In the early 1970s scientific calculators, often referred to as "slide rule calculators", became commonplace. Most of these calculators had the ability to display from eight to ten digits, depending on the manufacturer. Scientific calculations, however, often involve numbers that contain more than eight or ten digits. To overcome the limitation of an eight-or-ten-digit display, slide rule calculators depend on scientific notation. When a number becomes too large for the calculator to display, scientific notation is used automatically. Scientific Notations are used when numbers are too large or too small in the concise form.

Scientific notation is a way of writing numbers that look like this.
A number with just one digit to the left of the decimal point times 10 to a power. For example, the number 5.280 could be written in scientific notation as
5.280 x 10$^{3}$
the exponent is 3 because 10$^{3}$ = 1.000
1.000 x 5.280 = 5.280

In all cases, scientific notation represents figures in terms of a number that is 1 or greater and less than 10, multiplied by 10, and raised to a power.

Examples on Scientific Notation

1. 595 = 5.95 x 10$^{2}$
2. 88.500 = 8.85 x 10$^{4}$
3. 0.1590 = 1.590 x 10$^{-1}$

The exponent is negative since the decimal will have to go to the left to get it back to its original location.
0.000039 = 3.9 x 10$^{-4}$

What is Scientific Notation?

Scientific notations are used to denote very large or very small numbers in a concise form. The numbers written in scientific notation are of the form.

a × 10m, where 1 $\leq$ a < 10 and m is an integer.

Usually, a negative power is avoided except for scientific notations.

Scientific Notation Rules

Any number can be converted to a scientific notation by means of one of two rules.

Rule – 1: If the number is found to be a whole number or a whole number and a decimal, the decimal point is shifted enough number of places to the left and then placed immediately on the right side of the first non-zero digit.

Rule – 2: If the number is found to be a decimal, with digits only to the right hand side of the decimal point, then the decimal point is shifted enough number of places to the right to place it immediately on the right of the first non-zero digit.

Standard Notation

Scientific notation is particularly useful in applications which involve very large or very small mathematical quantities. Standard notation or decimal notation is the form we normally use to express numerical values. If a number is written in scientific notation, it is easy to write the same number in standard notation, because these both forms looks similar to each other.

The exponent on ten tells us the number of places and the direction to move the decimal point from one in order to express the number in decimal notation. If the exponent is positive, the movement is to the right, and if the exponent is negative, the movement is to the left.

In other words, if the exponent on ten is positive, multiply by that number of factors of ten. Similarly, if the exponent on ten is negative, divide that number of factors of ten.

Example for Standard Notation

Given below are examples based on standard notation

1. Write the number 7.8 x 105 in standard form.

Solution
The exponent (5) indicates how many times to move the decimal place.

7.8 x 105 = 780,000

2. Write the number 5.89 x 10^3 in its standard form

Solution
The exponent (3) indicates how many times to move the decimal place.

5.89 x 103 = 5890

Expanded Notation

The number which is represented in decimal notation can also be represented as expanded numeral called expanded notation.

Examples on Expanded Notation

Given below are some examples on expanded notation

1. How is the term 53 written in the expanded form?

Solution

The power (or exponent) of 3 means that the number is multiplied by itself three times.

So the expanded form is

= (5) x (5) x (5) = 125

2. How is the term ft2 written in an expanded form?

Solution

The power of exponent of 2 means that the term is multiplied by itself two times.
So the expanded form is

ft2 = (ft) x (ft)

Factorial Notation

Factorial values of a number have applicability in combinatorial analysis and the construction of probability density functions. The factorial value of a designated number is determined by multiplying that number by each of its preceding whole integers. The factorial function is expressed as n!, where n is the designated number.

The symbol n! is read as n factorial, and is a shorthand way of identifying the product of all positive numbers from 1 up to n.

The following examples illustrate the application of the factorial function. A student of logistics will find the factorial command function on an electronic calculator to be a valuable aid in achieving the learning objectives.

• The symbol n! has no meaning if n represents anything other than a positive whole number or zero. (-3 ! is undefined)
• The value of 0! is defined to be 1. (0! = 1)
• To obtain the factorial value from the calculator, enter the value followed by the x! or n! Key.

