Question

Prove that (3+5√11) /9 is irrational

Answer 1

$let \: the \: number \: is \: rational \\ \: so \: it \: can \: represents \: as \: \frac{r}{q} \\ where \: r \: and \: q \: are \: coprime \: \: \\ number \\ so \:$
$(3 + 5 \sqrt{11} ) \div9 = \frac{r}{q} \\ (3 + 5 \sqrt{11} ) = \frac{9r}{q}$
$5 \sqrt{11} = \frac{9r}{q} - 3 \\ \sqrt{11} = \frac{9r}{5q} - \frac{3}{5}$
$\sqrt{11} = \frac{9r - 15q}{5q}$
we now that √11 is an irrational number and it cannot represent in the form of r/q,but in this theory it is equal to the form of r/q.
so from the theory of conflict we can say that the number
$(3 + 5 \sqrt{11} ) \div 9 \: is \: irrational \: number$