Maths

14 MATHEMATICS
Example 10 : Show that 5–
3 is irrational.
Solution : Let us assume, to the contrary, that 5–
3 is rational.
That is, we can find coprime a and b (b 0) such that 5 3
a
Therefore, 5 ✁ ✂ ✄ 3
b
5
Rearranging this equation, we get 3 5 – a b a ✁
✂ ✂ ✄
b b
Since a and b are integers, we get 5– a is rational, and so 3
b
But this contradicts the fact that 3 is irrational.
a
b
✁ ✂ ✄
is rational.
This contradiction has arisen because of our incorrect assumption that 5 –
rational.
3 is
So, we conclude that 5 ☎ 3 is irrational.
Example 11 : Show that 3 2 is irrational.
Solution : Let us assume, to the contrary, that 3 2 is rational.
That is, we can find coprime a and b (b 0) such that 3 2
a
Rearranging, we get ✂ 2 ✄
3b
a
Since 3, a and b are integers, 3 b
But this contradicts the fact that
So, we conclude that 3 2 is irrational.
1. Prove that 5 is irrational.
2. Prove that 3 2 ✆ 5 is irrational.
is rational, and so 2
2 is irrational.
EXERCISE 1.3
3. Prove that the following are irrationals :
(i)
1
2
(ii) 7 5 (iii) 6 2 ✆
✂
a
b
is rational.
✄