## Understanding scientific notation of large numbers

19 Nov

( Disclaimer: I am saving this here for my future use as a backup. Copy pasted from onlinemath4all.com . link to the original post is ==> https://www.onlinemath4all.com/scientific-notation-rules.html )

Scientific notation is a standard way of writing very large and very small numbers so that they’re easier to both compare and use in computations.

In this section, we will learn the rules which are required to write a number in scientific notation.

## Scientific Notation Rules

Every number in the scientific notation must be in the form of

a x 10n

where ≤ a < 10 and n must be a positive or negative integer.

To convert a number into scientific notation, first we have to identify where the decimal point and non zero digit come.

There are two cases in it.

Case 1 : To move the decimal point to the left, we have to count number of digits as explained in the example given below. According to the example given above, we have to move the decimal point 3 digits to the left and exponent of 10 should be 3 (positive integer)

When we do so, we get the scientific notation of the given number.

Hence, 2301.8 = 2.3018 x 10³

Case 2 : To move the decimal point to the right, we have to count number of digits as explained in the example given below. According to the example given above, we have to move the decimal point 5 digits to the right and exponent of 10 should be -5 (negative integer)

When we do so, we get the scientific notation of the given number.

Hence, 0.000023 = 2.3 x 10

Important Note:

If we don’t find decimal point at anywhere of the given number, we have to assume that there is decimal point at the end of the number.

For example,

2300000 ————-> 2300000.

Here, the non zero digit comes first and decimal point comes next. So we have to apply case 1 to convert this number into scientific notation.

## Scientific Notation Rules – Examples

Example 1 :

Write the given number in scientific notation.

0.00006

Solution :

Here decimal point comes first at non zero digit comes next.

We have to move the decimal point to the right.

No. of digits from the decimal point up to the first non zero digit is 5.

So, the decimal point has to be moved 5 digits to the right and exponent of 10 should be -5 (negative integer)

Hence, the scientific notation of 0.00006 is

6 x 10-5

Example 2 :

Write the given number in scientific notation.

5400000

Solution :

Here we don’t find decimal point in 5400000. So we have to assume that there is decimal point at the end .

Then, 5400000 ———> 5400000.

Here non zero digit comes first and decimal point comes next.

We have to move the decimal point to the left.

No. of digits between the 1st non zero digit and the decimal point is 6.

So, the decimal point has to be moved 6 digits to the left and exponent of 10 should be 6 (positive integer)

5400000  =  5.400000 x 106

5400000  =  5.4 x 106

Hence, the scientific notation of 5400000 is

5.4 x 106

(Here zeros after the decimal point are not taken. Because, they are not valid zeros)

Example 3 :

Write the given number in scientific notation.

71 x 102

Solution :

Here we don’t find decimal point in 71x 102. So we have to assume that there is decimal point at the end of 71

Then,

71 x 102 ———> 71. x 102

Here non zero digit comes first and decimal point comes next.

We have to move the decimal point to the left.

No. of digits between the 1st non zero digit and the decimal point is 1.

So, the decimal point has to be moved 1 digit to the left and exponent of 10 should be 1 (positive integer)

71. x 102 = 7.1 x 101 x 102

71

Hence, the scientific notation of 71 x 102 is

7.1 x 103

Example 4 :

Write the given number in scientific notation.

33 x 10-3

Solution :

Here we don’t find decimal point in 33 x 10-3 So we have to assume that there is decimal point at the end of 33

Then,

33 x 10-3 ———> 33. x 10-3

Here non zero digit comes first and decimal point comes next.

We have to move the decimal point to the left.

No. of digits between the 1st non zero digit and the decimal point is 1.

So, the decimal point has to be moved 1 digit to the left and exponent of 10 should be 1 (positive integer)

33. x 10-3  = 3.3 x 101 x 10-3

33. x 10-3  = 3.3 x 10-2

Hence, the scientific notation of 33 x 10-3 is

3.3 x 10-2

Example 5 :

Write the given number in scientific notation.

0.63 x 10-3

Solution :

Here decimal point comes first and non zero digit comes next.

We have to move the decimal point to the right.

No. of digits from the decimal point up to the first non zero digit is 1.

So, the decimal point has to be moved 1 digit to the right and exponent of 10 should be -1 (negative integer)

0.63 x 10-3  =  6.3 x 10-1 x 10-3

0.63 x 10-3  =  6.3 x 10-4

Hence, the scientific notation of 0.63 x 10-3 is

6.3 x 10-4