Prove or disprove that the product of an irrational number a

Prove or disprove that the product of an irrational number and a non-zero rational number is irrational.

Solution

Consider the given statement

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

We prove this statement proof by contradiction.

Assume A be an irrational number and the non zero rational number is p/q   where q is non - zero and the product of A and p/q is rational a/b.where b is non - zero .

Then consider

A(p/q) = (a/c)

A = (a/c)/(p/q) = (aq)/(cq)

Since integers are closed under multiplication and a,c,p,q are integers ans so aq , cq are also integers, which makes a = (aq)/(cq) is rational by definition.

This is a contradiction to our assumption that A is irrational . hence our assumption that A is irrational is wrong.

So that , A is ratioanal. Hence , The product of an irrational number and a non-zero rational number is irrational