Great question. Let's let <em>r</em> be a rational number and <em>s</em> be irrational. Note <em>r</em> has to be nonzero for this to work. In other words, it's not true that when we multiply zero, a rational number, by an irrational number like π we get an irrational number. We of course get zero.
The question is: why is the product
irrational?
In math "why" questions are usually answered with an illuminating proof. Here the indirect proof is enlightening.
Suppose <em>p</em> was rational. Then
would be rational as well, being the ratio of two rational numbers, so ultimately the ratio of two integers.
But we're given that <em>s</em> is irrational so we have our contradiction and must conclude our assumption that <em>p</em> is rational is false, that is, we conclude <em>p</em> is irrational.