Question

Let x be rational and y be irrational. Is xy necessarily irrational? Justify your answer by an example

Answer 1

No, (xy) is necessarily an irrational only when x ≠0.

Let x be a non-zero rational and y be an irrational. Then, we have to show that xy be an irrational. If possible, let xy be a rational number. Since, quotient of two non-zero rational number is a rational number.

So,(xy/x) is a rational number => y is a rational number.

But, this contradicts the fact that y is an irrational number. Thus, our supposition is wrong.

Hence, xy is an irrational number. But, when x = 0, then xy = 0, a rational number.