Question

Prove that 2 − 3√5 is an irrational number.

Answer 1

Let’s assume on the contrary that 2 – 3√5 is a rational number. 

Then, there exist co prime positive integers a and b such that 

2 – 3√5 = \(\frac{a}{b}\)

⇒ 3√5 = 2 – \(\frac{a}{b}\)

⇒ √5 = \(\frac{(2b – a)}{(3b)}\)

⇒ √5 is rational [∵ 3, a and b are integers ∴ \(\frac{(2b – a)}{(3b)}\) is a rational number] 

This contradicts the fact that √5 is irrational. So, our assumption is incorrect. 

Hence, 2 – 3√5 is an irrational number.