Question

State true or false:

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

• A. True
• B. False

Answer 1

The correct option is A True

Let us assume that $\sqrt{2}$ is a rational number
$\Rightarrow \sqrt{2}=\frac{p}{q}\left[\frac{p}{q}isinsimplestform\right]$

$\Rightarrow 2q^2=p^2$

$p^2$ is even $\Rightarrow p$ is even $\Rightarrow p=2k$

$2q^2=4k^2\Rightarrow q^2=2k^2\Rightarrow q^2$ is even

$\Rightarrow q$ is even.

$\Rightarrow p,q$ have 2 as a common factor which is contradiction to assumption.

$\therefore \sqrt{2}=\frac{p}{q}$ is a false

$\Rightarrow \sqrt{2}$ is not a rational.