Question

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

Answer 1

Given: the number $\sqrt{5}$

We need to prove that $\sqrt{5}$ is irrational

Let us assume that $\sqrt{5}$ is a rational number.

So it can be expressed in the form p/q where $p,q$ are co-prime integers and $q\ne 0$
$\Rightarrow \sqrt{5}=\frac{p}{q}$

On squaring both the sides we get,
$$\begin{gathered} \Rightarrow 5=p²/q² \\ \Rightarrow 5q²=p²—————–\left(i\right) \\ p²/5=q² \\ So5divides{p}^2,pisamultipleof5 \\ \Rightarrow p=5m \\ \Rightarrow p²=25m²————-\left(ii\right) \\ Fromequations\left(i\right)and\left(ii\right),weget, \\ 5q²=25m² \\ \Rightarrow q²=5m² \\ q²isamultipleof5 \\ \Rightarrow qisamultipleof5 \end{gathered}$$
Thus,$p,q$ have a common factor$5$. This contradicts our assumption that they are co-primes. Therefore,$\frac{p}{q}$ is not a rational number

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