Question

if a is rational, b is irrational and ab is rational, find a

Answer 1

Answer:

The answer of the question is 0.

Answer 2

Answer:

First:

As, Zero is a rational number. So, by using the

counterexample a=0a=0 and b=xb=x (xx being an irrational number) we get ab=0ab=0. Thus it is not true that abab necessarily has to be irrational.

Second:

However, I have learned a technique that I have used to proof similar but slightly different combination (aa is rational and bb is irrational, is a+ba+b irrational?) of questions.

With this technique (Proof by contradiction):

Given aa is rational and bb is irrational. Let, abab be rational.

Suppose, a=mna=mn and ab=z=pqab=z=pq , where m,n,p,qm,n,p,q are intgers and n,q≠0n,q≠0

b=xb=x where xx is an irrational number.

a⋅b=za⋅b=z

mn⋅x=pq

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