Question

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

Answer 1

Use contradiction method

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

So, we can write it as

$\frac{1}{\sqrt{2}}=\frac{p}{q}$

where, $p$ and $q$ are two co-prime numbers.

On simplifying the above equation we get,

$\sqrt{2}=\frac{q}{p}$

Here, $q$ and $p$ are non zero integers.

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

Since, $\sqrt{2}=\frac{q}{p}$

So, $\sqrt{2}$ should also be a rational number.

But it contradicts the fact that $\sqrt{2}$ is irrational.

This contradiction has arisen because of our incorrect assumption that $\frac{1}{\sqrt{2}}$is rational.

Therefore, $\frac{1}{\sqrt{2}}$is irrational number.