the topological sphere $$S^2$$ and have a question about this. Observation. If you triangulate the sphere $$S^2$$ and color the vertices with three colors: then the number of 3-colored triangles is always even (or zero). In particular, there is no coloring with exactly one 3-colored triangle. For a proof, view $$S^2$$ as two triangulated disks with matching coloring of the boundaries that are glued together. As their boundaries have the same number of color changes, we know from Sperner’s Lemma that their triangulations have the same number (mod 2) of 3-colored triangles. So the total number of 3-colored triangles is even or zero. As an interesting corollary, we get the characterization: A triangulated sphere has zero 3-colored triangles iff all cycles of the triangulation have an even number of color changes. I looked at the torus, the Klein bottle, and the projective plane, and I find that the observation is also true for them. Edit: Just for contrast, adding an example below of a "soap bubble" surface, where the two soap bubbles share a common disk. This surface allows for triangulations with even and odd numbers of 3-colored triangles (but like the other surfaces I looked at, cannot have just one). Question. I wonder whether this also follows from more general theorems about triangulations of surfaces, or about maximal planar graphs? I have consulted algebraic topology and graph theory texts, but could not find any results in that direction. Would you have a suggestion where else to look, or maybe a reference for that? • This looks related to Sperner's lemma: en.wikipedia.org/wiki/Sperner%27s_lemma Jun 24, 2020 at 22:51 • @JanKyncl you are definitely right. It is like Sperner’s Lemma “without boundary”. Or the way I looked at it, 2 x Sperner’s Lemma and then glued together. Just in case you are interested, see this mathoverflow.net/q/362025/156936 Jun 25, 2020 at 3:47 • "Uneven" numbers are commonly called odd. Jun 28, 2020 at 19:57 • @VictorProtsak thanks a lot for your comment! I corrected the wording in the text now. Jun 29, 2020 at 11:32 A counting proof shows that this observation is unrelated to the global topology. Every edge is monochromatic or dichromatic. How many dichromatic edges are there? If each triangle tells you its number of dichromatic edges (either 0, 2, or 3), you can add these up and divide by two to get the total number of dichromatic edges (since every edge contributes to two triangles). So the number of trichromatic triangles must be even. This proof works for $$k$$-dimensional manifolds when $$k$$ is even, since the number of $$k$$-colored $$(k-1)$$-simplices bounding any $$k$$-simplex must be 0, 2, or $$k+1$$. Your corollary similarly transfers to higher even dimensions, at least for orientable manifolds, replacing "cycles of edges" with "hypersurfaces of $$(k-1)$$-simplices", and "even number of color changes" with "even number of $$k$$-colored $$(k-1)$$-simplices". • I'm not sure which direction you want to go in with this, but I think it falls pretty squarely in combinatorics and graph theory. What you (not incorrectly) call "triangulations of $S^2$" are often referred to as maximal planar graphs. My proof is very similar to the handshaking lemma. People have also considered coloring and planarity on tori and other manifolds. – Matt Jun 24, 2020 at 19:32 • thanks a lot for this helpful comment and great links. This is excellent Jun 24, 2020 at 20:20 Just to close the loop on this: the double-counting argument in the answer of user Matt allows for a nice visual proof of the (2-dim.) Lemma of Sperner. Just want to capture it here, as it connects nicely with the triangulation of the sphere / the maximal planar graph in my OP question. Start with a triangulated polygon in the plane, and label each vertex with one of 3 colors. The example just shows the boundary of such a triangulated, 3-colored polygon. Claim (Sperner’s Lemma): If the boundary has an odd number of color changes, then a 3-colored triangle exists in the polygon triangulation. In fact, more generally, an odd number of such 3-colored triangles exists. Proof: Go to 3-dimensional space, and build a “tent” over the polygon like in the diagram: add a colored vertex, and add the edges between this additional vertex and the boundary vertices of the polygon. This way, we have effectively created a triangulation of the topological sphere $$S^2$$. If the boundary of the polygon has an odd number of color changes, this gives an odd number of 3-colored triangles in the “tent” over the polygon. But from the double-counting argument in user Matt’s answer, we know an even number of 3-colored Sperner triangles must exist. Hence the polygon at the bottom must have an odd number of 3-colored triangles (at least one) in its triangulation, which completes the proof.	2022-05-23T15:11:13	{ "domain": "mathoverflow.net", "url": "https://mathoverflow.net/questions/364040/3-colored-triangulations-of-the-sphere-s2-and-sperners-lemma", "openwebmath_score": 0.7654329538345337, "openwebmath_perplexity": 368.1216546596065, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9828232940063591, "lm_q2_score": 0.8791467785920306, "lm_q1q2_score": 0.8640459328508988 }
https://math.stackexchange.com/questions/500511/dimension-of-the-sum-of-two-vector-subspaces	# Dimension of the sum of two vector subspaces $\dim(U_1+U_2) = \dim U_1 +\dim U_2 - \dim(U_1\cap U_2).$ I want to make sure that my intuition is correct. Suppose we have two planes $U_1,U_2$ though the origin in $\mathbb{R^3}$. Since the planes meet at the origin, they also intersect, which in this case is a one-dimensional line in $\mathbb{R^3}$. To obtain the dimension of $U_1$ and $U_2$, we add the dimensions of the planes (4), and the subtract the dimensions of the line (1), which results in (3). Can we generalize this notion to $\mathbb{F^{n}}$? Suppose we have an additional case where $U_1$ and $U_2$ are planes in $\mathbb{R^3}$, but $U_1 \subseteq U_2$. In this instance, $dim(U_1 + U_2) < 3$, because the first two-dimensional plane is contained in the second and as a result, the dimensions of the subspaces when summed cannot exceed two. Since both subspaces $U_1,U_2$ are two dimensional and $U_1 \subseteq U_2$, then their intersection is also two-dimensional, concluding $dim(U_1+U_2)=2+2-2 = 2$. Is this proper intuition? • The intuition is correct. Sep 21, 2013 at 17:17 In the latter case, they are actually the same plane, so their sum is again the same plane (as they are closed under addition). Here is another (analogous) way to think about it. Let's start with a basis $B_0$ for $U_1\cap U_2.$ We can extend $B_0$ to a basis $B_1$ for $U_1$ and a basis $B_2$ for $U_2$. Then $B_1\cup B_2$ is a basis for $U_1+U_2,$ and $B_1\cap B_2=B_0,$ so \begin{align}\dim(U_1+U_2) &= \|B_1\cup B_2\|\\ &= \|B_1\|+\|B_2\|-\|B_1\cap B_2\|\\ &= \|B_1\|+\|B_2\|-\|B_0\|\\ &= \dim(U_1)+\dim(U_2)-\dim(U_1\cap U_2).\end{align} This generalizes nicely to $\Bbb F^n$, and allows us to avoid geometric arguments that may be less sensible for an arbitrary field $\Bbb F$. Also, it will never be the case that the intersection of two planes in space is precisely $\{0\}.$ If there were two such planes $U_1$ and $U_2,$ then we would have $$\dim(U_1+U_2)=\dim(U_1)+\dim(U_2)-\dim(U_1\cap U_2)=2+2-0=4>3=\dim(\Bbb R^3),$$ which is not possible, since $U_1+U_2$ is a subspace of $\Bbb R^3$. • Thanks for the answer. Could it be the case that a pair of two dimensional planes in $\mathbb{R^4}$ intersect at a point? Sep 21, 2013 at 17:27 • Yes, indeed. Consider $$U_1=\{(w,x,y,z)\in\Bbb R^4:y=z=0\}$$ and $$U_2=\{(w,x,y,z)\in\Bbb R^4:w=x=0\}.$$ The upshot in $\Bbb R^3$ is that there "isn't enough room" for two planar subspaces to avoid each other that well. Sep 21, 2013 at 17:38 • Why can we conclude that $B_{1} \cap B_{2} = B_{0}$? It seems perfectly reasonable that the extensions share vectors outside of $B_{0}$. Sep 14, 2016 at 0:40 • @Jonathan: If there is some $\vec v\in B_1\cap B_2,$ then $\vec v\in U_1\cap U_2.$ What then can we conclude by definition of $B_0$? Sep 14, 2016 at 0:46 • Ok, then $v$ must be a linear combination of $B_{0}$. But, this forces both $B_{1}$ and $B_{2}$ to be linearly dependent, right? Sep 14, 2016 at 0:53 Cameron Buie's answer does answer the original question sufficiently. However, I'm adding the formal proof of the theorem in context (taken from "Linear Algebra Done Right" by Sheldon Axler). If $$U_1$$ and $$U_2$$ are subspaces of a finite dimensional vector space then: $$\dim(U_1+U_2)=\dim U_1 + \dim U_2 - \dim(U_1 \cap U_2)$$ Let $$u_1,...,u_m$$ be a basis of $$U_1\cap U_2$$; thus $$\dim (U_1\cap U_2)=m$$. Because $$u_1,...,u_m$$ is a basis in $$U_1\cap U_2$$, it is linearly independent in $$U_1$$. Hence this list can be extended to a basis $$u_1,...,u_m,v_1,...,v_j$$ of $$U_1$$. Thus, $$\dim U_1 = m+j$$. Also extend $$u_1,..,u_m$$ to a basis $$u_1,...,u_m, w_1,...,w_k$$ of $$U_2$$. $$\dim U_2 = m +k$$. We will show that $$u_1,...,u_m,v_1,...,v_j,w_1,...,w_k$$ is a basis of $$U_1+U_2$$. This will complete the proof because then we will have $$\dim(U_1+U_2)=m+j+k=\dim U_1 + \dim U_2 - \dim (U_1\cap U_2)$$ We just need to show that the list $$u_1,...,u_m,v_1,...,v_j,w_1,...,w_k$$ is linearly independent. To prove this, suppose: $$a_1u_1+...+au_m+b_1v_1+...+b_jv_j+c_1w_1+...+c_kw_k=0$$ where all $$a,b,c$$'s are scalars. We need to show that all the $$a,b$$ and $$c$$'s are $$0$$. The equation can be rewritten as $$c_1w_1+...+c_kw_k=-a_1u_1 - ... -a_mu_m -b_1v_1 - ... - b_j v_j$$ Which shows that $$c_1w_1+...+c_kw_k\in U_1$$. But actually all $$w$$'s are in $$U_2$$. So the LHS must be an element of $$U_1\cap U_2$$. $$c_1w_1+...+c_kw_k=d_1u_1+...+d_mu_m$$ for some choice of scalars $$d_1,d_2,...,d_m$$. But $$u_1,...,u_m,w_1,...,w_k$$ is linearly independent. So our last equation implies that all the $$c$$'s equal $$0$$. Thus our original equation involving $$a,b,c$$ becomes $$a_1u_1+...+a_mu_m+b_1v_1+...+b_jv_j=0$$ But we already knew that the list $$u_1,...,u_m,v_1,...,v_j$$ is linearly independent. This equation implies that all the $$a$$'s and $$b$$'s are $$0$$. We now know that all $$a,b$$ and $$c$$'s are $$0$$, hence proving our original claim.	2022-07-03T03:26:08	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/500511/dimension-of-the-sum-of-two-vector-subspaces", "openwebmath_score": 0.9851697087287903, "openwebmath_perplexity": 117.93825881355986, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9828232950125849, "lm_q2_score": 0.8791467675095294, "lm_q1q2_score": 0.8640459228433786 }
