## Introduction

This system is based upon the place value and face value of a digit in a number. We have learnt that a natural number can be written as the sum of the place values of all digits of the numbers. For example

3256 = 3 x 1000 + 2 x 100 + 5 x 10 + 6 x 1
Such a form of a natural number is known as its expanded form.

The expanded form of a number can also be expressed in terms of powers of 10 by using

$${10}^{0}$$ = 1, $${10}^{1}$$ = 10, $${10}^{2}$$ = 100, $${10}^{3}$$ = 1000 etc.
For example,

3256 = 3 x 1000 + 2 x 100 + 5 x 10 + 6 x 1
=>    3256 = 3 x $${10}^{3}$$ + 2 x $${10}^{2}$$ + 5 x $${10}^{1}$$ + 6 x $${10}^{0}$$

Clearly, each digit of the natural number is multiplied by $${10}^{n}$$, where n is the number of digits to its right and then they are added.

## Illustrative Examples

Example 1:    Write the following numbers in the expanded exponential forms:

(i) 32005    (ii) 56719    (iii) 8605192    (iv) 2500132
Solution.

(i) 32005 = 3 x $${10}^{4}$$ + 2 x $${10}^{3}$$ + 0 x $${10}^{2}$$ + 0 x $${10}^{1}$$ + 5 x $${10}^{0}$$

(ii) 560719 = 5 x $${10}^{5}$$ + 6 x $${10}^{4}$$ + 0 x $${10}^{3}$$ + 7 x $${10}^{2}$$ + 1 x $${10}^{1}$$ + 9 x $${10}^{0}$$

(iii) 8605192 = 8 x $${10}^{6}$$ + 6 x $${10}^{5}$$ + 0 x $${10}^{4}$$ + 5 x $${10}^{3}$$ + 1 x $${10}^{2}$$ + 9 x $${10}^{1}$$ + 2 x $${10}^{0}$$

(iv) 2500132 = 2 x $${10}^{6}$$ + 5 x $${10}^{5}$$ + 0 x $${10}^{4}$$ + 0 x $${10}^{3}$$ + 1 x $${10}^{2}$$ + 3 x $${10}^{1}$$ + 2 x $${10}^{0}$$

Example 2:    Find the number from each of the following expanded forms:

(i) 5 x $${10}^{4}$$ + 4 x $${10}^{3}$$ + 2 x $${10}^{2}$$ + 3 x $${10}^{1}$$ + 5 x $${10}^{0}$$

(ii) 7 x $${10}^{5}$$ + 6 x $${10}^{4}$$ + 0 x $${10}^{3}$$ + 9 x $${10}^{0}$$

(iii) 9 x $${10}^{5}$$ + 4 x $${10}^{2}$$ + 1 x $${10}^{1}$$

Solution.

(i) 5 x $${10}^{4}$$ + 4 x $${10}^{3}$$ + 2 x $${10}^{2}$$ + 3 x $${10}^{1}$$ + 5 x $${10}^{0}$$

= 5 x 10000 + 4 x 1000 + 2 x 100 + 3 x 10 + 5 x 1

= 50000 + 4000 + 200 + 30 + 5

= 54235

(ii) 7 x $${10}^{5}$$ + 6 x $${10}^{4}$$ + 0 x $${10}^{3}$$ + 9 x $${10}^{0}$$

=  7 x 100000 + 6 x 10000 + 0 + 9 x 1

= 700000 + 60000 + 9

= 760009

(iii) 9 x $${10}^{5}$$ + 4 x $${10}^{2}$$ + 1 x $${10}^{1}$$

= 9 x 100000 + 4 x 100 + 1 x 10

= 900000 + 400 + 10

= 900410