Question

Prove that 2+V3 is an irrational

number given that V3 is an
irrational numbers.​

Answer 1

$let \: us \: assume \: that \: \frac{2 + \sqrt{3} }{5} \: is \: a \\ rational \: number \: then \\ \\ = > \frac{2 + \sqrt{3} }{5} = \frac{p}{q} \\ \\ = > \frac{2}{5} + \frac{ \sqrt{3} }{5} = \frac{p}{q} \\ \\ = > \frac{ \sqrt{3} }{5} = \frac{p}{q} - \frac{2}{5} \\ \\ = > \sqrt{3} = 5( \frac{5p - 2q}{5q}) \\ \\ = > \sqrt{3} = \frac{5p - 2q}{q} \\ \\ here \: \frac{5p - 2q}{q} \: is \: rational \: as \: it \: is \\ in \: the \: form \: of \: \frac{p}{q} \: but \: it \: is \: equal \\ to \: \sqrt{3} \: which \: is \: an \: irrational \: \\ number \\ thus \: \frac{2 + \sqrt{3} }{5} \: is \: irrational$