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?
These things follow from some axioms, in particular those of a field. Choose different axioms and they will not follow. So the question is really why adopt field axioms. Because the system we are interested in follows them. What is BEDMAS? Order of operations? It is simply an standard order so that the more common expression is shorter that the less commom one. of expressions of the type 2?*?x?+?1 (2*x)+1 is more common than 2*(x+1) so we let 2*x+1=(2*x)+1
Multiplication of two binomials is accomplished with the distributive property and this can be demonstrated graphically.
Note also, that the example which you presented will give a sum which contains two additive inverses of eachother, therefore the two terms yield zero, giving you only two terms persisting in the simplified result.
I figured it was a field axiom that had to be preserved. But these are not the only questions I have. And its not sensible to post them all here. I'd like a book on this sort of stuff. Can anyone reccommend one? Or is this stuff I'd learn in number theory?
Howers: Most of what you ask is treated well in Introductory Algebra textbooks. You might be using an inferior textbook. Look for old used ones by authors such as Wright & New, Larson & Hostetler (& Edwards?), Lial & Miller, Drooyan, Barker (or Auffman & Barker). Some Intermediate Algebra books also deal with the field axioms, but best first to check the Introductory books. Many years ago, some(if not most) Algebra teachers would spend a couple of weeks instructing about the field axioms before moving in to different kinds of expressions, equations, and other problem-solving. Instruction dealth with reference to the real-number line for explaining the concepts about number properties and properties of equality and inequality.
Thanks. I've seen the field axioms for the first time in linear algebra, but never when deriving secondary school results. In highschool all we ever did is examples. I'll check out your list of introductory algebra books, but so far I'm having trouble finding most.