Question

Prove that 2√3 − 1 is an irrational number.

Answer 1

Let’s assume on the contrary that 2√3 – 1 is a rational number. 

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

2√3 – 1 = \(\frac{a}{b}\) 

⇒ 2√3 = \(\frac{a}{b}\) – 1 

⇒ √3 = \(\frac{(a – b)}{(2b)}\)

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

This contradicts the fact that √3 is irrational. So, our assumption is incorrect.

Hence, 2√3 – 1 is an irrational number.