# Prove that (2 Sqrt(3) – 1) is Irrational. - Mathematics

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

#### Solution

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, (2sqrt(3) – 1) is an irrational number.

Concept: Concept of Irrational Numbers
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