I need help with this proof. We have to prove the following statement by contradiction:
"Show that the product of a non-zero rational number and an irrational number is irrational."
This is what I have so far, based off of examples we did in class:
We proceed by way of contradiction. Assume that the product of a non-zero rational number and an irrational number is rational. Let p and r be rational numbers, and q be an irrational number. By definition, there exist integers a and b, b not equal to 0, and integers c and d, d not equal to 0, for which p = (a/b) and r = (c/d). So, pq = r, which implies (a/b)q = (c/d). Without loss of generality, we assume (a/b) and (c/d) are in lowest terms.
I don't know where to go from here. If anyone could help, I would greatly appreciate it.