Assume that $\sqrt{5}$ be a rational number. So,

$\sqrt{5},=\frac{\mathrm{p}}{\mathrm{q}}$, where p and q are co prime.

$5=\frac{{\mathrm{p}}^2}{{\mathrm{q}}^2}$

5q² =  p²                        (1)

This shows that p² is divisible by 5, then p is divisible by 5, then for any positive integer c, it can be said that p = 5c, p² = 25c².

Then equation (1) can be written as,

5q² = 25c²

q² = 5c²

This gives that q is divisible by 5.

So, p and q has a common factor 5 which is a contradiction to the assumption that they are co prime.

Hence, $\sqrt{5}$ is an irrational number.