Page 8 - SM inner class 8.cdr

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If a/b and c/d are any two rational numbers, and ì 2 - ç æ 3 - ÷ý = ì2 - 6 ( -5) ü
ý
í
í
æ a c ö î 3 è 5 2ø þ î3 10 þ
then ç - ÷ is also a rational number.
è b d ø æ2 1 ö
= ç - ÷
è3 10 ø
For example:
3
(20 - ) 17
3 2 1 = =
- = 30 30
4 3 12
æ ì 2 3ö 1ü ì 2 æ 3 1öü
Which is also a rational number. Therefore, ç í - ÷ - ý ¹ í - ç - ÷ý
î è 3 5ø 2 þ î 3 è 5 2ø þ
2. Commutative property
4. Property of 0
Subtraction is not commutative for rational
numbers. 0 is a rational number If 0 is subtracted from
any rational number leaves the number
Thus for any two rational numbers a/b and a
c/d, we have unchanged , so for every rational number
b
æ a c ö æ c a ö
ç - ÷ ¹ ç - ÷ æ a ö a
è b d ø è d b ø ç - ÷ 0 =
è b ø b
Look at the following example showing
that subtraction of rational numbers is But if a/b is subtracted from 0 we get its
not commutative. additive inverse. For a number a
5 2 1 b
- =
6 3 6 0 - a = - a
2 5 -1 b b
but - =
3 6 6
Solved examples of addition and
æ a c ö æ c a ö
So ç - ÷ ¹ ç - ÷ subtraction of Rational Numbers
è b d ø è d b ø
3 -5
Example 5. Add and .
3. Associative property 13 13
(
3 - 5 [3 + - )] (3 - )
5
5
Subtraction is not associative for rational Solution: + = =
13 13 13 13
numbers. Thus, for any three rational
a c e -2
numbers , and , we have = , [Since, 3 - 5 = -2]
b d f 13
3 - 5 -2
æ a c ö e a æ c eö Therefore, + = .
ç - ÷ - ¹ - ç - ÷ 13 13 13
è b d ø f b è d f ø 5 4
Example 6. Subtract from .
7 5
For example:
5 4 æ4 5 ö
2 3 Solution: Subtract from = ç - ÷
Consider three rational numbers , and 7 5 è 5 7 ø
3 5
5
1 = 4 + (additive inverse of )
Then,
2 5 7
5
æ4 - ö
æ ì 2 3ö 1ü ì10 - 9 1 ü = ç + ÷
ç í - ÷ - ý Þ í - ý è 5 7 ø
î è 3 5ø 2 þ î 15 2 þ
{28 + -25 )}
(
æ 1 1 ö =
= ç - ÷ 35
è15 2 ø 3
=
(2 -15 ) -13
= = 35
30 30
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Mathematics In Focus - 8