I would say by now, I'm an expert in manipulating equations and playing with algebra. However, I've also realized I have no idea why some of the operations I do are valid. For example... why is (x+2)(x-2) = x^2 - 4? Why does this expansion work? I'm guessing it preserves some kind of field definition. And why do the exponent laws hold? Why must BEDMAS be preserved? Why is a negative times a positive a negative? Why can you multiply two equations? These are all things I would like a firm theoretical grasp of. The problem is most of the algebra books and precalc books I've seen only emphasize memorization of the techiques, which is a skill I already have. I'd like theorems, proofs, and definitions of elementary math. The closest thing to such a book I've read is Courant's WIM, but even he already assumes you know a lot of this stuff, like exponents (which I do, but not why they work). Likewise, I'd like a firm grasp of Euclidean geometry for the mathematically mature. Can anyone reccommend titles?