https://math.stackexchange.com/questions/962638/taylor-approximation	# Taylor Approximation For $f(x)=e^x$, find a Taylor approximation that is in error by at most $10^-7$ on [-1,1]. Using this approximation, write a function program to evaluate $e^x$. Compare it to the standard value of $e^x$ obtained from the MATLAB function exp(x); calculate the difference between your approximation and exp(x) at 21 evenly spaced points in [-1,1]. So I am stuck on this problem. I know for the first part, I needed to find n smallest such that $e/(n+1)!$ $\leq$ $10^-7$, so I found n=10 to satisfy that part. So do I need to find a polynomial of degree 10 using a nested loop that evaluates $e^x$? What is my next movie in this problem? • Yes, a loop to evaluate $1+x+x^2/2+...$ up to 10. – lemon Oct 7 '14 at 18:49 • Thank you! and then I just compare my loop with the MATLAB function given and see where they are different? – user141745 Oct 7 '14 at 18:52 • Very relevant wiki example. – BeaumontTaz Oct 7 '14 at 18:59 • @user141745 Yes - the purpose of comparing is to make sure that your routine is accurate to within $10^{-7}$. – lemon Oct 7 '14 at 19:02 I posted the link in the comments, but I'll flush out the example on the wiki here with some added commentary. We have $f(x)=e^x$ and we know by Taylor'r theorem that $$f(x)=P_k(x)+R_k(x)=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\cdots+\frac{x^k}{k!}+R_k(x)$$ where $R_k(x)$ is our error term and $P_k(x)$ is the Taylor polynomial. We want to find the $k$ such that $R_k(x)<10^{-7}$ for $x\in[-1,1]$. We know that the remainder term $R_k(x)$ can be expressed as $$R_k(x)=\frac{e^\xi}{\left(k+1\right)!}x^{k+1}$$ for some $\xi\in[-1,1]$. We need to find the value of $\xi$ though. And since we don't know $e^x$ (otherwise we'd just use that function) we again approximate it using a lower order Taylor expansion $$e^x=P_1(x)+R_1(x)=1+x+\frac{e^\xi}{2}x^2$$ Since $e^x$ is an increasing function, we know that $e^\xi<e^x$ when $-1<\xi<x$. So we can say $$e^x=1+x+\frac{e^\xi}{2}x^2<1+x+\frac{e^x}{2}x^2$$ Which we can solve for $e^x$ and get $$e^x<2\frac{x+1}{2-x^2}$$ Some basic calculus gives us that the maximum of this expression for $x\in[-1,1]$ is at $x=1$ and $e^x<4$. Therefore $$R_k(x)=\frac{e^\xi}{\left(k+1\right)!}x^{k+1}<\frac{4x^{k+1}}{\left(k+1\right)!}$$ And since $x^{k+1}$ is a maximum on $[-1,1]$ at $x=1$ and we want $R_k(x)$ to be less than $10^{-7}$ we have $$R_k(x)<\frac{4}{\left(k+1\right)!}<10^{-7}$$ So basically, we just need to find $k$ such that $(k+1)!>4\cdot10^7$. And $11!=39916800=3.99\cdot10^7$ which is reallllly close. But, really we need $(k+1)!=12!$ so $k=11$. Now... this is a very modest upper bound. If we really think about it (and this is what you did), the upper bound on the interval $[-1,1]$ is $e$. So if we did this with $e^x<e$ on $x\in[-1,1]$ instead, then we'd end up with $$\frac{e}{\left(k+1\right)!}<10^{-7}\implies (k+1)!>2.718\cdot10^7\implies k=10$$ Same as you got. However, this relies on the fact that we know $e$ already. If we don't know $e$ we can use a higher order approximation than $2$. So we find that $$e^x<\frac{6+6x+3x^2}{6-x^3}$$ Which means it's either $e^x<\frac{15}{5}=3$ if it's at $x=1$ or it's at the maximum of the interval which involves find the roots of a fourth order polynomial $3x^4+12x^3+18x^2+36x+36$ whose critical points turn out not to be on our bound. So we can use this higher order approximation the same way we did before and get that $$\frac{3}{\left(k+1\right)!}<10^{-7}\implies (k+1)!>3\cdot10^7\implies k=10$$ From there, you can either hard code the approximation $$e^x\approx \text{myFunc}(x)=1+x+\frac{x^2}{2}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+\frac{x^6}{6!}+\frac{x^7}{7!}+\frac{x^8}{8!}+\frac{x^9}{9!}+\frac{x^{10}}{10!}$$ or do a sum like this in a for loop $$e^x\approx \text{myFunc}(x)=\sum_{i=0}^{10}\frac{x^i}{i!}$$ It appears as though Arkamis gave you some MATLAB code that'll do this for you, though. Just for fun, here's a matlab script that computes the answer in one line -- in MATLAB we almost never need to loop! The function takes three parameters: n, the maximum order of the polynomial to try; m, the number of data points in the interval we wish to evaluate; and tol, the tolerance we wish to achieve. function Order = TaylorExp(n,m,tol) Order = min(find(max(abs(tril(kron(ones(n,1),1./factorial(0:(n-1))))*power(kron(ones(n,1),linspace(-1,1,m)),kron(ones(1,m),(0:(n-1))')) - kron(ones(n,1),exp(linspace(-1,1,m)))),[],2) < tol)) - 1;	2021-06-12T12:19:39	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/962638/taylor-approximation", "openwebmath_score": 0.8216952681541443, "openwebmath_perplexity": 221.43075925590142, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9688561685659695, "lm_q2_score": 0.8918110454379297, "lm_q1q2_score": 0.8640366325678043 }
https://math.stackexchange.com/questions/799362/line-equation-of-a-tangent-line-of-fx-x-cos3x	# Line equation of a tangent line of $f(x) = x\cos(3x)$ I'm new here so maybe I'll need some help with formatting with MathJax, as well. So question asks for tangent line of $f(x) = x\cos(3x), x= \pi$ So: $$f(x) = x.\cos(3x)$$ $$f'(x) = -x.\sin(3x)+\cos(3x).3$$ $$f(\pi) = -\pi$$ $$f'(\pi) = -3$$ $$y = -3x - 2\pi$$ The answer of the book: $$y = -x$$ Considering: $$y = ax + b$$ $$y = f(x)$$ $$a = f'(x)$$ What am I missing? • you miscalculated the derivative. – mesel May 17 '14 at 17:51 You made a couple of mistakes in using the product rule. $$f(x) = x\cos(3x) \implies f'(x) = x \cdot \frac{d}{dx}(\cos(3x)) + \dfrac{d}{dx}(x)\cdot \cos(3x)$$ $$= x\cdot (-\sin (3x))\cdot 3 + \cos 3x$$ $$= -3x\sin(3x)+ \cos (3x)$$ Now substitute $x = \pi$ into $f'(x)$ to obtain slope at that point: $-3\pi\cdot \sin(3\pi) + \cos (3\pi) = -1$ So the slope of the desired line needs to be $-1$. Now, given $x_0 = \pi$, $y_0 = f(x_0) = f(\pi) = -\pi$. So we have the point on the tangent line (the point of tangency): $(\pi, -\pi)$. That gives you the line \begin{align} y - y_0 = -1(x - x_0) &\iff y - (-\pi) = -1(x - \pi)\\ \\ & \iff y + \pi = -x + \pi \\ \\ & \iff y = -x\end{align} • Your answer showed me something that I was repeating question after question and haven't given attention. Derivative was wrong and how to find line equation was misunderstood. Thank you a lot. – Fabiano Araujo May 17 '14 at 18:15 • You're welcome, Fabiano! – amWhy May 17 '14 at 18:15 HINT: $$f'(x)=\frac{d(x\cos3x)}{dx}=\frac{dx}{dx}\cdot\cos3x+x\cdot\frac{d(\cos3x)}{dx}=\cos3x-3x\sin3x$$	2021-07-24T22:40:20	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/799362/line-equation-of-a-tangent-line-of-fx-x-cos3x", "openwebmath_score": 0.9988675117492676, "openwebmath_perplexity": 731.5645384417, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9688561712637256, "lm_q2_score": 0.8918110418436166, "lm_q1q2_score": 0.8640366314913206 }
https://gmatclub.com/forum/two-important-formulas-for-mixture-problems-82701.html	GMAT Question of the Day: Daily via email \| Daily via Instagram New to GMAT Club? Watch this Video It is currently 23 Jan 2020, 21:16 ### 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 # Two important formulas for mixture problems Author Message TAGS: ### Hide Tags Intern Joined: 11 Aug 2009 Posts: 10 Two important formulas for mixture problems [#permalink] ### Show Tags Updated on: 19 Aug 2009, 06:52 6 9 I came across two methods which I found very handy in solving some types of mixture problems. So I thought of sharing it with the gc community. here they are Rule of Alligation It is the rule that enables us to find the ratio in which two or more ingredients at the given price must be mixed to produce a mixture of a desired price. Mean Price: The cost price of a unit quantity of the mixture is called mean price. Rule of Alligation If two quantities are mixed, then $$\frac{Quantity of cheaper}{Quantity of dearer}=\frac{(C.P.) of dearer - (Mean Price)}{(Mean Price) - (C.P.)