Is it possible then that the product of and is rational, that is
But that means
But since are all integers and the integers are closed under multiplication, must be rational since it is the quotient of integers, contradicting the original assumption that it was irrational.
Therefore, reductio ad absurdum, the product of a non-zero rational number and an irrational number cannot be rational. Q.E.D.
John
My calculator said it, I believe it, that settles it
