Question

Prove the following are irrational.
$\sqrt{5}$

Answer 1

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

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

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

$5q^2=p^2$ (1)

This shows that $p^2$ is divisible by 5, then $p$ is divisible by 5, then for any positive integer $c$, it can be said that $p=5c$, $p^2=25c^2$.

Then equation (1) can be written as,

$5q^2=25c^2$

$q^2=5c^2$

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.