Prove that (2 √3 – 1) is irrational.

1.	Prove that (2 √3 – 1) is irrational.
Answer» Let x = 2 √3 – 1 be a rational number. x = 2 √3 – 1 ⇒ x² = (2 √3 – 1)² ⇒ x² = (2 √3 )² + (1)² – 2(2 √3)(1) ⇒ x² = 12 + 1 - 4 √3 ⇒ x² – 13 = - 4 √3 ⇒ \(\frac{13-x^2}4\) = √3 Since x is rational number, x² is also a rational number. ⇒ 13 - x² is a rational number ⇒ \(\frac{13-x^2}4\) is a rational number ⇒ √3 is a rational number But √3 is an irrational number, which is a contradiction. Hence, our assumption is wrong. Thus, (2 √3 – 1) is an irrational number.

Answer»

Let x = 2 √3 – 1 be a rational number.

x = 2 √3 – 1

⇒ x² = (2 √3 – 1)²

⇒ x² = (2 √3 )² + (1)² – 2(2 √3)(1)

⇒ x² = 12 + 1 - 4 √3

⇒ x² – 13 = - 4 √3

⇒ \(\frac{13-x^2}4\) = √3

Since x is rational number, x² is also a rational number.

⇒ 13 - x² is a rational number

⇒ \(\frac{13-x^2}4\) is a rational number

⇒ √3 is a rational number

But √3 is an irrational number, which is a contradiction.

Hence, our assumption is wrong.

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

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