Page 6 - SM inner class 8.cdr

c
a
2. Without Representing on Number line: If and are any two rational numbers, then
b d
Without representing the rational number on
æ a c ö
a number line, we can compare them by the ç + ÷ is also a rational number.
è b d ø
method similar to the one used for fractional
numbers. For example:
If two rational numbers have the same 1 3
(i) Consider the rational numbers and
positive denominator, the number with the 3 4
larger numerator will be greater than the one Then,
9
with smaller numerator. æ 1 3ö (4 + ) 13
ç + ÷ = =
2 5 è 3 4 ø 12 12
Example 3. Compare and .
7 7 which is also a rational number.
2 5 -5 -1
Solution: The rational numbers and have (ii) Consider the rational numbers and
7 7 12 4
same denominator, therefore, smaller the æ -5 - ö 1 { - + - )} -8
3
5
(
Then, ç + ÷ = =
numerator, smaller will be the rational è12 4 ø 12 12
2 5
number. Since 2 < 5 therefore < . -2
7 7 = , is a also rational number.
3
If two rational numbers have different 2. Commutative property
denominators then first make denominators
equal and then compare. Two rational numbers can be added in any
7 8 order.
Example 4 Compare and .
5 7 Thus for any two rational numbers a/b and
Solution: First, convert the rational numbers c/d, we have
æ c
to have the same positive denominator, æ a + c ö ÷ = ç + a ö
÷
ç
7 7 ´ 7 49 è b d ø èd b ø
= =
5 5 ´ 7 35 For example:
8 8 ´ 5 40 + )
3
5
= = æ 1 + 3ö (2 =
7 7 ´ 5 35 (i) ç 2 4 ø ÷ = 4 4
è
49 40
2
Now, compare and æ 3 1ö (3 + ) 5
÷ =
35 35 and ç 4 + 2ø 4 = 4
è
49 40
As 49 > 40, therefore > æ 1 3ö æ3 1 ö
35 35 Therefore, ç + ÷ = ç + ÷
7 8 è 2 4 ø è4 2 ø
Hence, >
5 7 5
=
4
Properties of Addition of Rational
æ 3 - 5ö {9 + -20 )} -11
(
Numbers (ii) ç 8 + 6 ø ÷ = 24 = 24
è
9
1. Closure property æ -5 3 ö - { 20 + } -11
and ç + ÷ = =
The sum of two rational numbers is always a è 6 8 ø 24 24
rational number. æ 3 - 5ö æ -5 3 ö
Therefore, ç + ÷ = ç + ÷
è 8 6 ø è 6 8 ø
-11
=
24

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Mathematics In Focus - 8