Question

Question 2
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 $x\ne 0$.
Let x be a non -zero rational and y be irrational. Then we have to show that xy is rational. If possible, let xy be a rational number.
Since, Quotient of two non-zero rational So,$\left(\frac{xy}{x}\right)$ 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.