Name: Mr William
Question: I have taken a proof from the notes for a course I teach. The proof is that root 2 is irrational but that should illustrate the procedure. We use what is called a proof by contradiction. The essential idea is that we make an assumption and if we follow that up with a number of logical steps and end up with a false statement, it must be that our assumption was false. So, let's assume that is rational, in particular assume that = ^{p}/_{q} where p and q are relatively prime integers. Thus,
Equivalently,
Here's where we use our number theory sense (on divisibility properties)  we know 2  2q^{2}, thus 2  p^{2}. But this means that p^{2} must be even. The only way this can happen is that p itself is even. For simplicity's sake let's write p = 2m. Then we have
But now 2  2m^{2} leading to 2  q^{2}. The only way this can happen is that q is even also! But then we have both p and q being even when we've already assumed that they were relatively prime. This can't be. Thus, something has gone wrong! What? It can only be our assumption that = ^{p}/_{q} where p and q are relatively prime integers. cannot be rational and must be irrational. Cheers,Harley
