The product of a non-zero rational and an irr | vedclass

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(B) Let $r$ be a non-zero rational number and $i$ be an irrational number.Assume the product $p = r 	imes i$ is rational.Then $p = \frac{a}{b}$ for s

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always irrational

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(B) Let $r$ be a non-zero rational number and $i$ be an irrational number.Assume the product $p = r 	imes i$ is rational.Then $p = \frac{a}{b}$ for some integers $a, b$ where $b 
eq 0$.Since $r$ is a non-zero rational,$r = \frac{p}{q}$ for some integers $p, q$ where $p, q 
eq 0$.Then $i = \frac{p}{r} = \frac{p}{q} 	imes \frac{b}{a} = \frac{pb}{qa}$.Since $p, b, q, a$ are integers,$\frac{pb}{qa}$ is a rational number.This contradicts the fact that $i$ is irrational.Therefore,the product of a non-zero rational and an irrational number is always irrational.For example,$\frac{3}{4} 	imes \sqrt{2} = \frac{3\sqrt{2}}{4}$,which is an irrational number.

The product of a non$-$zero rational and an irrational number is

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