Prove that 1√5 is an irrational number.

Let $\frac{1}{\sqrt{5}}$ be a rational number. $\therefore \frac{1}{\sqrt{5}} = \frac{p}{q}$, where $q \neq 0$ and $p, q$ are co-primes. $5p^2 = q^2 \implies q^2$ is divisible by $5 \implies q$ is divisible by $5$. Let $q = 5a$. $25a^2 = 5p^2 \implies p^2 = 5a^2 \implies p^2$ is divisible by $5 \implies p$ is divisible by $5$. This leads to contradiction as $p, q$ are co-primes. $\therefore \frac{1}{\sqrt{5}}$ is an irrational number.