Question

Prove that 3/5 root 2 is irrational, given that root 2 is irrational?

Answer 1

$\purple{Prove \: that \: \frac{3}{5} \sqrt{2} \: is \: irrational}$

$\green{Given:}$

$\sqrt{2} \: is \: irrational \:$

$\green{To \: \: prove}$

$\frac{3}{5} \sqrt{2} \: is \: an \: irrational \: number.$

$\green{proof}$

$Let \: us \: assume \: that \: \frac{3}{5 \ } \sqrt{2} \: is \: a \: rational number.$

So, it can be written in the form a/b

$\frac{3}{5} \sqrt{2} = \frac{a}{b}$

Here a and b are coprime numbers and b ≠ 0

$\frac{3}{2} \sqrt{2} = \frac{a}{b} \\ \sqrt{2 } = \frac{2a}{3}$

$This \: shows \: \frac{3}{5} \sqrt{2} \: is \: a \: rational \: number. \: But \: we \: know \: that \: √2 \: is \: an \: irrational \: number.$

So, it contradicts our assumption. Our assumption is a rational number is incorrect.

$\frac{3}{5} \sqrt{2} \: is \: an \: irrational \: number$

Hence proved

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