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Commutative Property

Let us multiply a few pairs of rational numbers in different orders and check the results.

5 4 20 4 5 20 5 4 4 5
(a) × = or × = Clearly, × = × .
7 5 35 5 7 35 7 5 5 7

(b) −3 × 2 = −6 or 2 ×   − 3  = − 6 Clearly, −3 × 2 = 2 ×   −  3  .
8 5 40 5  8  40 8 5 5  8 
−3  −  5 15 −5  −  3 15 −3  −  5 −5  −  3
(c) ×   = or ×   = Clearly, ×   = ×   .
7  8  56 8  7  56 7  8  8  7 
We observe that two rational numbers can be multiplied in any order. This shows that multiplication
is commutative for rational numbers. In other words, the commutative property holds for the
multiplication of rational numbers.

In general,
p r p r r p
if and are two rational numbers, then × = × .
q s q s s q

Associative Property

−3 2 3
Let us take three rational numbers , and , and multiply them in group.
5 3 4

−3  2  3 −3  6  −3 6 −18 −3
We have, ×  ×  = ×   = × = =
5  3  4 5  12  5 12 60 10

 −3  2 3  −  6 3 −6 3 −18 −3
Or,   5 ×   3 × 4 =   × = 15 × 4 = 60 = 10

 15
4
−3  2  3  −3  2 3
Clearly, ×  ×  =  ×  ×
5  3  4  5  3 4
We observe that the rational numbers are grouped in any order when multiplied does not affect
the product. Hence, multiplication is associative for rational numbers. In other words, the
associative property holds for the multiplication of rational numbers.

In general,
p r u p  r u  p r  u
if , and are three rational numbers, then × × = × × .


q s v q   s v  q s v

Existence of Multiplicative Identity

When we multiply any rational number with 1, we get the same rational number.
3 −3
Let us consider two rational numbers and .
4 4
3 3 3 3 3 3 3
We have, ×= Also, 1 × = Thus, ×= 1 × =
1
1
4 4 4 4 4 4 4
−3 −3 −3
1
1
Similarly, ×= × =
4 4 4
Thus, ‘1’ is also the multiplicative identity of rational numbers.
23 Rational Numbers