Question

prove that irrational
$1 \div \sqrt{2}$

Answer 1

$we \: have \: to \: prove \: \frac{1}{ \sqrt{2} } is \: irrational$
$let \: us \: assume \: the \: opposite$
$assume \: that \: \frac{1}{ \sqrt{2} } is \: rational$
$hence \: \frac{1}{ \sqrt{2} } can \: be \: written \: in \: the \: form \: \frac{a}{b}$
$where \: a \: and \: b \: are \: co \: prime \: (no \: commom \: factor \: other \: than \: 1)$
$hence \: \frac{1}{ \sqrt{2} } = \frac{a}{b}$
$\frac{b}{a} = \sqrt{2}$
$here \: \frac{b}{a} is \: a \: rational \: number$
$but \: \sqrt{2} is \: irrational \:$
$since \: rational \: is \: not \: equal \: to \: irrational$
$this \: is \: a \: contradiction$
$hence \: our \: assumption \: is \: incorrect$
$hence \: \frac{1}{ \sqrt{2} } is \: irrational$
$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: hence \: proved.....$