Question

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

Answer 1

Let's assume that $\frac{1}{\sqrt{2}}$ is a rational number.
So, we can write this number as
$\frac{1}{\sqrt{2}}=\frac{a}{b}$
Here, a and b are two co prime numbers and b is not equal to 0.
Multiply by $\sqrt{2}$ both sides we get,
$1=\frac{a\sqrt{2}}{b}$
Now multiply by b,
$b=a\sqrt{2}$
Divide by a we get,
$\frac{b}{a}=\sqrt{2}$ ---(i)
Here, a and b are integers, that means $\frac{b}{a}$ is a rational number. Then, $\sqrt{2}$ should also be a rational number as $\frac{b}{a}=\sqrt{2}$.
But, we know that $\sqrt{2}$ is an irrational number.
So, our assumption is wrong.
Hence, $\frac{1}{\sqrt{2}}$ is an irrational number.