Example-8. Show that 5 - √3 is irrational.

Solution : Let us assume that 5 – √3 is rational.

That is, we can find coprimes a and b (b ≠ 0) such that 5 – √3 = a/b.

Therefore, 5 – a/b= √3

we get √3 = 5 -a/b,

Since a and b are integers, 5 -a/b, is rational, and √3 is also rational.

But this contradicts the fact that √3 is irrational.

This contradiction has arisen because of our assumption that 5 – √3 is rational.

So, we conclude that 5 – √3 is irrational.

Example-9.Show that 3√2 is irrational.

Solution :Let us assume, the contrary, that 3 √2 is rational.

i.e., we can find co-primes a and b (b ¹ 0) such that 3 √2 =a/b.

we get √2 = a/3b.

Since 3, a and b are integers, a/3b is rational, and so √2 is rational.

But this contradicts the fact that √2 is irrational.

So, we conclude that 3√ 2 is irrational.

Example-10.Prove that √2 + √3 is irrational.

Solution : Let us suppose that √2 + √3 is rational.

Let √2 + √3 = a/b , where a, b are integers and b ≠ 0

Therefore, √2 = a/b - √3