Question

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

Answer 1

Let us assume that 2 $\sqrt{3}$ - 1 is rational

i.e. we can find coprime

integers a and b ( b ≠ 0 ) such that

2$\sqrt{3}-1=\frac{a}{b}$

2$\sqrt{3}-1=\frac{a}{b}$+ 1

2$\sqrt{3}-1=\frac{a+b}{b}$

$\sqrt{3}=\frac{a+2b}{2b}$

Since a and b are integers, we get

$\frac{a+2b}{2b}$ is rational , and so √3 is

rational.

But this contradicts the fact that √3

is irrational.

This contradiction has arisen

because of our incorrect assumption

that 2√3 - 1 is rational.

So , we conclude that 2√3 -1 is

irrational.