I'm on one my last Math problems of the assignment, and I can't seem to figure out where to start or where to go with it. The problem is:
-Use Proof by contradition to prove the following:
Assume p is rational and q is irrational. Prove if p =/= 0, then p*q is irrational.
Any help would be appreciated, thanks
I appreciate the assistance, so I put in what you said and now have
q= s/tq which should mean p*q= s^2/t^2q, but wouldn't that mean since that is a ratio that p*q is a rational nymber would be true, therefore not giving a contradition? Or am I missing something still?
I don't think you read my post correctly.
I said: "... express in the form " not "express in the form ".
Look, it goes a bit like this.
You want to show that pq is irrational if p is rational and q irrational. So what we're going to do is say: if pq were rational, then we would show that q also has to be rational, which contradicts our premise that q is irrational, hence demonstrating a contradiction.
So, suppose .
Then .
But are all integers so and are integers ...
... which means ...?
Have you already proved that the rational numbers are closed under multiplication? If so, you can use a proof by contradiction. Suppose, with p rational, not equal to 0 and q irrational, pq = r is a rational number, then p= r(1/q) is a product of rational numbers.