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.