Question

How to prove that 2/√3-1 is irrational

Answer 1

to prove

$\frac{2}{ \sqrt{3 } - 1 } is \: irrational$

let us assume to the contrary that the given number is rational, So there exist the co-prime a and b . where b is not equal to 0

$\frac{2}{ \sqrt{3} - 1} = \frac{a}{b} \\ \frac{ \sqrt{3} - 1}{2} = \frac{b}{a} \\ \frac{ \sqrt{3} }{2} = \frac{b}{a} - 1 \\ \sqrt{3} = \frac{b - a}{2} \\ since \: a \: and \: b \: are \: integers \: so \: \frac{b - a}{2} \: is \: rational \: and \: \\ so \: \sqrt{3} \: is \: rational \: \\ but \: this \: contradicts \: the \: fact \: that \: \sqrt{3} \: is \: irrational \: \\ hence \: \frac{2}{ \sqrt{3} - 1 } \: is \: irrational$