Question

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

Answer 1

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

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

5 – 2√3 = \(\frac{a}{b}\) 

⇒ 2√3 = 5 – \(\frac{a}{b}\) 

⇒ √2 = \(\frac{(5b – a)}{(2b)}\) 

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

This contradicts the fact that √2 is irrational. So, our assumption is incorrect. 

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