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

Prove that one of any three consecutive positive integers must be divisible by $3$.

Prove that the following number is irrational: $\sqrt{5}-\sqrt{3}$.

Show that the square of any odd integer is of the form $4q + 1$ for some integer $q$.

Four bells toll at intervals of $8$,$12$,$15$,and $18$ seconds respectively. All four begin to toll together. How many times will they toll together in one hour,excluding the one at the start?

Find the $\text{g.c.d.}$ (Greatest Common Divisor) of $196$ and $38220$ using Euclid's division algorithm.

Vedclass Test Series

Exam Paper Generator

Online Exam Module