}$$ Taking a simple example In what ratio must rice at 9.30 /kg be mixed with rice at 10.80/kg so that the mixture be worth 10/kg? C.P of cheaper=9.30 C.P of dearer=10.80 C.P of mean or mean price = 10.0 so putting the values in the formula $$\frac{Qc}{Qd}=\frac{Cd-Cm}{Cm-Cc}$$ =$$\frac{10.80-10}{10-9.3}$$ =8:7 Ans Originally posted by joebloggs on 19 Aug 2009, 06:10. Last edited by joebloggs on 19 Aug 2009, 06:52, edited 2 times in total. Intern Joined: 11 Aug 2009 Posts: 10 ### Show Tags 19 Aug 2009, 06:25 9 11 Suppose a container contains x units of liquid from which y units are taken out and replaced with water. After n operations, the quantity of pure liquid = $$[x(1-\frac{y}{x})^n]$$ Example: A container contains 40 liters of milk. From this container 4 litres of milk was taken and replaced by water. This process was repeated further two times. How much milk is now contained by the container? Amount of milk left after 3 operations = $$[40(1-\frac{4}{40})^3]liters = (40 *\frac{9}{10}^3)$$ = 29.16 liters ##### General Discussion Intern Joined: 11 Aug 2009 Posts: 10 Re: Two important formulas for mixture problems [#permalink] ### Show Tags 19 Aug 2009, 06:37 3 you also might wanna see a short method for mixture problems used by Economist check out this problem http://gmatclub.com/forum/how-much-of-the-mixture-is-to-be-removed-and-replaced-82423.html Hope this post helps Soon I'll be posting some mixture problems which I couldn't solve in one go ... see ya .... Senior Manager Joined: 18 Jun 2009 Posts: 310 Location: San Francisco Re: Two important formulas for mixture problems [#permalink] ### Show Tags 19 Aug 2009, 12:40 nice post .. thanks .. +1 Manager Joined: 16 Apr 2009 Posts: 152 Schools: Ross Re: Two important formulas for mixture problems [#permalink] ### Show Tags 19 Aug 2009, 12:53 Keep up the good work ! Intern Joined: 14 Aug 2019 Posts: 3 Re: Two important formulas for mixture problems [#permalink] ### Show Tags 14 Aug 2019, 02:59 Manager Joined: 01 Jan 2017 Posts: 57 WE: General Management (Consulting) Two important formulas for mixture problems [#permalink] ### Show Tags 28 Aug 2019, 00:40 Quote: In what ratio must rice at 9.30 /kg be mixed with rice at 10.80/kg so that the mixture be worth 10/kg? C.P of cheaper=9.30 C.P of dearer=10.80 C.P of mean or mean price = 10.0 For those who love a more visual approach, consider the following, which may be even simpler. Just subtract cheaper and dearer from the mean and find the difference. ---9.3----10.8-- -----\----/------ -------10------- ------/---\------ ---0.8---0.7--- Two important formulas for mixture problems [#permalink] 28 Aug 2019, 00:40 Display posts from previous: Sort by	2020-01-24T04:18:37	{ "domain": "gmatclub.com", "url": "https://gmatclub.com/forum/two-important-formulas-for-mixture-problems-82701.html", "openwebmath_score": 0.8512399792671204, "openwebmath_perplexity": 6930.326394164262, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9773708026035287, "lm_q2_score": 0.8840392832736084, "lm_q1q2_score": 0.8640341838261749 }
http://openstudy.com/updates/556876efe4b0ae03fc81fc6b	## anonymous one year ago Find circle which is tangent to x-axis and path through points (1,-2) and (3,-4). I worked it through analytic geometry and it gives two possible circles!!! and I can not figure how 3 conditions for a circle gives 2 answers or how to draw it geometrically? 1. anonymous @ganeshie8 2. welshfella general equation is (x - a)^2 + ( y - b)^2 = r^2 you can create 2 equations in a , b and r by substituting the 2 given points 3. anonymous Circle is clearly defined by 3 conditions. How can I draw 2 circles which path through two points and tangent to a line!! 4. welshfella The circle touches the x -axis so that another point on (x , 0) 5. welshfella - good question - I'm trying to figure that 6. anonymous Can you give a picture? that is puzzling me 7. welshfella \|dw:1432910060214:dw\| 8. welshfella lo1 - not a great diagram 9. anonymous I'm sorry I meant two possible cirlces 10. welshfella \|dw:1432910234711:dw\| 11. welshfella thats worse than the first 12. anonymous :) Yeah that is amazing. 13. ganeshie8 two circles can have the same common tangent, yes ? 14. welshfella yes 15. ganeshie8 may be forget about x-axis and look at below diagram \|dw:1432910495765:dw\| 16. ganeshie8 both circles meet below 3 conditions : 1, 2) pass through two points (intersection) 3) having that blue line as tangent 17. welshfella ah - yes the common tangenytis the x-axis 18. anonymous I got it now thanks 19. ganeshie8 your question about "why 3 conditions are not giving me an unique circle" is very interesting, im still thinking of a better explanation 20. anonymous I don't know. In such cases I just change that thinking. I think the tangent line condition isn't tough enough ((unless the tangent point is given)) 21. anonymous anyway that for your helping :) 22. ganeshie8 hmm the tangent point is not so random, we only have two choices so its still a bit mysterious 23. ganeshie8 \|dw:1432911213933:dw\| 24. anonymous But the result is the same tangent line condition isn't unique as you have said. 25. ganeshie8 geometrically how do you know there doesn't exist a third circle that meets the given conditions ? 26. ganeshie8 \|dw:1432911454842:dw\| 27. ganeshie8 how do i convince more than two circles are never possible given 1) two points 2) tangent line 28. anonymous the center isn't on locus of chord made by these 2 points 29. anonymous I mean when you bisect it and make a perpendicular which is the locus of the center 30. ganeshie8 perpendicular bisector of the chord passes through the center but again how do you know the 3 centers are not collinear ? 31. ganeshie8 \|dw:1432911763458:dw\| 32. welshfella the line joining the points of contact to the centers of the circles are perpendicular to the tangent - can that be used to answer the question? 33. ganeshie8 BINGO!!! 34. ganeshie8 \|dw:1432912228871:dw\| 35. ganeshie8 hmm idk 36. welshfella yes - its a puzzle 37. welshfella \|dw:1432912469311:dw\| 38. welshfella why can't there be an other larger circle like the above? 39. anonymous for any triangle it has only one unique circle paths through its vertices \|dw:1432912656462:dw\| so if we moved the point along the locus the lengths won't be equal and if we shifted it up or down, the the distance to 2pionts won't be equal. 40. ganeshie8 brilliant! 41. welshfella i'm afraid i gotta go and leave this interesting discussion .. 42. myininaya I know you wanted geometric stuff or whatever but I think I have found two answers with a combination of algebra and calculus. 43. myininaya It is pretty long that way. :p 44. myininaya $\text{ our circle has points: } (1,-2);(3,-4);(a,0) \\ \text{ we have horizontal tangent at } (a,0) \\ (a-h)^2+(0-k)^2=r^2 \\ (1-h)^2+(k+2)^2=r^2 \\ (3-h)^2+(k+4)^2=r^2 \\ y'=\frac{h-x}{y-k} \\ y'\|_{x=a}=0=\frac{h-a}{0-k} \implies h=a \\ (a-a)^2+k^2=r^2 \text{ so } k=-r \text{ or } k=r \\ \text{ so going with the } k=r \text{ thing we have } \\ (1-h)^2+(r+2)^2=r^2 \\ (3-h)^2+(r+4)^2=r^2 \\ (1-h)^2+4r+4=0 \\ (3-h)^2+8r+16=0 \\ 2(1-h)^2+8=(3-h)^2+16 \\ h^2+2h-15=0 \\ (h+5)(h-3)=0 \\ h=-5 \text{ or } h=3$ so pluggin some things back in we can find r then k $h=-5 \\ a=-5 \\ (1+5)^2+4r+4=0 \\ r=10 \\ k=10,-10$ $h=3 \\ a=3 \\ (1-3)^2+4r+4=0 \\ r=2 \\ k=2 ,-2$ some of these solutions need to be checked you will see only 2 of the 4 will work and like i said I know this isn't the approach you wanted but I thought it would be nice to try another approach 45. welshfella looks good to me	2017-01-23T12:43:05	{ "domain": "openstudy.com", "url": "http://openstudy.com/updates/556876efe4b0ae03fc81fc6b", "openwebmath_score": 0.6302857995033264, "openwebmath_perplexity": 1560.9655067903057, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9773707976599674, "lm_q2_score": 0.8840392863287585, "lm_q1q2_score": 0.864034182441887 }
https://mathhelpboards.com/threads/diagonal-crossing.3856/	# Diagonal Crossing #### mathmaniac ##### Active member Challenge: How many boxes are crossed by a diagonal of an 800 X 200 rectangle? Solution with proof required... #### caffeinemachine ##### Well-known member MHB Math Scholar Challenge: How many boxes are crossed by a diagonal of an 800 X 200 rectangle? Solution with proof required... Hello Mathmaniac I don't understand the question. What us meant by 'boxes are crossed'. Also, if this is a challenge problem then shouldn't this go in the 'Challenge and Puzzles' forum? #### MarkFL Staff member Yes, I sent the OP a VM asking if this is a challenge or if it is a problem for which he needs help shortly after it was posted, but have not gotten a response yet. Once the matter is settled, I will move it if need be, then remove this post so that the topic is not cluttered. #### mathmaniac ##### Active member I don't understand the question. What us meant by 'boxes are crossed'. Here is an example: The boxes inside the rectangle are meant to be squares.... #### MarkFL Staff member Its a challenge,Mark... I have thus moved the topic to the Challenge Questions and Puzzles sub-forum. I know you used the word "Challenge" but wanted to make sure it fit the criteria, i.e., you have the correct solution ready to post in the event no one solves it. Last edited: Yes!!! #### Bacterius ##### Well-known member MHB Math Helper [JUSTIFY]The rectangle is made of $800 \times 200$ "boxes". The diagonal thus has gradient $\pm \frac{200}{800} = \pm \frac{1}{4}$, where a "box" has unit dimensions. Note the diagonal starts at the top left corner of the top-left-most box. This is important. So, after 4 units of width travelled, the diagonal will intersect the top left corner of another box:[/JUSTIFY] [JUSTIFY]And this section of the diagonal intersects four boxes. Since the rectangle is 800 boxes wide, this section of the diagonal will repeat $\frac{800}{4} = 200$ times, and so the diagonal intersects $4 \times 200 = 800$ boxes.