CONVERT BETWEEN STANDARD AND SCIENTIFIC NOTATION

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. 

Standard Form to Scientific Notation 

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 to 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. 

So,

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. 

So,

0.000023 = 2.3 x 10-5

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 to Standard Form

To convert a number from scientific notation to standard notation, first we have to notice the exponent of 10 in scientific notation.

If the exponent of 10 is positive, we have to move the decimal point to the right. 

For example, if you have 103, you have to move the decimal point 3 digits to the right. 

If the exponent of 10 is negative, we have to move the decimal point to the right. 

For example, if you have 10-5, you have to move the decimal point 5 digits to the left. 

Solved Examples

Examples 1-3 : Write the given number in standard form.

Example 1 :

5.236 x 105

Solution :

Here, the exponent of 10 is positive 5. So we have to move the decimal point five digits to the right.

In 5.236, we have only three digits after the decimal point.

So, we have to add two zeros to move the decimal point five digits to the right.   

Therefore, the standard form of 5.236 x 105 is 

523600

Example 2 :

6.415 x 10-6

Solution :

Here, the exponent of 10 is negative 6. So we have to move the decimal point six digits to the left.

In 6.415, we have only 1 digit before the decimal point.

So, we have to add five zeros to move the decimal point six digits to the left.   

Therefore, the standard form of 6.415 x 10-6 is 

0.000006415

Example 3 :

2.0 x 107

Solution :

Here, the exponent of 10 is positive 7. So we have to move the decimal point seven digits to the right.

In 2.0, we have one zero to the right of the decimal point.

So, we have to add six more zeros to move the decimal point seven digits to the right.   

Therefore, the standard form of 2.0 x 107 is 

20000000

Examples 4-6 : Write the given number in scientific notation.

Example 4 :

4580245

Solution :

Here, we don't find decimal point in 4580245. So we have to assume that there is decimal point at the end . 

Then,

4580245 ---------> 4580245.

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

We have to move the decimal point to the left. 

There are six digits between the 1st non zero digit and the decimal point.

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

Therefore, the scientific notation of 4580245 is

4.4580245 x 106

Example 5 :

325.8645

Solution :

325.8645

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

We have to move the decimal point to the left. 

There are two digits between the 1st non zero digit and the decimal point.

So, the decimal point has to be moved two digits to the left and exponent of 10 should be 2 (positive integer).

Therefore, the scientific notation of 325.8645 is

3.258645 x 102

Example 6 :

0.0008743

Solution :

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

We have to move the decimal point to the right. 

There are for digits from the decimal point up to the first 1st non zero digit.

So, the decimal point has to be moved four digits to the right and exponent of 10 should be -4 (negative integer).

Therefore, the scientific notation of 0.0008743 is

8.743 x 10-4

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