Question

State whether the following statement is true or not:

$7-\sqrt{2}$ is irrational.

• A. True
• B. False

Answer 1

The correct option is A True

Let us suppose $7-\sqrt{2}$ is rational.

Therefore,

$=>7-\sqrt{2}$ is in the form of $\frac{p}{q}$ where $p$ and $q$ are integers and $q\ne 0$

$=>\sqrt{2}=-\frac{p}{q}+7$

​$=>\sqrt{2}=\frac{-p+7q}{q}$

As $p,q$ and $7$ are integers therefore, $\frac{-p+7q}{q}$ is a rational number.

Hence,

$\sqrt{2}$ is a rational number.

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

So this is a contradiction.

This contradiction has arisen because of our wrong assumption that $7-\sqrt{2}$ is a rational number.

Hence, $7-\sqrt{2}$ is an irrational number.