Question

Show that 3 − √5 is an irrational number.

Answer 1

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

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

3 - √5 = \(\frac{a}{b}\)

⇒ √5 = \(\frac{a}{b}\) + 3 

⇒ √5 = \(\frac{(a + 3b)}{b}\)

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

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

Hence, 3 - √5 is an irrational number.