Question

Prove that $(5-2\sqrt{3})$ is an irrational number.

Answer 1

Let $5-2\sqrt{3}$ be a rational number.

$\therefore 5-2\sqrt{3}=\frac{p}{q}$ [ where p and q are integer, $q\ne 0$ and q and p are integers ]

$\Rightarrow -2\sqrt{3}=\frac{p}{q}-5$

$\Rightarrow -2\sqrt{3}=\frac{p-5q}{q}$

$\Rightarrow \sqrt{3}=\frac{p-5q}{-2q}$

We know that p and q are integers, so $\frac{p-5q}{-2q}$ is a rational number.

So, if RHS is a rational number, LHS will also be a rational number, which means $\sqrt{3}$ is a rational number.

But $\sqrt{3}$ is an irrational number.

This contradicts our assumption

Hence, $5-2\sqrt{3}$ is an irrational number.