Let $x$ be a rational number and $y$ be an irrational number. Is $xy$ necessarily irrational? Justify your answer with an example.

(NO) Let $x = 0$ (a rational number) and $y = \sqrt{3}$ (an irrational number).
Then,the product $xy = 0 \times \sqrt{3} = 0$.
Since $0$ can be expressed as $\frac{0}{1}$,it is a rational number.
Therefore,$xy$ is not necessarily an irrational number.