Question

Prove that the following are irrational.
$\frac{1}{\sqrt{2}}$

Answer 1

Let us assume that $\frac{1}{\sqrt{2}}$ is rational number

Hence $\frac{1}{\sqrt{2}}$ can be written in the form of $\frac{a}{b}$ where $a,b(b\ne 0)$ are co-prime

$⟹\frac{1}{\sqrt{2}}=\frac{a}{b}$

$⟹\frac{b}{a}=\sqrt{2}$

But here $\sqrt{2}$ is irrational and $\frac{b}{a}$ is rational

as $Rational\ne Irrational$

This is a contradiction so $\frac{1}{\sqrt{2}}$ is a irrational number