Let us assume, that (2 + √3) is a rational, it can be expressed in the form of a/b, where a and b are coprime positive integers and b ≠ 0.
∴ 2 + √3 = a/b
(∵ HCF of a and b is 1)
⇒ a/b = √2
⇒ a−2b/b = √3
a−2b/b = rational number
(∵ a and b are positive integers) Thus, from eqn. (i) is a rational number.
But this contradicts the fact that is an irrational number. So, our assumption that 2 + √3 is a rational number, is wrong.
Hence, (2 + √3) is an irrational number.
