Standard Index Form – Numbers Greater than 1

What is Standard Index Form?

Standard index form or Standard form is a way of expressing numbers as a product of two terms: The first term is a number between 1 and 10 and the second part is a power of 10.

In general, it is expressed as \( \scriptsize \; a \; \times \; 10^n \!\!\;\)where a is a number between 1 and 10 and n is an integer which can be positive, negative or zero.

Numbers Greater Than 1:

Every number has a decimal point usually at the back of the last digit of the number. E.g. 15, the decimal point is after 5. Also, 706 the decimal point is after 6.

Worked Example 7.1.1:

Express these numbers in standard form: 

a. 25 000 000
b. 5 000
c. 7
d. 270 000
e. 15

Solution

a. 25 000 000

Step 1: Move the decimal point to the back of the first significant digit 

Step 2: Count the number of movements made from the initial position of the decimal point.

=   2.5 × 10⁷

b. 5 000

= 5 × 10³

c. 7

There is 0 movement

therefore the power of 10, in this case, is 0.

= 7 × 10⁰

d. 270 000

=  2.7 × 10⁵

e. 15

1.5 × 10¹

Worked Example 7.1.2:

Express the following in standard form. 

a. 18.902
b. 40 830
c. 5123.59
d. 193.25

Solution

a. 18.902

=  1.8902 × 10¹

b. 40 830

=  4.0830 × 10⁴

c. 5123.59

= 5.12359 × 10³

d. 193.25

= 1.9325 × 10²

Worked Example 7.1.3:

Change these standard forms to ordinary numbers.

a. 6 × 10⁵
b. 2.3 × 10⁴
c. 1.05 × 10⁶
d. 7003 × 10⁴

Solution

a. 6 × 10⁵ 

= 6 × 10 × 10 × 10 × 10 × 10

= 6 × 100 000 

= 600 000

b. 2.3 × 10⁴ 

= 23 000 

c. 1.05 × 10⁶   

= 1 050 000

d. 7003 × 10⁴

= 70 030 000