Prove that 2/√7 is an irrational number.

Question

Prove that \(\frac{2}{\sqrt7}\) is an irrational number.

Answer 1

Let’s assume on the contrary that \(\frac{2}{\sqrt7}\) is a rational number. Then, there exist co-prime positive integers a and b such that 

\(\frac{2}{\sqrt7}\) = \(\frac{a}{b}\)

⇒ √7 =\(\frac{2b}{a}\)

⇒ √7 is rational [∵ 2, a and b are integers ∴ \(\frac{2b}{a}\) is a rational number] 

This contradicts the fact that √7 is irrational. So, our assumption is incorrect. 

Hence, \(\frac{2}{\sqrt7}\) is an irrational number.