Can the product of an irrational and any real number be a rational?
What you are asking and what you really want to know might be two different things.
Remember $\displaystyle \mathbb{Q} \subset \mathbb{R}$
I can think of an irrational number that is multiplied by a real that gives a rational. But this does not work for all cases. Is this what you want to know?
OK. Hope the following is better.
Pickslides had said: "I can think of an irrational number that is multiplied by a real that gives a rational. But this does not work for all cases."
So, since the product of an irrational and a real is a rational in some cases but not in other cases, are there rules by which we can know in advance whether the outcome would be an irrational or a rational?
(e.g. irrational x rational = irrational ... or... irrational x irrational = rational,... or... irrational x itself = rational... etc.)