Page 12 - SM inner class 8.cdr

Example 14. Verify (a + ) b ´ c -6 3 -6 3
(ii) by = ¸
c
c
= (a ´ ) + (b ´ ) 25 5 25 5
6
-1 2 3 -6 5 {( - ) ´ }
5
If a = , b = and c = . = ´ =
8 9 5 25 3 25 ´ 3
æ -1 2 ö 3 -30
Solution: L.H.S = (a + ) ´ c = ç + ÷ ´ =
b
è 8 9 ø 5 75
9
æ - + 16 ö 3 -2
= ç ÷ ´ =
è 72 ø 5 5
7 3 7 ´ 3 21 7
= ´ = = = 1. Closure property
72 5 72 ´ 5 36 120
The collection of non-zero rational numbers is
æ 1 3 ö æ2 3 ö
c
R.H.S = (a ´ ) + (b ´ ) = - ´ ÷ + ç ´ ÷ closed under division.
c
ç
è 8 5 ø è9 5 ø
a c
- ´ ö
3
æ 1 3 æ2 ´ ö If and are two rational numbers, such that
= ç ÷ + ç ÷ b d
5
è 8 ´ 5 ø è9 ´ ø c a c
¹ 0, then ¸ is always a rational number.
- ö
æ 3 æ 2 ö d b d
= ç ÷ + ç ÷
è 40 ø è15 ø For example:
9
- + 16 7 2 1 2 3
= = (i) ¸ = ´ = 2 is a rational number.
120 120 3 3 3 1
\ LHS = RHS 4 3 4 2 8
(ii) ¸ = ´ = is a rational number.
Thus, (a + ) b ´ = (a ´ ) c + (b ´ ) c 5 2 5 3 15
c
Division of Rational Numbers 2. Commutative property
If m and n two rational numbers such that n ¹ Division of rational numbers is not
0, then the result of dividing m by n is the com mu ta tive.
rational number obtained on multiplying m by a c
If and are any two rational number, then
the reciprocal of n. b d
a c c a
When m is divided by n, we write m ÷ n. Thus ¸ ¹ ¸
1 b d d b
m ÷ n = m × .
n For example:
w y 4 3
If and are two rational numbers such that For two rational numbers and , we have
x z 5 8
y 4 3 3 4
¹ 0, then ¸ ¹ ¸
z 5 8 8 5
-1 4 8 32 3 5 15
w y w æ ö y w z LHS = ´ = RHS = ´ =
÷ = ´ ç ÷ = ´ 5 3 15 8 4 32
x z x è ø x y
z
\ LHS ¹ RHS
Example 15. Divide:
\ Commutative property is not true for division.
9 5
(i) by
16 8 3. Associative property
8
9 5 9 8 (9 ´ )
Solution: ¸ = ´ = Division of rational numbers is not associative.
5
16 8 16 5 (16 ´ ) a c e
72 If , and are any three rational numbers,
= b d f
80 a æ c eö æ a c ö e
9 then ¸ ç ¸ ÷ ¹ ç ¸ ÷ ¸ .
= b è d f ø è b d ø f
10
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Mathematics In Focus - 8