Question

prove that 2 - root 3 is irrational give that root 3 is irrational ​

Answer 1

$\bold{let \: 2 - \sqrt{3} \: is \: a \: rational \: number} \\ \ \bold{\therefore2 - \sqrt{3} = \frac{a}{b} ( \: where \: a \: and \: b \: are \: co - prime \: and \: b≠0)} \\\bold{ \implies - \sqrt{3} = \frac{a}{b} - 2} \\ \bold{\implies - \sqrt{3} = \frac{a - 2b}{b} } \\ \\\bold{ since \: \: \frac{a - 2b}{b} \: is \: a \: rational \: number \: so \: - \sqrt{3} \: is \: also \: rational. \: } \\\bold{ but \: this \: is \: not \: possible \: because \: an \: irrational \: number \: cannot \: } \bold{ be \: equal \: to \: a \: rational \: number.} \\ \\ \\ \bold{\therefore \: 2 - \sqrt{3} is \: not \: rational. \: hence \: it \: is \: irrational.}$