Question

Prove that `(2 sqrt(3) – 1)` is irrational.

Answer 1

Let x = 2 `sqrt(3)` – 1 be a rational number.
x = 2 `sqrt(3)` – 1
⇒` x^2 = (2  sqrt(3)  – 1)^2`
⇒ `x^2 = (2 sqrt(3))^2 + (1)^2 – 2(2  sqrt(3) )(1)`
⇒ `x^2 = 12 + 1 - 4 sqrt(3)`

⇒ `x^2 – 13 = - 4 sqrt(3)`
⇒ `(13− x ^2 )/4 = sqrt(3)`
Since x is rational number, x2 is also a rational number.
⇒ 13 -` x^2` is a rational number
⇒`(13−x^2)/4` is a rational number
⇒ `sqrt(3)`  is a rational number
But √3 is an irrational number, which is a contradiction.
Hence, our assumption is wrong.
Thus, (2`sqrt(3)` – 1) is an irrational number.