Question

Prove that $(2\sqrt{3}-1)$ is an irrational number.

Answer 1

Let $(2\sqrt{3}-1)$ be a rational number.

A rational number can be written in the form of $\frac{p}{q}$ where p,q are integers.

$2\sqrt{3}-1=\frac{p}{q}$

$2\sqrt{3}=\frac{p}{q}+1$

$2\sqrt{3}=\frac{p+q}{q}$

$\sqrt{3}=\frac{p+q}{2q}$

p,q are integers then $\frac{p+q}{2q}$ is a rational number.

Then,$\sqrt{3}$ is also a rational number.

But this contradicts the fact that $\sqrt{3}$ is an irrational number.

Therefore, our supposition is false.

So,$2\sqrt{3}-1$ is an irrational number.