# Q3  Prove that the following are irrationals : (i) $1/ \sqrt 2$

Let us assume $\frac{1}{\sqrt{2}}$ is rational.
This means $\frac{1}{\sqrt{2}}$ can be written in the form $\frac{p}{q}$ where p and q are co-prime integers.
$\\\frac{1}{\sqrt{2}}=\frac{p}{q}\\ \sqrt{2}=\frac{q}{p}$
Since p and q are co-prime integers $\frac{q}{p}$ will be rational, this contradicts the fact that  $\sqrt{2}$ is irrational. This contradiction arises because our initial assumption that $\frac{1}{\sqrt{2}}$  is rational was wrong. Therefore $\frac{1}{\sqrt{2}}$ is irrational.