Question

Show that 4$\sqrt{2}$ is an irrational number.

Answer 1

Let us assume that 4$\sqrt{2}$ is a rational number.

$\Rightarrow$4$\sqrt{2}$ = $\frac{p}{q}$, where p and q are the integers and q $\ne$ 0.

$\Rightarrow$$\sqrt{2}$ = $\frac{p}{4q}$

Since, p, q and 4 are integers. So, $\frac{p}{4q}$ is a rational number.

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

but this contradicts the fact that $\sqrt{2}$ is an irrational number.

This contradiction has arisen due to the wrong assumption that 4$\sqrt{2}$ is a rational number.

Hence, 4$\sqrt{2}$ is an irrational number.