Is irrational times rational always irrational? Is an irrational number times a rational number always irrational?
If the rational number is zero, then the result will be rational. So can we conclude that in general, we can't decide, and it depends on the rational number?

If $a$ is irrational and $b\ne0$ is rational, then $a\,b$ is irrational. Proof: if $a\,b$ were equal to a rational $r$, then we would have $a=r/b$ rational.