[/JUSTIFY] [HR][/HR] [JUSTIFY]A more interesting problem is to consider a rectangle of dimensions $p \times q$ where $p$ and $q$ are distinct primes. Then the diagonal never intersects the top-left corner of a box within the rectangle, and a different approach is called for:[/JUSTIFY] Last edited: #### mathmaniac ##### Active member Thats right,Bacterius... And can you show your approach for distinct primes? #### Bacterius ##### Well-known member MHB Math Helper Thats right,Bacterius... And can you show your approach for distinct primes? [JUSTIFY]It's simple enough, in fact. Because we know that the diagonal will, in this case, never intersect a corner, we can count the number of boxes crossed by simply counting the number of times the diagonal intersects both the vertical sides and the horizontal sides of the boxes. The diagonal will intersect $p - 1$ horizontal sides, and $q - 1$ vertical sides, simply by virtue of being a line crossing the rectangle from top-left to bottom-right (or top-right to bottom-left). But not so fast - the diagonal always starts inside the rectangle, and so automatically intersects the top-left box (or whichever corner of the rectangle you start your diagonal from), which we haven't yet considered, giving a total of $(p - 1) + (q - 1) + 1 = p + q - 1$ boxes intersected. Checking with the diagram above gives $11 = 7 + 5 - 1$ boxes intersected, as expected. Not very rigorous, just a proof sketch showing the general approach.[/JUSTIFY] Last edited: #### mathmaniac ##### Active member Here we have a winner!!!	2020-11-28T23:01:53	{ "domain": "mathhelpboards.com", "url": "https://mathhelpboards.com/threads/diagonal-crossing.3856/", "openwebmath_score": 0.7453159093856812, "openwebmath_perplexity": 1109.8177279388092, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9865717460476701, "lm_q2_score": 0.8757870029950159, "lm_q1q2_score": 0.8640267127106489 }
http://math.stackexchange.com/questions/112416/finding-domain-of-sqrt-fracx2-1x2-3x2-5x2-2x2-4x2-6	# Finding domain of $\sqrt{ \frac{(x^2-1)(x^2-3)(x^2-5)}{(x^2-2)(x^2-4)(x^2-6)} }$ How can I find the domain of: $$\sqrt{ \frac{(x^2-1)(x^2-3)(x^2-5)}{(x^2-2)(x^2-4)(x^2-6)} }$$ I think the hard part will be to find: $$\frac{(x^2-1)(x^2-3)(x^2-5)}{(x^2-2)(x^2-4)(x^2-6)} \ge 0$$ So far I have: not sure how to preceed: $$\sqrt{ \frac{(x^2-1)(x^2-3)(x^2-5)}{(x^2-2)(x^2-4)(x^2-6)} }$$ For $\sqrt{g(x)}$ to be valid, $g(x) \ge 0$ For $f(x)$ to be valid, $(x^2-2)(x^2-4)(x^2-6) \ne 0$ Thus, $x \ne \sqrt 2, 2, \sqrt 3$ $$g(x) = { \frac{(x^2-1)(x^2-3)(x^2-5)}{(x^2-2)(x^2-4)(x^2-6)}} \ge 0$$ - A friendly note: You have posting pictures in which you presumably had written the Math. I'd suggest you learn $\TeX$. Use the community to learn the commands. You can see the basic codes by clicking on them as seeing Math as $\TeX$ commands here. It will pay you in the long run and you'll become effective without having to write those crappy pics. Will you? (I like the way write your questions showing us what you have done! +1 for that!) – user21436 Feb 23 '12 at 12:09 @KannappanSampath, I know TeX but it takes longer to type them :) perhaps I need more practice. I also have put the more important parts of the question in TeX, the not as important stuff, I thought providing an image will suffice – Jiew Meng Feb 23 '12 at 12:12 I see you already know $\TeX$ from the first part of your question. Why don't you do the same thing for the Math in the pic as well? – user21436 Feb 23 '12 at 12:13 Is the numerator $(x^2-1)(x^2-3)(x^2-5)$ or $(x^2-1)(x^3-1)(x^5-1)$? – 01000100 Feb 23 '12 at 12:13 Its the 1st, $(x^2-1)(x^2-3)(x^2-5)$ – Jiew Meng Feb 23 '12 at 12:14 The first thing you should note is that the expression $$\Phi=\frac{(x^2-1)(x^2-3)(x^2-5)}{(x^2-2)(x^2-4)(x^2-6)}$$ is undefined at any zero of the denominator. So, the points $x=\pm \sqrt 2$, $\pm 2$, and $\pm\sqrt 6$ are not in the domain of $\sqrt\Phi$. Now, to find the domain of $\sqrt\Phi$, we need to find when $\Phi$ is nonnegative. Towards solving $\Phi\ge0$, the fundamental observation to make is that the only points "across which" $\Phi$ can change signs are at the zeros of its numerator or at the zeroes of its denominator. The zeroes of the numerator are: $$\pm 1, \pm \sqrt 3, \pm \sqrt 5$$ and the zeroes of the denominator are $$\pm \sqrt 2, \pm 2 \pm\sqrt 6.$$ As already mentioned, the only points across which $\Phi$ can change sign are at one of the zeroes above. Let's also note the expression $\Phi$ is "even": if $\Phi(x)\ge 0$, then $\Phi(-x)\ge 0$. So, let's find the $x$ values on the nonnegative $x$-axis that satisfy $\Phi(x)\ge0$. Then by "reflection" we'll obtain the points on the negative $x$-axis where $\Phi(x)\ge0$. So, draw a number line with the zeroes listed above: The zeroes subdivide the nonnegative $x$-axis into intervals, the endpoints of which, except the endpoint 0, are the zeroes of either the numerator or the denominator of $\Phi$. In each subinterval, if you pick a point $x_0$ (not an endpoint, save for 0), evaluate $\Phi(x_0)$, and note its sign, then across that entire subinterval, the expression $\Phi$ will have that sign. For example in $[0,1)$, picking $x=1/2$ it is easy to see that the sign of $\Phi(1/2)$ is $$\frac{((1/2)^2-1)((1/2)^2-3)((1/2)^2-5)}{((1/2)^2-2)((1/2)^2-4)((1/2)^2-6)} ={ (-)(-)(-)\over(-)(-)(-)} \ge 0$$ So on all of $[0,1)$, the expression $\Phi$ is positive. (Note, you only need to find the sign, there is no need to do the actual arithmetic.) The complete "sign chart" is shown below: And we see that $\Phi>0$ on $$[0, 1)\cup(\sqrt2,\sqrt3)\cup(2,\sqrt5)\cup(\sqrt6,\infty).$$ By symmetry, $\Phi>0$ on $$( -\infty,-\sqrt6)\cup(-\sqrt5,2)\cup (-\sqrt3,-\sqrt2)\cup (-1,0].$$ So, the domain of $\sqrt\Phi$ is the union of the two sets above, together with the zeroes of the numerator of $\Phi$ that aren't zeros of the denominator of $\Phi$: $x=\pm1$, $x=\pm \sqrt3$, and $x=\pm \sqrt5$. Below is a plot of $y=\Phi$ and $y=\sqrt\Phi$. Note that the zeroes of the denominator of $\Phi$ are vertical asymptotes of the graph of $y=\Phi$ and the zeroes of the numerator of $\Phi$ are the zeroes of $\Phi$. - 1. $\frac{f(x)}{g(x)} \ge 0$ is not so different from $f(x)\times g(x) \ge 0$ (except for zeros of $g$.) 2. With $(x^2-1) = (x-1)(x+1)$ etc, your problem reduces to the form of $(x-a)(x-b)(x-c)(x-d)...(x-z) \ge 0$ Edit: oops I only read the hand-written part! Anyways thanks to the monotonicity of $x^5-1$ etc, you can still use similar argument. Edit 2: Plot it in google. - +1 for the link. I didn't know Google could plot. – user23211 Feb 23 '12 at 13:37 $(x^2-1)$ is negative iff $x\in(-1,1)$. Similarly, $(x^2-2)$ is negative for $x\in(-\sqrt{2},\sqrt{2})$, $(x^2-3)$ for $x\in(-\sqrt{3},\sqrt{3}), \dots,(x^2-6)$ for $x\in(-\sqrt{6},\sqrt{6})$. Consider the open intervals $(-\infty,-\sqrt{6}), (-\sqrt{6},-\sqrt{5}),(-\sqrt{5},-2),\dots$ separately. Don't forget to look at how $f(x)$ behaves on the boundaries. E.g $x=\pm\sqrt{n}$ for $n=1,2,3,4,5,6$. - Ok, now I get the domain as $-\sqrt{5} \lt x \lt -2$, $-\sqrt{3} \lt x \lt -\sqrt{2}$, $-1 \lt x \lt 1$ ... but how do I find out without graph about what happens $x \lt -\sqrt{6}$ and $x \gt \sqrt{6}$ – Jiew Meng Feb 23 '12 at 14:23 When x>$\sqrt{6}$ all the factors in the numerator and denominator are positive, so the whole expression is positive. The situation is the same for $x<−\sqrt{6}$ – 01000100 Feb 23 '12 at 14:49	2015-07-07T22:46:16	{ "domain": "stackexchange.com", "url": "http://math.stackexchange.com/questions/112416/finding-domain-of-sqrt-fracx2-1x2-3x2-5x2-2x2-4x2-6", "openwebmath_score": 0.9384967088699341, "openwebmath_perplexity": 249.44829644022835, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.98657174604767, "lm_q2_score": 0.8757869948899665, "lm_q1q2_score": 0.864026704714436 }
https://cozilikethinking.wordpress.com/2013/09/01/71/	Let $f:X\to Y$ be a mapping. We will prove that $f^{-1}(Y-f(X-W))\subseteq W$, with equality when $f$ is injective. Note that $f$ does not have to be closed, open, or even continuous for this to be true. It can be any mapping. Let $W\subseteq X$. The mapping of $W$ in $Y$ is $f(W)$. As for $f(X-W)$, it may overlap with $f(W)$, we the mapping be not be injective. Hence, $Y-f(X-W)\subseteq f(W)$. >Taking $f^{-1}$ on both sides, we get $f^{-1}(Y-f(X-W))\subseteq W$. How can we take the inverse on both sides and determine this fact? Is the reasoning valid? Yes. All the points in $X$ that map to $Y-f(X-W)$ also map to $W$. However, there may be some points in $f^{-1}(W)$ that do not map to $Y-f(X-W)$. Are there other analogous points about mappings in general? In $Y$, select two sets $A$ and $B$ such that $A\subseteq B$. Then $f^{-1}(A)\subseteq f^{-1}(B)$	2017-09-23T20:12:11	{ "domain": "wordpress.com", "url": "https://cozilikethinking.wordpress.com/2013/09/01/71/", "openwebmath_score": 0.9339148998260498, "openwebmath_perplexity": 74.47047190814499, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. Yes\n2. Yes", "lm_q1_score": 0.9884918502576514, "lm_q2_score": 0.8740772384450967, "lm_q1q2_score": 0.864018226698692 }
