There are many many elegant algebraic (more generally, non-geometric) proofs, but fewer of them are both accessible and interesting to a pre-calculus student. Three nice examples of what I'm looking for are the proof of the irrationality of the square root of 2, Cantor's diagonal argument (although I don't know if that's considered an actual proof), and the proof of Euler's formula by algebraic manipulation of the expansions of exp, sin, and cos. I'm seeking a dozen or so more like these, i.e., mostly algebraic, elegant, fun (if you're a math geek :-), and short (you could write them down in ~1 page). Thanks in advance!
(Clarification: If the proof has geometric aspects, like the diagonal argument, that's fair game. What I want to avoid is primarily geometric proofs, like those you do in your first geometry course. It's not that I'm against those, it's just that those are really easy to find as there are books and books of them -- like, specifically, 13 of them! :-) Also, I'm trying to avoid "proofs without words", which I find personally annoying because, although they may not literally have words once you see how the proof works, getting there usually involves a LOT of explanation! So the label "proofs without words" is somewhat disingenuous to my mind.)