Question

Prove that the following is irrational : $\frac{1}{\sqrt{2}}$

Answer 1

We can prove $\frac{1}{\sqrt{2}}$ irrational by contradiction.
Lets suppose that $\frac{1}{\sqrt{2}}$ is rational. It means we have some co-prime integers a and b (b $\ne$ 0) such that $\frac{1}{\sqrt{2}}$=$\frac{a}{b}$
$\Rightarrow \sqrt{2}=\frac{b}{a}$ (1)
R.H.S of (1) is rational but we know that $\sqrt{2}$ is irrational. It is not possible which means our supposition is wrong. Therefore, $\frac{1}{\sqrt{2}}$ cannot be rational. Hence, it is irrational.