https://math.codidact.com/questions/278629	Q&A # Why does the decimal expansion of $1/(10n - 1)$ have this neat pattern? +10 −0 I was playing around with the reciprocals of some positive integers and found these interesting patterns: $$\frac{1}{19} = 0.\overline{052631578947368421}$$ Now, this repeating decimal can also be obtained by "concatenating" the powers of $2$ successively to the left as follows: \begin{align} 1& \\ 21& \\ 421& \\ 8421& \\ {\color{red}{1}}68421 \\ {\color{red}{3}}{\color{green}{3}}68421 \\ \vdots\quad & \end{align} Notice that here I think of the red $\color{red}{1}$ of $16$ and $\color{red}{3}$ of $32$ as being "carried over", so the units digit of $32$ and the tens digit of $16$ add to give me the green $\color{green}{3}$. The sequence gets increasingly complicated as more terms are concatenated in this manner, but it definitely looks like it's converging to $\dotsc 052631578947368421$, where the ellipsis indicates that the pattern repeats forever towards the left. This is a bit surprising because (informally) this would mean that if I perform this pattern infinitely far out to the right of the decimal point, then I get the decimal expansion of $1/19$. How can I go about making this idea precise? Some more data for this pattern: \begin{align} \vdots\quad & \\ {\color{red}{3}}{\color{green}{3}}68421 \\ {\color{red}{6}}{\color{green}{7}}368421 \\ {\color{red}{1}}{\color{green}{34}}7368421 \\ {\color{red}{2}}{\color{green}{69}}47368421 \\ {\color{red}{5}}{\color{green}{38}}947368421 \\ {\color{red}{10}}{\color{green}{77}}8947368421 \\ \vdots\quad & \end{align} In this pattern, one can see that at each stage one new digit from the required sequence is correctly added, despite all the carrying-over. I have also observed this to be the case for the decimal expansion of $1/29$: $$\frac{1}{29} = 0.\overline{0344827586206896551724137931}$$ In this case, we concatenate the powers of $3$ in a similar fashion. Is there any significance to the fact that $19 = {\color{red}{2}}10 - 1$ and $29 = {\color{red}{3}}10 - 1$ to the use of powers of $2$ and $3$, respectively, in these computations? I also checked for $1/39$, and though the repeating part of its decimal expansion is not as impressively long as for the other two cases, the pattern still holds up: $$\frac{1}{39} = 0.\overline{025641}$$ and concatenating the powers of $4$ indeed gives $\dotsc 025641$. The last case I checked was $1/49$, and it also obeys this pattern: $$\frac{1}{49} = 0.\overline{020408163265306122448979591836734693877551}$$ and concatenating the powers of $5$ gives us this very repeating set of digits. Of course, the pattern for $1/49$ makes it obvious that there is a concatenation that works in the forward direction: namely, the powers of $2$ concatenated in bunches of two digits. But this just means that $$\frac{1}{49} = \frac{2}{100} + \frac{2^2}{100^2} + \frac{2^3}{100^3} + \dotsb$$ which is true by the formula for the sum of an infinite geometric series. So, "forward concatenating" patterns can be explained in this manner, but I'm stumped regarding how to explain the "backward concatenating" ones. If anyone can throw some light on this, it would be greatly appreciated! Why does this post require moderator attention? Why should this post be closed? +5 −1 First of all, there is indeed a pattern. To figure out why, there's an easier example than what you've found: $$0.\bar{1} = \frac{1}{9}.$$ How does that work? $1$ is the only power of $1$: $1^n = 1$. Now, how do we prove this? if $x = 0.\bar{1}$, then $10x = 1.\bar{1}$ and so, $9x=1$. Hidden in that proof lies the answer for the other cases - you have some repeating number and in order to show what fraction that repeating number is equal to, here $1/\left(10n-1\right)$, you multiply it by $10n$ and subtract the number (again, I'll just call it $x$). To take your example of $1/19$, we have that the fraction is $x = 0.\overline{052631578947368421}$, so we multiply by $20$ to get $20x = 1.\overline{052631578947368421}$, subtract $x$ and $19x = 1$. Crucially, the number left on the RHS after subtracting is exactly $1$, as everything after the decimal point cancels. It must do this, because the fraction is some integer divided by some other integer. That is, if $x=n/m$, then $mx=n$. As we're in a decimal system, multiplying by $10$ is shifting one digit to the left. So, in doing this multiplication by $10n$, you're shifting the digit to the left, then multiplying by $n$, which is exactly the procedure outlined in the question. Now, each particular digit (after this multiplication procedure) must cancel with the digit to the left so at this point, they must be equal and we have the exact behaviour outlined in your question. To explain this generally and in more detail (taking $x<1$ wlog.) I'll use a fraction $$x = \frac{m}{10n-1}.$$ When written in decimal format, this is $x = 0.x_1x_2\ldots x_k\ldots$, with $x_{k+r}=x_k$ for some $r$, where each $x_k$ is a single digit (i.e. an integer in the range $0-9$ as we're in a decimal basis). Now, we know that $\left(10n-1\right)x = m$, or $10nx - x = m$. However, m is an integer, so all the digits after the decimal point (the 'fractional part', in other words) are $0$. This means that the digits in the fractional part of $10nx - x$ must also be $0$. So, we look at the digits $x_r$ and $x_1\ldots x_{r-1}$. The fractional part of $10nx -$ the fractional part of $x$ equals $0$, or the fractional part of $10nx$ equals the fractional part of $x$. For this to be the case, all the digits must be equal. So, we take the digit $x_r$. To calculate $x_{r-1}$, we use the properties of the fractional parts being equal and multiplication by 10 as shifting the digit to the left to give that the fractional part of $10n\left(0.x_1\ldots x_r\right) - 0.x_1\ldots x_r = 0.0\ldots x_r$ (where the $0$s after the point are repeated $r-1$ times). This is exactly what's written in the question, just in a different way: $nx_r \mod 10 = x_{r-1}$. Or, the digit to the left of $x_{r-1}$ is $nx_r$, with change. Accounting for this change gives that $x_{r-j} = n^jx_r$, which is where the powers of $n$ come from, the change being what needs to be added to digits further to the left. To explain what I mean by phrases such as "Accounting for this change", let's take the initial stream of digits again and write them out more explicitly, where the number in brackets to the right of each line represents the number of times the 0 is repeated after the decimal point: Now, we apply this procedure of multiplying by $10n$. It is a requirement that, because the fractional part of $x$ must equal the fractional part of $10nx$, $10nx-m=x$, that is \begin{align}10nx - m &= 0.0\ldots x_k \qquad(k-1)\\ &+ 0.0\ldots x_{k-1} \qquad(k-2)\\ &+ 0.0\ldots x_{k-2} \qquad(k-3) + \ldots\end{align} Taking this line by line (where the term in brackets on the LHS has $j-1$ $0$s after the decimal point and the superscripts denote that this series arises from the $j^{th}$ term in the similar expansion of $x$) gives that We can add this to the next term in the expansion of $x$ to get that (unfortunately at this point, notation gets very confusing but the first term on the left hand side has $j-1$ $0$s and the second, $j-2$) \begin{align}10n\left(0.0\ldots x_j + 0.0\ldots x_{j-1}\right) &= 0.0\ldots x_{k-1}^{(j)} \qquad(k-2)\\ &+ 0.0\ldots x_{k-2}^{(j)} \qquad(k-3)\\ &+ 0.0\ldots x_{k-2}^{(j-1)} \qquad(k-3)\\ &+ 0.0\ldots x_{k-3}^{(j)} \qquad(k-4)\\ &+ 0.0\ldots x_{k-3}^{(j-1)} \qquad(k-4) + \ldots\end{align} Now, having done this, the $k^{th}$ term is exactly $$\sum_{j>k}0.0\ldots x_k^{(j)} \qquad(k-1),$$ which is the sum over $10n\times$ all the digits to the right of the $k^{th}$ digit, where you're only summing the resulting $k^{th}$ digit. However, you can recursively apply this to get that the $k^{th}$ digit is the sum over $\left(10n\right)^p\times$ all the digits, $p$ positions to the right of the $k^{th}$ digit, where you're only summing the resulting $k^{th}$ digit. This is precisely what's done in the question, although written in a different way and shows that the behaviour arises because the terms are required to be sums of powers of $10n$ of previous terms because this is a fraction with a denominator of $10n-1$. Why does this post require moderator attention? I'm confused how this answers the question of the pattern of digits. You only use the fact that x = 1/19 in your answer and don't explain what the significance of the powers of n is. (Or maybe my reading comprehension skills are poor...) Quintec‭ about 1 month ago Thank you for your answer! I hope you won't mind if I say that I somehow feel this answer doesn't go all the way towards explaining why this "backwards concatenation" process appears. Specifically, I do understand that if $x = 0.\overline{x_1 x_2 x_3 \dotso x_r}$, then $x$ is a rational number equal to some $n/m$ that can be found as you outlined. However, why does this method of specifically concatenating the powers of $n$ work when considering $1/(10n - 1)$? Abheri Ragam‭ about 1 month ago Secondly, though I say this "works", how do I adequately make sense of that? After all, the string that I get by my method of concatenation strictly speaking gives an "infinite number" such as $\dotsc 052631578947368421$. So perhaps, my claim is something like $$\frac{1}{10n-1} = \lim_{k \to \infty} \frac{(10n)^0 + (10n)^1 + (10n)^2 + \dotsb + (10n)^k}{10^k}\ ?