Question

Show that 2root3-1 is irrational

Answer 1

$2 \sqrt{3 } - 1 = r \: where \: r \: is \: a \: non \: zero \: integer \\ \sqrt{3} = r - 1 \div 2 \\ since \: r - 1 \: and \: 2 \: are \: both \: rational \: no \: r - 1 \div 2\: is \: also \: a \: rational \: no \: \\ \ \sqrt{3 } = rational \: \: no \\ but \: this \: is \: a \: contradiction \: to \: the \: fact \: that \: \sqrt{3 } is \: an \: irrational \: no \: \\ therefore \: 2 \sqrt{3} - 1 \: is \: an \: irrational \: no$