Question

Prove that $6+\sqrt{2}$ is irrational:

Answer 1

STEP : Proving that $6+\sqrt{2}$ is irrational

Let us assume $6+\sqrt{2}$ is rational.

So, we can write it as $6+\sqrt{2}=\frac{p}{q}$

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

On simplifying the above equation we get,

$\sqrt{2}=\frac{p}{q}-6$

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

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

Therefore, $\frac{p}{q}-6$ is also a rational number.

Since, $\sqrt{2}=\frac{p}{q}-6$

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 $6+\sqrt{2}$ is rational.

Therefore, $6+\sqrt{2}$ is an irrational number.