$$ I'm not sure... But I do think there's a bit more to be said from where you leave off in your post :) Abheri Ragam‭ about 1 month ago I've updated my answer (although honestly, I feel that the first bit is about 100 times clearer). I've included the bit about the powers of $n$ and what you're referring to as 'backwards concatenation' is what I'm referring to as 'multiplying a digit by $n$ to get the digit to the left'. It works because the '-1' is the bit that causes the fractional part to cancel, hence be equal Mithrandir24601‭ about 1 month ago But, $x_{r-j}$ is not equal to $n^j x_r$! (I presume you meant modulo $10$?) For instance, for the decimal expansion of $1/19$, $x_{r-5} = 3$ but $2^5 x_r = 32 \neq 3 \pmod{10}$. In fact, they are not equal precisely because of the carrying-over happening in the concatenation process. I'm sorry, but I'm still not satisfied that your answer fully explains this phenomenon. Abheri Ragam‭ about 1 month ago +2 −0 I was pretty curious about this, so I posted a question on Math.SE. Most of this answer will draw from the idea of J. W. Tanner's answer there. There is actually a simple reason why this pattern holds. Lets start by examining how the digits are generated: each digit is a power of two (with carrying). We can withhold decimal shifting for a second and simply look at a simplified expression with the same digits, only generated to the left of the decimal point: \begin{aligned}\sum_{k=0}^{n-1} 2^k\cdot 10^k&=1+20+400+16000+320000+\cdots+2^{n-1}\cdot10^{n-1}\\ &=\cdots7368421\end{aligned} You can see how the pattern generated is the same. Let's look at it another way, using the geometric series formula. \begin{aligned}\sum_{k=0}^{n-1} 2^k\cdot10^k&=\frac{1(1-20^n)}{1-20}\\ &=\frac{20^n-1}{19}\end{aligned} At this point, it's pretty clear why the digits generated become the repeated decimal of $1/19$. You can see the denominator of $19$ right there! If you're not satisfied with that observation, we can further note the following: \begin{aligned}\frac{20^n-1}{19}&=\frac{20^n}{19}-\frac{1}{19}\\ &=\frac{2^n}{19}\cdot10^n-\frac{1}{19}\end{aligned} Let's specifically focus on $2^n/19$. Since we are interested in repeated groups of digits, let's let $n$ be a multiple of the totient of $19$, which is $18$. The reason we chose the totient is because of a useful property of modular arithmetic: $a^b\equiv 1\mod c$ if $a$ and $c$ are coprime and $b$ is a multiple of the totient of $c$ (Euler's theorem). Clearly, $2$ is coprime to $19$, so letting $n$ be a multiple of $18$ gives us a remainder of 1 when dividing $2^n$ by $19$. Therefore, the digits after the decimal point of $2^n/19$ are the same as the digits of $1/19$. Going back to the whole expression, we then multiply by $10^n$, bringing $n$ of those repeated digits over to the left of the decimal point. Everything to the right is canceled by the subtraction of $1/19$ (the digits must cancel, since the expression is an integer). The result is the final number has its last $n$ digits be the first $n$ digits of $1/19$ when $n$ is a multiple of $18$, which is what we wanted to prove. (It's not difficult to generalize this for any $n$, but that's beyond this answer). A similar result can be produced for any $10c-1$, where the digits will repeat every totient of $10c-1$ (though it may repeat before that) and the repeating portion will be the same as the digits of $1/(10c-1)$. Since it's basically the same process, I won't repeat it here. Why does this post require moderator attention? #### 1 comment Thank you! I've been considering what you say in your answer, and I'm now satisfied with the explanation :) Abheri Ragam‭ about 1 month ago +0 −0 I think the simplest way to see how it works is as follows: Given $x=\frac{1}{10n-1}$, it is easy to check that $\frac{x+1}{10}=nx$. Now $\frac{x+1}{10}$ means shifting the digits to the right, with an $1$ added as the first decimal. For example, in the case of $x=1/19$, we have $$\frac{x+1}{10} = 0.1\overline{052631578947368421}.$$ But now we see that the additional digit matches the one at the end of the period, so we just can shift the period one to the left. At the same time on the left we use the equation derived above: $$2\cdot\frac{1}{19} = 0.\overline{105263157894736842}.$$ Thus multiplying by $2$ (or, in the general case, by $n$) is equivalent to shifting the digits one position to the right (and adding a $1$ at the first decimal, which gives the carry from the multiplication). Thus since the last digit of the original period was $1$, the digits to the left of it are successive powers of $2$ (with carry), or successive powers of $n$ in the general case. So the only remaining question is why the period always ends in $1$. Well, since multiplying the number with $19$ (or, in the general case, with $10n-1 = 10(n-1)+9$) has to give $1=0.\overline{999999999999999999}$, the last digit of the period, when multiplied with the last digit of the denominator, that is $9$, must give $10k+9$ for some integer $k$. But that is clearly only possible if that last digit is $1$. Why does this post require moderator attention?	2020-11-30T20:41:38	{ "domain": "codidact.com", "url": "https://math.codidact.com/questions/278629", "openwebmath_score": 0.9742663502693176, "openwebmath_perplexity": 243.84589595101738, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9904406014994152, "lm_q2_score": 0.8723473746782093, "lm_q1q2_score": 0.8640082584927213 }
https://math.stackexchange.com/questions/1236347/if-a-and-b-are-positive-integers-and-4ab-1-mid-4a2-1-then-a-b/1236370	# If $a$ and $b$ are positive integers and $4ab-1 \mid 4a^2-1$ then $a=b$. Prove that if $a$ and $b$ are positive integers and $$(4ab-1) \mid (4a^2-1)$$ then $a=b$. I am stuck with question, no idea. Is there any way to prove this using Polynomial Division Algorithm? Would appreciate any help. • This actually holds even with the weaker assumption that $4ab - 1 \mid (4a^2 - 1)^2$. – George V. Williams Apr 15 '15 at 19:48 • I added a generalization to my answer which makes it crystal clear how this is nothing but the uniqueness of inverses. – Bill Dubuque Apr 15 '15 at 22:02 Note that $4ab\equiv 1\pmod {4ab-1}$ and if $4ab-1\mid 4a^2-1$ then $4a^2\equiv 1\pmod{4ab-1}$. So $$b\equiv (4a^2)b=a(4ab) \equiv a\pmod{4ab-1}.$$ Is that possible if $a\neq b$? Without using modular arithmetic, you can write this as: $$a-b = b(4a^2-1)-a(4ab-1)$$ So $4ab-1\mid a-b$. • The proof is just a special case of the uniqueness of inverses - see my answer. – Bill Dubuque Apr 15 '15 at 20:01 • Well, yes, but you still arrive at $4ab-1\mid a-b$. @BillDubuque – Thomas Andrews Apr 15 '15 at 20:05 • The point is that viewing it this way makes everything obvious, even the requisite inequality (omitted in your answer). I expanded my remark to emphasize this - giving the natural generalization of the OP's theorem. Hopefully that makes my point clearer. – Bill Dubuque Apr 15 '15 at 21:57 Clearly, $a \geq b$, since we need $4a^2-1 \geq 4ab-1$. We have $$\dfrac{4a^2-1}{4ab-1} = k \in \mathbb{Z}$$ Hence, $$k = \dfrac{4a^2-1}{4ab-1} = \dfrac{4a^2-4ab+4ab-1}{4ab-1} = 1 + \dfrac{4b(a-b)}{4ab-1}$$ Now note that $\gcd(4ab-1,4b)=1$, since $(4b)a - (4ab-1) = 1$. Hence, $4ab-1$ divides $a-b$. However $0 \leq a-b < 4ab-1$. Hence, $a-b=0 \implies a=b$. Note $\,\ {\rm mod}\,\ 4ab\!-\!1\!:\,\ \overbrace{(4a)\color{#c00}a\equiv 1\equiv(4a)\color{#c00}b}^{\ \ \ \large \color{#c00}a\, \equiv\, (4a)^{-1}\equiv\,\color{#c00} b}\,\Rightarrow\, \color{#c00}{a\equiv b}\$ by uniqueness of inverses (of $\,4a\,$ here) Remark $\$ For completeness, here is the standard proof of uniqueness of inverses: $$\ ca\equiv 1\equiv cb \,\Rightarrow\, a\equiv a(cb)\equiv (ac)b\equiv b\quad\ \$$ The OP Theorem becomes obvious when expressed in this general form, amounting simply to the uniqueness of inverses mod $\,n\,$ within the standard rep system $\,\{0,1,\ldots,n\!-\!1\},\,$ namely Theorem $\$ If $\, \color{#0a0}{a< bc}\,$ and $\, bc\!-\!1\mid ac\!-\!1\$ then $\ a = b,\$ for integers $\,a,b,c > 0$ Proof $\ \ {\rm mod}\,\ n\!=\!bc\!-\!1\!:\,\ bc\equiv 1\equiv ac\,\Rightarrow\, a\equiv c^{-1}\!\equiv b.\$ $\,bc\!-\!1\mid ac\!-\!1\,\Rightarrow\, b\le a \le \color{#0a0}{bc\!-\!2}\,$ (by $\,b\not\equiv 0).\,$ So $\,a\equiv b\pmod{n}\,$ and $\,a,b\in \{0,1,\ldots,\color{#0a0}{n\!-\!1}\}\Rightarrow\, a=b\,$ (else $\,n\,$ divides the smaller natural $\,a\!-\!b> 0,\,$ contradiction). $\ \$ QED The OP is the special case $\ c = 4a,\$ where $\ a < 4ab = bc,\,$ so the Theorem applies. Note how translating the theorem into the language of congruences has simplified it so much that we immediately recognize it as a special case of a well-known result about uniqueness of inverses. This is yet another example of a ubiquitous principle that I frequently emphasize here, namely uniqueness theorems provide powerful tools for proving equalities. Note that any integer $m \ge 2$ will do, instead of 4. For example, take Thomas Andrew's solution, and put $m$ for $4$ everywhere: Note that $mab\equiv 1\pmod {mab-1}$ and if $mab-1\mid ma^2-1$ then $ma^2\equiv 1\pmod{mab-1}$. So $$b\equiv (ma^2)b=a(mab) \equiv a\pmod{mab-1}.$$ Is that possible if $a\neq b$? Without using modular arithmetic, you can write this as: $$a-b = b(ma^2-1)-a(mab-1)$$ So $mab-1\mid a-b$. My note: To show that $mab-1 > a-b$ for $m \ge 2$, $ab-a+b-1 =(a+1)(b-1) \ge 0$. Note that if $m=b=1$, then $mab-1 = a-1$. • Yes: $\ (ma)a\equiv 1\equiv (ma)b\,\Rightarrow\, a\equiv (ma)^{-1}\equiv b\,$ by uniqueness of inverses, see my answer. To get the general result replace $\,ma\,$ by $\,c\,$ above. – Bill Dubuque Apr 15 '15 at 20:23 • Update: even the inequality has a natural interpretation when viewed from this standpoint: it is simply what's needed for the inverses to lie in the standard rep range. So the theorem is precisely equivalent to the uniqueness of inverses in the standard rep range, except the congruence language has been removed, being replaced by equivalent divisibility language, which obscures the essence of the matter: inverse unqueness (and it is further obfuscated by choice of a specific modulus). See the edit to my remark.for details. – Bill Dubuque Apr 16 '15 at 0:01	2019-08-26T06:49:38	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/1236347/if-a-and-b-are-positive-integers-and-4ab-1-mid-4a2-1-then-a-b/1236370", "openwebmath_score": 0.9476520419120789, "openwebmath_perplexity": 303.29098091829053, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9904405999474861, "lm_q2_score": 0.8723473713594991, "lm_q1q2_score": 0.8640082538519148 }
