Question

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

Answer 1

To prove $\frac{1}{\sqrt{2}}$ is irrational

Let us assume that $\sqrt{2}$ is irrational

$\frac{1}{\sqrt{2}}=\frac{p}{q}$ (where $p$ and $q$ are co prime)

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

$q=\sqrt{2}p$

Squaring both sides

$q^2=2p^2$.....................(1)

By theorem - If $p$ is a prime no. and $p$ divides $a^2$, then $p$ divides $a$ also, where $a$ is a positive integer}

$q$ is divisible by $2$

$\therefore q=2c$ (where $c$ is an integer)

Putting the value of q in equitation (1)

$2p^2=q^2=4c^2$

$p^2=\frac{4c^2}{2}=2c^2$

$\frac{p^2}{2}=c^2$

$p$ is also divisible by $2$

But $p$ and $q$ are coprime

This is a contradiction which has arisen due to our wrong assumption

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