TITLE: Is a non-zero integral multiple of an irrational number guaranteed to be irrational?
QUESTION [5 upvotes]: I got to wondering if a non-zero integral multiple of any irrational number is guaranteed to be irrational?   This seems intuitive but I can't prove it to myself.
There is an answer on this site with regard to factors, but not multiples of irrationals.   
If someone could sketch a quick proof, my day would be golden. Thanks!

REPLY [16 votes]: Yes, it is guaranteed. Suppose for the sake of contradiction that $\alpha$ is irrational, and that $n\alpha$ is rational for some non-zero integer $n$. In other words, suppose
$$n\alpha=\frac{a}{b}$$
for some non-zero integer $n$ and rational number $\frac{a}{b}$. Then this implies
$$\alpha=\frac{a}{bn}$$
contradicting the definition of $\alpha$ being irrational.