http://creativenet.ca/24dcs5jl/python-distance-between-two-coordinates-92a9fd	The bearing outputs negative but should be between 0 – 360 degrees. Distance Calculator is use to calculate the distance between coordinates and distance between cities. An $$m_B$$ by $$n$$ array of $$m_B$$ original observations in an $$n$$-dimensional space. To calculate the distance between two points we use the inv function, which calculates an inverse transformation and returns forward and back azimuths and distance. As per wiki definition. geopy makes it easy for Python developers to locate the coordinates of addresses, cities, countries, and landmarks across the globe using third-party geocoders and other data sources. , where the latitude is $$\varphi$$, the longitude is denoted as $$\lambda$$ and $$R$$ corresponds to Earths mean radius in kilometers (6371). pip install geopy Geodesic Distance: It is the length of the shortest path between 2 points on any surface. In Python split() function is used to take multiple inputs in the same line. For example there is the Great-circle distance, which is the shortest distance between two points on the surface of a sphere. MATH. It is a great package to work with map projections, but in there you have also the Geod class which offers various geodesic computations. Examples: Input : x1, y1 = (3, 4) x2, y2 = (7, 7) Output : 5 Input : x1, y1 = (3, 4) x2, y2 = (4, 3) Output : 1.41421 Inputs are converted to float type. Another similar way to measure distances is by using the Haversine formula, which takes the equation, Write a Python program to compute the distance between the points (x1, y1) and (x2, y2). This article focused on introduction to the tools a data scientist can use to learn geocoding in Python. 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In this article, we will see how to calculate the distance between 2 points on the earth in two ways. You can also use geopy to measure distances. An $$m_A$$ by $$n$$ array of $$m_A$$ original observations in an $$n$$-dimensional space. The two points must have the same dimension. TL;DR - Vincenty's inverse formula provides an accurate method for calcualting the distance between two latitude/longitude pairs. Python provides several packages for data manipulation that are easy to use and are supported by a large community of contributors. We can take this function now and apply distances to different cities. Python \| Get a google map image of specified location using Google Static Maps API. Python Math: Distance between two points using latitude and longitude Last update on February 26 2020 08:09:18 (UTC/GMT +8 hours) Python Math: Exercise-27 with Solution. It is a method of changing an entity from one data type to another. 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Project of visualizing geo-location data ( i.e theoretical distance returned by GeoDjango ( shp ) and x2... Getaway Movie 2020 Release Date, Sound Effects App For Android, Ramshorn Snail Anatomy, Cara Menghitung Expected Return Dengan Excel, Red Degeneration Of Fibroid Histology, Afton Family Past Vs Present Singing Battle Gacha Life, Chief Sales Officer Salary, Gaucho Rock In 12v Jeep, Fallout 76 Gold Bullion Items List,	2021-05-08T22:47:05	{ "domain": "creativenet.ca", "url": "http://creativenet.ca/24dcs5jl/python-distance-between-two-coordinates-92a9fd", "openwebmath_score": 0.28711771965026855, "openwebmath_perplexity": 1859.2880088230952, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9755769113660689, "lm_q2_score": 0.8856314813647587, "lm_q1q2_score": 0.8640016251983875 }
https://math.stackexchange.com/questions/944568/modular-arithmetic	# Modular arithmetic Hello, What is the remainder when the following sum is divided by 4? $1^5 + 2^5 + 3^5 +...+ 99^5 + 100^5$ I feel like it has to do with modular arithmetic... I am trying to decompose every number but it seems to long and unnecessary. Any ideas? P.S. thank you for your ideas. I got it. Please don't post solutions • It's $0$, I believe. – Akiva Weinberger Sep 24 '14 at 17:04 • There's no need to compute all the numbers. Modular Arithmetic is the way, but you have also to get some regularity. Usually, the first thing to do is to try smaller numbers, to see if there are patterns – Exodd Sep 24 '14 at 17:07 • Hint: Any even number squared is divisible by 4 and any odd number power will give remainder 1. So count how many odds there are. – Ali Caglayan Sep 24 '14 at 17:08 HINT : Note that in mod $4$, $$1^5\equiv1,\ \ 2^5\equiv 0,\ \ 3^5\equiv (-1)^5=-1\equiv 3,\ \ 4^5\equiv 0$$ and that $$1+0+3+0\equiv 0,\ \ 100=4\times 25.$$ hint :take {1,2,3,4},{5,6,7,8} .....{97,98,99,100} as a set now each corresponding term in each set has same remainder when divided by 4 , so you effectively need to calculate only the remainder of $1^5+2^5+3^5+4^5$ w.r.t 4 and then multiply it by 25 and again find that numbers remainder w.r.t 4 $$1^5 \equiv 1 \pmod{4}$$ $$4\mid 2^5 \Rightarrow 2^5 \equiv 0 \pmod{4}$$ $$3^5 \equiv (-1)^5 \equiv -1 \pmod{4}$$ $$4^5 \equiv 0^5 \equiv 0 \pmod{4}$$ This means that every even number, when raised to the fifth power, is $0 \pmod{4}$. Thus the sum you asked about is equal to the number of $1 \pmod{4}$ numbers in $\{1,2,\dots 100\}$ minus the number of $3 \pmod{4}$ numbers in $\{1,2,\dots 100\}$. Every fourth number from $1$ to $97$, inclusive, is $1 \pmod{4}$. There are $25$ of these. Every fourth number from $3$ to $99$, inclusive, is $3 \pmod{4}$. There are $25$ of these as well, so $$\displaystyle\sum\limits_{n=1}^{100} n^5 \equiv 0 \pmod{4}$$	2019-09-18T21:49:49	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/944568/modular-arithmetic", "openwebmath_score": 0.7862219214439392, "openwebmath_perplexity": 262.99803592972114, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9755769071055402, "lm_q2_score": 0.8856314813647587, "lm_q1q2_score": 0.8640016214251292 }
https://math.stackexchange.com/questions/2238134/matrix-size-for-error-correcting-linear-code	# Matrix size for error-correcting linear code This is Exercise 21 of the textbook "Abstract Algebra: Theory and Applications" by Thomas W. Judson, 2016; Page 111. Chapter 8 "Algebraic Coding Theory" mainly deals with binary linear code with Hamming code as an example. Exercise 8.21: If we are to use an error-correcting linear code $C$ to transmit the 128 ASCII characters, what size matrix must be used? The screenshot is as follows: My Attempt: I consider the parity check matrix $H_{(n-k) \times n} = [P_{(n-k)\times k}\mid I_{n-k}]$. Let $m = n - k$, the number of parity bits. First of all, the data bit is $k = \log_2 128 = 7$. Let the error-correcting capability be $t = 1$. The minimum Hamming distance of the linear code $C$ is at least $3$. Therefore, the zero column and $e_i$'s columns are not in $P_{(n-k) \times k}$ (because $H$ contains no idential columns). Thus, we have $2^{m} - m - 1 \ge k$, which gives $m \ge 4$. That is, the matrix size is $4 \times 11$ (Note: not $4 \times 7$ as pointed out in the comment by @Jyrki Lahtonen). Is the argument above correct? Also, how to solve this problem for any parameter $t$, the error-correcting capability of $C$? • The dimensions of $H$ are $(n-k)\times n$. So the number of columns should be $n=k+m$. Not $k$ as your final answer suggests. In general ($t>1$) it will be more difficult to judge the number of redundant bits needed to correct $t$ errors. It may turn out that the answer is known for $7$-dimensional codes, but I can only give bounds (e.g. what you would get using a BCH-code). Apr 17 '17 at 9:45 • @JyrkiLahtonen Thanks. I will modify my answer. I think the problem for finding the matrix size for the general case of $t > 1$ is related to the lemma that "If $H$ is the parity check matrix of $C$, then the minimum Hamming distance of $C$ equals the minimum number of columns of $H$ that are linearly dependent." Is it right? By the way, a bound on the matrix size is also appreciated. Apr 17 '17 at 9:52 • " what matrix size must be used?" Is that really the full text of the exercise? What are the error detection-correction capabilities desired? Is the code binary? There seems to be some missing data. BTW, the link does not work for me (what about a screenshot?) Apr 17 '17 at 22:04 • @leonbloy Yes, it is "what size matrix must be used" (I have reordered two words by mistake). The code is binary. This is implied by the text in the chapter. Thanks for pointing this out. I will update this post right now. Apr 18 '17 at 3:03 I assume that by "an error-correcting linear code", they meant a code capable of correcting one (bit) error (per coded symbol). In that case your reasoning is on the right track (and it's the reasoning used to construct a Hamming code). We know $k=7$, the matrix $H$ will have up to $2^m-1=n$ columns. The smallest Hamming code in our case is then $(15,11)$ ($11$ bits of information plus $4$ of redundacy). Because this give a code with higher $k$ than needed, we can trim the $11-7=4$ unused data bits, and we are left with a $(11,7)$ code ($7$ bits of information plus $4$ of redundacy). (Equivalently, the Hamming construction gives the -sufficient- condition of $n\le 2^{n-k}-1$; it's easy to check that, for $k=7$, $n=11$ is the minimum number that fulfill the condition) If we only need one error detection, a single parity bit is enough, so we have a $(8,7)$ code. • Thanks. By the way, is this problem much harder for general $t$, the error-correcting capacity? Apr 18 '17 at 3:35 • Yes, see BCH codes for example. Apr 18 '17 at 4:57 Answer my question: I realized that my argument in the post relies on the standard parity-check matrix $H$. I show another argument here and ask for reviews. First, the information bit $k = \log_2 128 = 7$. Let $r = (n-k)$. The parity-check matrix $H_{(n-k) \times n}$ has $r$ rows. Therefore, $H$ has at most $2^{r} - 1$ columns (the zero column is excluded). To achieve the an-error-correcting capability, the code should be able to correct all the possible $n = 7 + r$ one-bit errors, each corresponding to one column of $H$. Therefore, we have $$2^r - 1 \ge 7 + r.