Let x be rational and y be irrational. I

No, $(x y)$ is necessarily an irrational only when $x \neq 0$.

Let $x$ be a non-zero rational and $y$ be an irrational. Then, we have to show that $x y$ be an irrational. If possible,

let $x y$ be a rational number. Since, quotient of two non-zero rational number is a rational number.

So, $(x y / x)$ 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, $x y$ is an

irrational number. But, when $x=0$, then $x y=0$, a rational number.