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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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

Vedclass · Accepted Answer

always irrational

Vedclass · Answer

(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

Similar Questions

On a morning walk,three persons step off together and their steps measure $40\, cm$,$42\, cm$ and $45\, cm$,respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps? (in $cm$)

Prove that,if $a$ and $b$ are odd positive integers,then $a^{2} + b^{2}$ is even but not divisible by $4$.

According to Euclid's division lemma,for given positive integers $a$ and $b,$ there exist unique non-negative integers $q$ and $r$ such that $a = bq + r,$ where ...........

Find the square root of: $3 - \frac{1}{3} \sqrt{56}$

If two positive integers $a$ and $b$ are written as $a = x^3 y^2$ and $b = x y^3$,where $x$ and $y$ are prime numbers,then $HCF(a, b)$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module