Page 23 - ICSE Math 8

IV. Existence of Multiplicative Identity: Role of ‘One’
p
“A rational number multiplied by 1 is the rational number itself.” Thus, if q is any rational number, then
p p p
q ×=1 q =×1 q , where ‘1’ is the multiplicative identity of the rational numbers.
3 3 −4  4
− 
11
Example 21: Find if: (a) ×= (b) ×= ×  
11×
7 7 5  5 
3 3 3 −4 −4  4
− 
1
1
1
Solution: (a) ×= =× (b) ×= =×  
1
7 7 7 5 5  5 
V. Existence of Multiplicative Inverse (or Reciprocal) of a non-zero rational number
p
For every non-zero rational number q , where q ≠ 0, there exists its Maths Info
p p q
multiplicative inverse such that × = 1.
q q p • Zero has no reciprocal.
q p p • Reciprocal of ‘1’ is ‘1’ and
If p is the multiplicative inverse of q , then q is also the multiplicative reciprocal of (–1) is (–1).
inverse of q .
p
−13
Example 22: Find the multiplicative inverse of: (a) (b) –6 Try This
19

Solution: (a) The multiplicative inverse of −13 = −19 . Find the multiplicative inverse
19 13 of:
−1 (a) –8 × –7 (b) –4 ÷ –5
(b) The multiplicative inverse of −=6 . 5 3 –5 2
6
p
VI. Multiplication by Zero: “The product of any rational number with ‘0’ is always ‘0’.” Thus, if q is a
p
p
rational number, then q ×= =×0 0 0 q .
Example 23: Find: (a) −3 × 0 (b) 4 × 0 (c) −9 × 0
5 7 −11
−3 4 −9
0
Solution: (a) ×= 0 (b) ×= 0 (c) ×= 0
0
0
5 7 −11
VII. Distributive Property of Multiplication Over Addition: “Multiplication of rational numbers is distributive
p r u
over addition.” Thus, if , and are any three rational numbers, then
q s v
p r  u   p r   p u 
q ×   s + v  =  q × s   +   q × v   .


The multiplication of rational numbers is also distributive over subtraction.
p r u p r  u   p r   p u 
If q s v are three rational numbers, then q ×   s − v   =   q × s   −  q × v   .
,,

3  − 1 5   3  − 1  3 5 
Example 24: Verify ×  +  =  ×   +  ×  .
4  3 6   4  3   4 6 

1
()
15
3  − 1 5  3 { 2×− +× } 2 3, 6
Solution: LHS = ×  +  = ×
4  3 6  4 6 3 3, 3
1, 1
3  −+ 3 3 9 3
25 
= ×   = × = = LCM = 6
4  6  4 6 24 8
11