Question

Prove the following number is irrational.
$\frac{1}{\sqrt{2}}$

Answer 1

Le us consider that the $\frac{1}{\sqrt{2}}$ is rational.

Now we write it in the form of $\frac{a}{b}$

where $a$ and $b$ are co-prime

Therefore, $\frac{1}{\sqrt{2}}=\frac{a}{b}$

Hence, $\frac{b}{a}=\sqrt{2}$

where $\frac{b}{a}$ is rational, $\sqrt{2}$ is irrational.

i.e. $Rational\ne Irrational$

which is a contradiction,

and our consideration is incorrect.

Therefore

$\frac{1}{\sqrt{2}}$ is irrational.