$$ That is, $r \ge 4$ and the size of $H$ is at least $4 \times 11$.	2022-01-23T10:59:19	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/2238134/matrix-size-for-error-correcting-linear-code", "openwebmath_score": 0.8746059536933899, "openwebmath_perplexity": 202.5856900283433, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.975576912786245, "lm_q2_score": 0.8856314632529871, "lm_q1q2_score": 0.8640016087867141 }
https://ipso-facto.joseaniceto.com/pi-monte-carlo.html	# The Monte Carlo method Monte Carlo methods are a group of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in optimization problems, numerical integration, and to generate draws from a probability distribution.1 # Estimating the value of $$\pi$$ Consider the circle in the diagram. If we generate a sufficiently large number of $$(x, y)$$ points in the domain $$-1 \leqslant x \leqslant 1$$ and $$-1 \leqslant y \leqslant 1$$, then the fraction of points that fall inside the circle will be equal to the ratio between the area of the circle and area of the square. $$\frac{\textrm{number of points that fall inside the circle}}{\textrm{total number of points}} = \frac{\textrm{circle area}}{\textrm{square area}} =$$ $$= \frac{\pi R^2}{(2R)^2} = \frac{\pi}{4}$$ A point falls inside the circle if it complies to the following condition: $$x^2 + y^2 \leqslant R^2$$ By counting the number of random points that fall within the circle we can estimate $$\pi$$ as: $$\pi = 4 \times \frac{\textrm{number of points that fall inside the circle}}{\textrm{total number of points}}$$ ## The algorithm: We can thus devise an algorithm that will: 1. Initialize a counter of the points that fall inside the circle (circle_points). 2. Generate a random point $$(x, y)$$. 3. Check if the point falls within the circle by evaluating $$x^2 + y^2 \leqslant R^2$$. 4. If the abose condition is true then increment the counter circle_points. 5. Calculate $$\pi$$ . 6. Repeat from step 2 for the desired number of runs. In principle, the higher the number of runs the better the estimation will be (closer to the real value of $$\pi$$). ## The code Putting the above algorithm to pratice we get the following code2: import random def run_simulation(runs=1000): simulation = [] circle_points = 0 for i in range(runs): # Create a (x, y) point at random x, y = random.uniform(-1, 1), random.uniform(-1, 1) # Check if the point falls within the circle if x2 + y2 <= 1: circle_points += 1 # Calculate current pi value pi_estimate = 4 * circle_points/(i+1) # Save simulation step simulation.append([x, y, pi_estimate]) # Print final pi value print('Result: Pi =', simulation[-1][2]) return simulation sim = run_simulation(runs=10000) Here is the result of the simulation from 1000 runs to 100000 runs. At 100000 runs the estimation ($$\pi = 3.142720$$) is already very close the actual value ($$\pi = 3.141593$$). And if you are curious, here is how the estimation approaches the real value from the first run to 10000th run. A final remark: We do not need to perform the simulation above using a full circle. By selecting a quarter of a circle and the corresponding square we could achieve the same result with a fourth of the computational effort.	2022-11-30T19:42:37	{ "domain": "joseaniceto.com", "url": "https://ipso-facto.joseaniceto.com/pi-monte-carlo.html", "openwebmath_score": 0.7709189057350159, "openwebmath_perplexity": 502.95441580252526, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9924227580598142, "lm_q2_score": 0.8705972600147106, "lm_q1q2_score": 0.8640005339431163 }
https://tex.stackexchange.com/questions/264425/reducing-space-between-for-some-of-the-equations	# Reducing space between for some of the equations I need to display a few equations as below The code I have written is :- \documentclass{report} \usepackage{amsmath} \begin{document} $$\label{Equ. 3.9} x_{i+1}=\cos(a_i) \cdot [x_i-y_i \cdot 2^{-i} \cdot d_i] \tag{3.9}$$ $$\label{Equ. 3.10} y_{i+1}=\cos(a_i) \cdot [y_i+x_i \cdot 2^{-i} \cdot d_i] \tag{3.10}$$ $$\label{Equ. 3.11} \cos(\alpha)=\dfrac{1}{\sqrt [2]{1+{\tan(\alpha)}^2}} \tag{3.11}$$ $$\label{Equ. 3.12} K_i=\cos(arctan(2^{-i}))= \dfrac{1}{\sqrt [2]{1+\tan(arctan(2^{- i}))}}=\dfrac{1}{\sqrt[2]{1+2^{-2i}}} \tag{3.12}$$ The product of $K_i$ represents the so-called K factor (Equ. \ref{equ: 3.13}) $$\label{equ: 3.13} K=\prod K_i=\prod_{i=0}^{n-1} \dfrac{1}{\sqrt{1+2^{-2i}}} \tag{3.13}$$ \end{document} The result I am getting with the above code is as below I need to reduce the space between the equations so the result looks like the 1st screenshot. Please guide how I can achieve this? The equation that caused the numbering problem is: $$\label{equ:matrix} V= \begin{bmatrix} x'\\ y'\\ \end{bmatrix} \begin{bmatrix} x \cdot \cos(a) - y \cdot \sin(a)\\ y \cdot \cos(a) + x \cdot \sin(a)\\ \end{bmatrix} \tag{3.6}$$ When i remove the tag command from this it throws an error • The equations displayed are also aligned by =, so the align environment would be best. You can still label and tag each equation individually, – John Kormylo Sep 2 '15 at 19:05 • Please don't post a substantially new query as an addendum to an existing posting. For one, relatively few people may realize that you've modified an existing query to ask a follow-up question. Instead, please post a new query. – Mico Sep 4 '15 at 16:20 • @Mico Sure... Sorry about that.. I am on it – Shray Sharan Sep 4 '15 at 16:22 Use the gather environment and the \intertext command. Btw, you don't have to put the equation numbers by yourself. Also, I propose to load the cleveref package: in cross references, you won't even have to type ‘equ.’, since cleveref knows (most) counters, and adds the counter name before its value. \documentclass{report} \usepackage{mathtools} \begin{document} \setcounter{chapter}{3}\setcounter{equation}{8} \begin{gather} \label{Equ. 3.9} x_{i+1}=\cos(a_i) \cdot [x_i-y_i \cdot 2^{-i} \cdot d_i] \\ \label{Equ. 3.10} y_{i+1}=\cos(a_i) \cdot [y_i+x_i \cdot 2^{-i} \cdot d_i] \\ \label{Equ. 3.11} \cos(\alpha)=\dfrac{1}{\sqrt [2]{1+{\tan(\alpha)}^2}}\\ \label{Equ. 3.12} K_i=\cos(\arctan(2^{-i}))= \dfrac{1}{\sqrt [2]{1+\tan(\arctan(2^{- i}))}}=\dfrac{1}{\sqrt[2]{1+2^{-2i}}}\\ \intertext{The product of $K_i$ represents the so-called K factor (equ. \eqref{equ: 3.13})} \label{equ: 3.13} K=\prod K_i=\prod_{i=0}^{n-1} \dfrac{1}{\sqrt{1+2^{-2i}}} \end{gather} \end{document} • I used the tag command in one of the equations in between and from their it has stopped taking care of the numbering... Can i post the code for that equation in this post for you check it? – Shray Sharan Sep 4 '15 at 15:27 • Please do, since it is linked to this question. – Bernard Sep 4 '15 at 15:29 • I added the code for that equation.. – Shray Sharan Sep 4 '15 at 15:32 • I've inserted your equation code amidst the other equations and the result is normal, except number 3.6 between 3.12 and 3.13 looks weird. Could you post a complete code showing the problem (perhaps you should initiate a new thread)? – Bernard Sep 4 '15 at 16:07 Especially since you're already loading the amsmath package, you should look into using that package's gather environment to typeset a collection of displayed, numbered equations. Use \intertext{...} to intersperse text between some of the equations. Some side remarks: For the sake of good (math) typography, consider getting rid of all \cdot directives and getting rid of the 2 radicands for the square root ops. \documentclass{report} \usepackage{amsmath} \allowdisplaybreaks \begin{document} \begin{gather} x_{i+1}=\cos(a_i) \cdot [x_i-y_i \cdot 2^{-i} \cdot d_i] \tag{3.9} \label{Equ. 3.9}\\ y_{i+1}=\cos(a_i) \cdot [y_i+x_i \cdot 2^{-i} \cdot d_i] \tag{3.10} \label{Equ. 3.10}\\ \cos(\alpha)=\dfrac{1}{\sqrt [2]{1+{\tan(\alpha)}^2}} \tag{3.11} \label{Equ. 3.11} \\ K_i=\cos(\arctan(2^{-i}))= \dfrac{1}{\sqrt [2]{1+\tan(\arctan(2^{- i}))}}=\dfrac{1}{\sqrt[2]{1+2^{-2i}}} \tag{3.12} \label{Equ. 3.12}\\ \intertext{The product of $K_i$ represents the so-called $K$ factor (Equ. \ref{equ: 3.13})} K=\prod K_i=\prod_{i=0}^{n-1} \dfrac{1}{\sqrt{1+2^{-2i}}} \tag{3.13} \label{equ: 3.13} \end{gather} \end{document}	2021-05-16T08:12:32	{ "domain": "stackexchange.com", "url": "https://tex.stackexchange.com/questions/264425/reducing-space-between-for-some-of-the-equations", "openwebmath_score": 0.9909743070602417, "openwebmath_perplexity": 4512.701767026933, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9579122696813394, "lm_q2_score": 0.9019206804839998, "lm_q1q2_score": 0.8639608861149662 }