Page 20 - Start Up Mathematics_8 (Non CCE)

II. Commutative Property
“Two rational numbers can be multiplied in any order.”
p r p r r p
So, if and are two rational numbers, then × = × .
q s q s s q
 8 −   4 − 
−   8
−   4
×
=
×
Example 25: Check if         ? (NCERT)
 9   7   7   9 
−   4
 8 −  ( −× −8) ( 4) 32  −  4  −  8 ( −4) × −8( ) 32
Solution:     = = and   ×   = =
×
 9   7  97 63  7   9  79 63
×
×
−   8
−   4
 8 −   4 − 
=
×
So,        
×
 9   7   7   9 
III. Associative Property
“The three rational numbers can be multiplied in any order.”
p r u  p r  u p r  u 
So, if , and v are three rational numbers, then   q × s   × v = q ×   s × v  .
q s

×
×
Example 26: Check if  2 × − 6  × 4 = 2  − 6 4  ? (NCERT)




 3 7  5 3  7 5 
Solution:  2 × − 6   × 4 = { 2×−( 6) } × 4

 3 7  5 37× 5
−12 4 ( −12) × 4 −48 −16  −48 ( −48) ÷3 −16 
= × = = =   = = 
21 5 21 ×5 105 35  105 105 ÷ 3 35 
2  − 6 4  2 { ( −× 4 }
6)
and ×  ×  = ×
3  7 5  3 75×
2  − 24  2×−( 24) − 48 − 16
= ×   = = =
3  35  335× 105 35
 2 − 6  4 2  − 6 4 
So,  ×  × = ×  × 
 3 7  5 3  7 5 
IV. Existence of Multiplicative Identity: Role of ‘One’
“A rational number multiplied by 1 is the rational number itself.”
p p p p
1
So, if is any rational number, then ×= =× , where ‘1’ is the multiplicative identity of the rational
1
q q q q
numbers.
3 3 −4  4 12 12
− 
11
11×
Example 27: Find if: (a) ×= (b) ×= ×   (c) ×=
11×
7 7 5  5  35 35
3 3 3 −4 −4  4 12 12 12
− 
1
1
1
1
1
1
Solution: (a) ×= =× (b) ×= =×   (c) ×= =×
7 7 7 5 5  5  35 35 35
V. Existence of Multiplicative Inverse (Reciprocal)
“To find the multiplicative inverse (or reciprocal) of a non-zero rational number, interchange the positions of
the numerator and denominator.”
p q
So, if is a non-zero rational number, where q ≠ 0 then there exists a rational number such that
q p
12