Prove that $\sqrt{5}$ is irrational number.

Let $\sqrt{5}$ be rational.

Thus, $\sqrt{5}$ can be expressed in the form of $\dfrac{p}{q}$.

$\Rightarrow \sqrt{5} = \dfrac{p}{q} \\[1em] \Rightarrow \sqrt{5}q = p \\[1em] \text{Squaring both sides, we get : } \\[1em] \Rightarrow (\sqrt{5}q)^2 = p^2 \\[1em] \Rightarrow 5q^2 = p^2 \text{ …….(1)}$

As 5 divides 5q², so 5 divides p² but 5 is prime,

Thus, 5 divides p.

Let p = 5m for some positive integer m.

Then, p = 5m

Substituting this value of p in (1), we get :

$\Rightarrow 5q^2 = (5m)^2 \\[1em] \Rightarrow 5q^2 = 25m^2 \\[1em] \Rightarrow q^2 = 5m^2$

As 5 divides 5m², so 5 divides q² but 5 is prime.

Thus, 5 divides q.

This shows that 5 is a common factor of p and q. This contradicts the hypothesis that p and q have no common factor, other than 1.

$\therefore \sqrt{5}$ is not a rational number.

Hence, proved that $\sqrt{5}$ is a irrational number.