Algebra review for Spivak Calculus

Question

I got a bit bored with High School maths so I picked up a copy of Calculus by Spivak. I am really enjoying the book and have found that the proofs and theorems aren't as hard as others have made them out to be. However, I am struggling a bit with the algebra. The education I have received thus far has been very focused on the methods for manipulating equations, meaning that the instructor will get up in front of the class and say "Oh you just flip them around" without ever explaining why. Is there any book that very quickly and rigorously goes over algebra, focusing on very extreme questions where it is impossible to rely on a shallow understanding of the rules?

Answer 1

For algebra, I might recommend Algebra by Gelfand and Shen. But to be honest, for most people attacking rigorous calculus out of high school, the greatest difficulty is working with inequalities. A good book for that would be An Introduction to Inequalities by Beckenbach and Bellman.

Answer 2

I would recommend going to Khan Academy's list of math courses. They have three courses on algebra at the high school level and it looks like pretty thorough collection of topics. You can brush up on your prealgebra, trig, precalculus skills if necessary as well, and you might also find it useful in having the Khan calculus videos as you go through Spivak.

focusing on very extreme questions where it is impossible to rely on a shallow understanding of the rules

For this, pick up a number theory textbook. I think Dover has one on Amazon for cheap. Try to do the exercises. They will test your ability to solve problems algebraically.