Question

Show that 2 − √3 is an irrational number.

Answer 1

Let’s assume on the contrary that 2 – √3 is a rational number. Then, there exist co prime positive integers a and b such that 

2 – √3= \(\frac{a}{b}\)

⇒ √3 = \(2-\frac{a}{b}\)

⇒ √3 = \(\frac{(2b – a)}{b}\)

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

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

Hence, 2 – √3 is an irrational number.