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Division of Rational Numbers

We know that division is the opposite of multiplication. To divide a rational number by another
rational number, we multiply the dividend by the multiplicative inverse or reciprocal of the divisor.

In general,
p r p r p s
if is a rational number, and is a non-zero rational number, then ÷ = × .
q s q s q r
Reciprocal

Example 18: Divide the following:
−3 3 −9 −2
(a) by (b) by
15 5 24 6

1
−3 3 −3 5 − ( ) × 1 −1 3 5
Solution: (a) ÷ = × = = (Q Reciprocal of is )
15 5 15 3 31 3 5 3
×
9
−9 − ( ) 2 − ( ) 9 6 − ( ) × 1 −9 9 1 − ( ) 2 6
(b) ÷ = × = = = = 1 (Q Reciprocal of is )
24 6 24 − ( ) 2 4 ×− ( ) 2 −8 8 8 6 − ( 2)
Properties of Division of Rational Numbers

Closure Property

Observe the following divisions.
5 3 5 4 20
÷ = × = , a rational number
7 4 7 3 21
−3 3 −3 4 −4
÷ = × = , a rational number
7 4 7 3 7
5
But ÷= ? , we know that division by 0 is not defined.
0
9
Therefore, the division of rational numbers is not closed when it involves 0 as a divisor. Hence,
rational numbers are not closed under division.
All non-zero rational numbers are closed under division.
Commutative Property

Let us verify the commutative property of division for rational numbers.
3
5
5 ÷ 3 = 5 × 9 = 5 , and ÷ 5 = 3 × 9 = 3 . Since, 5 ≠ 3 , therefore ÷ 3 ≠ 3 ÷ 5
9 9 9 3 3 9 9 9 5 5 3 5 9 9 9 9
This shows that division is not commutative for rational numbers. In other words, the commutative
property does not hold for the division of rational numbers.
Associative Property

−5 3 7
Let us take any three rational numbers, , and .
7 4 9
−5  3 7  −5  3  9 −5 27 −5 28 −20
We have, ÷  ÷  = ÷  ×  = ÷ = × =
7  4  9 7  4 7  7 28 7 27 27

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