Show that $2+\sqrt{2}$ is not rational.

Given :

The given number is $2+\sqrt{2}$
 

To do :

We have to prove that $2+\sqrt{2}$ is not rational.

Solution :

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

Hence, it can be written in the form of $\frac{a}{b}$, where a, b are co-prime, and b is not equal to 0.

$2+\sqrt{2}=\frac{a}{b}$

$\sqrt{2} = \frac{a}{b} - 2$

$\sqrt{2} = \frac{a -2b}{b}$

Here, a, b and $-2$ are integers.

So, $\frac{a -2b}{b}$ is a rational number.

But, we already know that, $\sqrt{2}$ is an irrational number.

This contradicts the assumption, $2+\sqrt{2}$ is rational.

Therefore, $2+\sqrt{2}$ is not a rational number.