Page 11 - SM inner class 8.cdr

For example: Multiplication by 0
3 -8
For two rational number and , we have Every rational number multiplied with 0 gives
5 11 0.
3 æ - 8ö æ 8 3
- ö
´ ç ÷ = ç ÷ ´ a
5 è 11 ø è 11 ø 5 If is any rational number, then
b
3
3 æ 8 -8 æ ö a a
- ö
LHS = ´ ç ÷ RHS = ´ ç ÷ ´ 0 = 0 = 0 ´ .
5 è 11 ø 11 è ø b b
5
-24 -24
= = For example:
55 55 æ 7
- ö
\ LHS = RHS (i) - ´5 0 = 0 (ii) ç ÷ ´ 0 = 0
è 11 ø
Commutative property is true for
multiplication. Multiplicative Inverse or Reciprocal
a
,
3. Associative property For every rational number , a ¹ 0 there exists
b
Multiplication of rational numbers is b a b
a rational number such that ´ =1. Then
associative. a b a
b a
a c e is called the multiplicative inverse of .
If , and are any three rational numbers, a b
b d f
a b
a æ c eö æ a c ö e If is a rational number, then is the
then ´ ç ´ ÷ = ç ´ ÷ ´ b a
b è d f ø è b d ø f
multiplicative inverse or reciprocal of it.
For example:
For example:
1 æ 1 1
- ö
For three rational number , ç ÷ and , we 1
2 è 4 ø 3 (i) The reciprocal of 2 is .
2
have æ 3 æ 5
- ö
- ö
1 æ - 1 1ö æ 1 æ - 1öö 1 (ii) The multiplicative inverse of ç ÷ is ç ÷.
´ ç ´ ÷ = ç ´ ç ÷÷ ´ è 5 ø è 3 ø
2 è 4 3ø è 2 è 4 øø 3
Note: (i) 0 has no reciprocal.
1 æ - ö 1 -1 æ - ö 1 1 -1
LHS = ´ ç ÷ = RHS = ç ÷ ´ =
2 è12 ø 24 è 8 ø 3 24 (ii) 1 and – 1 are the only rational
numbers which are their own reciprocals.
\ LHS = RHS
Example 13. If the product of two numbers is
Multiplicative identity -5 and one of them is -3 . Find the other.
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The product of any rational number and 1 is
the rational number itself. ‘One’ is the Solution: Let the other number be x.
multiplicative identity for rational numbers. According to the question.
a -3 ´ x = -5
If is any rational number, then
b 4 8
a a a x = -5 ¸ -3
1
´1 = = ´
b b b 8 4
-5 -4
For example: = ´
5 5 8 3
(i) ´ 1 = 5
7 7 =
6
- ö
æ 3 -3
(ii) ç ÷ ´ =1 5
è 8 ø 8 Hence, the other number is .
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Mathematics In Focus - 8