prove that 1/(2+root3) is an irrational number - Maths - Introduction to Trigonometry

Question

prove that 1/(2+root3) is an irrational number

Chetna Khanna · Accepted Answer

Let, if possible $\frac{1}{2 + \sqrt{3}}$ be rational.

Then, there exist positive co-primes a and b such that

$\frac{1}{2 + \sqrt{3}} = \frac{a}{b} \Rightarrow b = a (2 + \sqrt{3}) \Rightarrow b = 2 a + a \sqrt{3} \Rightarrow \sqrt{3} = \frac{b - 2 a}{a}$

$\Rightarrow \sqrt{3}$ is rational, which is a contradiction.

Hence, $\frac{1}{2 + \sqrt{3}}$ is irrational.

prove that 1/(2+root3) is an irrational number.