Is irrational times rational always irrational?

Any nonzero rational number times an irrational number is irrational. Let $r$ be nonzero and rational and $x$ be irrational. If $rx=q$ and $q$ is rational, then $x=q/r$, which is rational. This is a contradiction.

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.

Claim: If $x$ is irrational and $r \ne 0$ is rational, then $xr$ is irrational.

Proof: Suppose that $xr$ were rational. Then, $x = \frac{xr}{r}$ would be rational (as the quotient of two rationals). This clearly contradicts the assumption that $x$ is irrational. Therefore, $xr$ is irrational.

The $r = 0$ case is special, and the above argument doesn't work.