Let assume that is rational

Therefore it can be expressed in the form of , where p and q are integers and q≠0

Therefore we can write √5 = - √3

(√5)² = ( - √3)²

5 = - +3

5 - 3 = -

- 2 =

= √3

is a rational number as p and q are integers. This contradicts the fact that √3 is irrational, so our assumption is incorrect. Therefore is irrational.