Which books are recommended for learning math from the ground up and review the basics - from middle school to graduate school math?
I am about to finish my masters of science in computer science and I can use and understand complex math, but I feel like my basics are quite poor.
On
If you want to get into it, you might consider downloading (free) the lecture notes of a real analysis course given by Vaughan Jones - a Fields Medal winner. I'm 66 and always wanted to do real math. Last spring I jumped right in with them from a standing start. These notes are self contained, elegant and very accessible. This could be considered a foundational course for much of math and includes a development of derivatives and integrals as well.
I've been turned on ever since. I would venture that with your background you would be in a good position to see if this is appealing to you. Here is the link:
On
If you already know some advanced stuff but want to spend some time reviewing high school mathematics, you might like the books from the now sadly defunct Gelfand Correspondence Program in Mathematics. See here.
On
My favorite "beginner" book is Michael Spivak's Calculus book:
Don't let the title fool you. It's actually a completely rigorous introduction to single variable real analysis. It starts by axiomatizing the real numbers, i.e. with basic concepts of grade school algebra (less the least upper bound property), and rigorously develops many interesting results, including:
The book is certainly not easy, but you'll learn a lot and have a great time working through it.
I read the 2nd edition, published in 1996, but it looks like little has changed in the recent 3rd edition (note that they publish a new edition after 12 years, not every year like for the average crappy calculus book).
The complete Table of Contents:
On
The best beginning algebra book I've read is Algebra by Israel M. Gelfand. It explains many things that are glossed over in introductory algebra texts, like why $x^{0}=1$ and $x^{-n}=\frac{1}{x^{n}}$. However, the Kindle edition is riddled with formatting errors.
On
I strongly recommend NO BULLSHIT guide to Math and Physics. Impressive work, excellent for people scared of maths, too.
On
A book that comes pretty close to covering this vast area of mathematics is Mathematics Form and Function by Saunders MacLane. The book explains basically everything from basic trigonometry to sheaves. Naturally, one needs a good dose of mathematical maturity for reading it and not everything is done in full detail. But the book is great for filling gaps and providing a map to find ones way in mathematics.
On
The best way to teach yourself basic math through pre-algebra is to get a nursing student work book for calculating dosages. The workbooks are designed to start you at the beginning and give you examples, problems and answers that you can check yourself eventually leading to conversions and some basic algebra. I used one when I returned to college prior to entering elementary algebra, and it served me well, then and now.
Get Mathematics: Its Content, Methods and Meaning by Kolmogorov et al. This is a readable summary by the top Soviet mathematicians, and as the Soviets had no copyright it is incredibly inexpensive. If you have mastered this, you are pretty well prepared for anything.