Examples on Factorial Notation

Below are some examples based on factorial notation

1. Given that n=5, then find n!.

Solution

n! = 5!
= (5) (4) (3) (2) (1)
= 120

2. Given that n=8, then find n!.

Solution

n! = 8!
= (8) (7) (6) (5) (4) (3) (2) (1)
= 40,320

3. Given that n=10, then find n!.

Solution

n! = 10!
= (10) (9) (8) (7) (6) (5) (4) (3) (2) (1)
= 3,628,800

In each integral increment in the value of the designated whole number n>=0, the corresponding factorial value increases rapidly. Also note that 0! = 1.

The characteristics of factorial is explained clearly in the following tabular column.

Scientific Notation to Standard Notation

To express a number given in scientific notation as a decimal number, shift the decimal point in the reverse direction and attach required zeros. Move the decimal point according to the exponent of 10. With positive exponents the decimal point is moved to the right with negative exponents it is moved to the left.

It is also possible to convert scientific notation to standard notation. For example, while converting 2.3 x 104 to standard notation, write the first factor of 2.3 and then move the decimal four places to the right. Add zeros as placeholders wherever necessary. Thus, the number becomes 23,000 in standard notation.

If the exponent were negative in scientific notation, the decimal would move that number of places to the left instead of the right. Remember that a negative exponent indicates a number less than 1.

Examples for Scientific Notation to Standard Notation

Express the following scientific notation forms to standard forms.

1. 4.3 x 103

2. 8.907 x 105

3. 5.123 x 104

Conversion of a Number into Scientific Notation

1. First we have to move the decimal point so that we get a number between 0 and 10.

2. Then count the number of places to which the decimal point must be moved. This is the value of the power or exponent to which ten must be raised.

3. If the given number is larger than 10, then the exponent is positive. When we write the number in the scientific form.

4. If the given number is smaller than 1, then the exponent or power must be negative when the number is represented in scientific form.

How to do Scientific Notation :

Write 40,000,000,000,000,000 m in scientific form.

The given number is larger than 10, hence the exponent is positive. The number of places to which the decimal point must be shifted is 16. Hence, we have it as

4 × 1016 m

The mass of electron is 0.00000000000000000000000000091kg. This can be written in scientific notation as 9.1 × 10-28 kg

Adding numbers in scientific notation can be done in two different ways.

One way is to change the numbers out of scientific notation and work the problem normally.

Subtracting Scientific Notation

For the subtraction of two numbers written in scientific notation, it is necessary to express both to the same power of 10. After subtracting it may be necessary to shift the decimal point to express the result in scientific notation.

The other way to subtract the given numbers in scientific notation is to change the given number out of scientific notation and work it as a normal subtraction problem.

Examples for Subtracting Scientific Notation

Here and some example based on subtracting scientific notation

Subtract the following :- 5.3 x 103 - 4.62 x 102

Solution

Step – 1: Change the given numbers normally to 5,300 - 462
Step – 2: Then subtract normally to get 4,838
Step – 3: If the problem asks for the answer in scientific notation then change 4,838 into 4.838 x 103

This way of working the problem is easiest when the numbers are not very large or not very small.

Multiplying Scientific Notation

To multiply the numbers in scientific notation,

Step 1: Multiply the decimal parts.
Step 2: Add the exponents while retaining the base 10.
Step 3: Adjust the answer to see that it is still in the scientific notation. If the decimal part is 10 or greater than 10 then we move the decimal point to the left and if the decimal part is less than 1, then move the decimal point to the right so that the decimal part is a number greater than or equal to one and less than 10.

