Math Help - If 'a' is rational and 'b' is irrational is 'ab' neccesarrily irrational?

1. If 'a' is rational and 'b' is irrational is 'ab' neccesarrily irrational?

So, this is a question in my textbook. I'm sure it's a very simple answer, that's right under my nose, but that I'm just missing for some reason. Anyway, if we have two numbers $a$ and $b$ and $a$ is rational, but $b$ is irrational, then is $ab$ neccesarily irrational? Please provide a proof with your answer, thanks in advance.

2. Suppose a*b=t (rational!)

a rational so a=p/q

t is rational so t=m/n

Hence, ab=m/n ==> b=m/n * q/p ==> b rational! A contradiction!

3. Unless a = 0...

4. And a single counterexample is sufficient to prove a general statement false!