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

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

Concept: Concept of Irrational Numbers

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