If the exponent is positive, then:
a) If the decimal point is moved to the left, increase the exponent by the number of decimal places moved. For example, 32.34 × 104 = 3.234× 105
b) If the decimal point is moved towards the right, then decrease the exponent by the number of decimal places moved.
For example, 0.03234 × 104 = 3.234 × 102

If the exponent is negative, then:
a) If the decimal point is moved to the right, then increase the exponent by the number of decimal places moved.
For example, 0.3234 × 10-4 = 3.234 × 10-5
b) If the decimal point is moved to the left, then decrease the exponent by the number of decimal places moved.
For example, 32.3 × 10-4 = 3.23 × 10-3

The product of 4 × 104 and 3.2 × 107 is
Step 1: Multiply the coefficients 4 × 3.2 = 12.8
Step 2: Add the exponents while the base remains 10, so we get 104 + 7
The product is 12.8 × 1011 i.e. 1.28 × 1012

Examples on Multiplying Scientific Notation

Given below are some examples based up on multiplying scientific notation

Multiply 9 × 10-3 and 3 × 10-7

Solution:

Step 1: 9 × 3 = 27 Multiply the decimal parts.
Step 2: (-3) + (-7) = -10 Add the exponents.
Step 3: 27 × 10-10 = 2.7 × 10-9 Adjust the answer to scientific notation.
Hence, the product is 2.7 × 10-9

Dividing Scientific Notation

To divide the numbers in scientific notation:

Step 1: Divide the decimal parts.
Step 2: Subtract the exponents while the base remains 10. We have to subtract the second exponent from the first exponent.
Step 3: Adjust the answer so that it is still in the scientific notation. If the decimal part is 10 or greater than 10 then we move the decimal point to the left and if the decimal part is less than 1, then move the decimal point to the right so that the decimal part is a number greater than or equal to one and less than 10.

If the exponent is positive, then

a) If the decimal point is moved to the left, increase the exponent by the number of decimal places moved.
For example, 32.34 × 104 = 3.234 × 105

b) If the decimal point is moved towards the right, then decrease the exponent by the number of decimal places moved.
For example, 0.03234 × 104 = 3.234 × 102

If the exponent is negative, then

a) If the decimal point is moved to the right, then increase the exponent by the number of decimal places moved. For example, 0.3234 × 10-4 = 3.234 × 10-5

b) If the decimal point is moved to the left, then decrease the exponent by the number of decimal places moved.
For example, 32.3 × 10-4 = 3.23 × 10-3

For example: The quotient of 3.2 107 and 1.2 × 103 is

Step 1: Divide the decimal parts $\frac{3.2}{1.2}$ = 2.667
Step 2: Subtract the exponents 107 - 3

Hence, the quotient is 2.667 × 104

Examples on Dividing Scientific Notation

Below you could see example based on dividing scientific notation

Divide 6 × 103 by 2 × 102

Solution:

Step 1: 6 ÷ 2 = 3 Divide the decimal parts

Step 2: 3 - 2 = 1 Subtract the exponents

Step 3: 3 × 101 The quotient is already in the scientific notation.

Hence, the quotient is 3 × 10.

Scientific Notation Examples

Below are some examples on scientific notation

Scientific Notation Problems

1. Add 3.49 × 1010 and 9.8 × 1011

Solution: The given numbers are too large to convert into normal form and then add. So we will change the power of 3.49× 1010 into 0.349 × 1011

Now both the numbers have the same power. Add the numbers while keeping the powers as it is.

0.349 × 1011 + 9.8 × 1011 = 10.149 × 1011

2. Subtract 7.68 × 10-3 from 8 × 10-3

The easiest way to solve this problem is by converting the scientific numbers into normal form and then subtract.

7.68 × 10-3 = 0.00768 and 8 × 10-2 = 0.08

Now subtract. 0.08 - 0.00768 = 0.07232

Now we have to write it in the scientific form = 7.232 × 10-2

3. Multiply 2 × 104 and 7 × 106

Solution:

Step 1: 2 × 7 = 14 Multiply the decimal parts
Step 2: 4 + 6 = 10 Add the exponents
Step 3: 14 × 1010 = 1.4 × 1011 Adjust the answer to scientific notation
Hence, the product is 1.4 × 1011

4. Divide 6.6 × 10-6 by 1.1 × 10 -8

Solution:

Step 1: 6.6 ÷ 1.1 = 6 Divide the decimal parts
Step 2: (-6) - (-8) = 2 Subtract the exponents
Step 3: 6 × 102 The solution is already in the scientific notation.
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