Prove that $\frac{1}{\sqrt{5}}$ is irrational.

Given :

The given number is $\frac{1}{\sqrt{5}}$  

To do :

We have to prove that $\frac{1}{\sqrt{5}}$ is irrational.

Solution :

Let us assume $\frac{1}{\sqrt{5}}$ is rational.

Hence, it can be written in the form of $\frac{a}{b}$, where a, b are co-prime, and b is not equal to 0.

$\frac{1}{\sqrt{5}}=\frac{a}{b}$

$\frac{b}{a} = \sqrt{5}$

Here, a and b are integers.

So, $\frac{b}{a}$ is a rational number.

We know that $\sqrt{5}$ is an irrational number.

This contradicts the assumption, $\frac{1}{\sqrt{5}}$ is rational.

Therefore, $\frac{1}{\sqrt{5}}$ is an irrational number.