Concepts of Modern Mathematics by Stewart
Dover Version with Google Preview Button
How to Prove It by Velleman
>Never done any proofs, but I will ask my math teacher if he can recommend any books or resources that will help me learn.
If you are interested in getting started with "real" math, i.e. moving from solving computational problems to writing proofs, I highly recommend picking up a copy of How to Prove It . The author assumes basically no knowledge of higher level math and teaches various techniques in reading and writing proofs, learning quite a bit of basic set theory along to way to practice your proof writing.
EDIT: wanted to also mention that set theory is the "core" of modern math, or rather its foundation.
I've always enjoyed all types of math but all throughout (engineering) undergrad and grad school all I ever got to do was computational-based math, i.e. solving problems. This was enjoyable but it wasn't until I learned how to read and write proofs (by self-studying How to Prove It ) that I really fell in love with it. Proofs are much more interesting because each one is like a logic puzzle, which I have always greatly enjoyed. I also love the duality of intuition and rigorous reasoning, both of which are often necessary to create a solid proof. Right now I'm going back and self-studying Control Theory (need it for my EE PhD candidacy but never took it because I was a CEG undergrad) and working those problems is just so mechanical and uninteresting relative to the real analysis I study for fun.
EDIT: I also love how math is like a giant logical structure resting on a small number of axioms and you can study various parts of it at various levels. I liken it to how a computer works, which levels with each higher level resting on those below it. There's the transistor level (loosely analogous to the axioms), the logic gate level, (loosely analogous set theory), and finally the high level programming language level (loosely analogous to pretty much everything else in math like analysis or algebra).