Proof by Contradiction
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 (Happy)
The way I'd look at it is:
Originally Posted by kallivis
If pq were rational it could be expressed in the form s/t where s and t are integers.
Now express p in the same form (m/n, say) and multiply pq by 1/p.
Continue from there.
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?
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.