What is a good book for learning math, from middle school level?

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.

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:

http://sites.google.com/site/math104sp2011/lecture-notes

My favorite "beginner" book is Michael Spivak's Calculus book:

• http://books.google.com/books/about/Calculus.html?id=7JKVu_9InRUC
• http://www.amazon.com/Calculus-Michael-Spivak/dp/0521867444

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:

• all the calculus you would see in a first course (the completely rigorous development of Taylor series is the highlight here for me)
• irrationality of $\pi$
• transcendence of $e$
• logarithms and trigonometric functions from first principles (e.g. he derives that the derivative of log x must be c/x for some c, and so choosing c = 1 arrives at the natural log, naturally!)
• that all complex polynomials in a single complex variable can be factored

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:

• Preface
• Part I. Prologue:
  • 1. Basic properties of numbers
  • 2. Numbers of various sorts
• Part II. Foundations:
  • 3. Functions
  • 4. Graphs
  • 5. Limits
  • 6. Continuous functions
  • 7. Three hard theorems
  • 8. Least upper bounds
• Part III. Derivatives and Integrals:
  • 9. Derivatives
  • 10. Differentiation
  • 11. Significance of the derivative
  • 12. Inverse functions
  • 13. Integrals
  • 14. The fundamental theorem of calculus
  • 15. The trigonometric functions
  • 16. Pi is irrational
  • 17. Planetary motion
  • 18. The logarithm and exponential functions
  • 19. Integration in elementary terms
• Part IV. Infinite Sequences and Infinite Series:
  • 20. Approximation by polynomial functions
  • 21. e is transcendental
  • 22. Infinite sequences
  • 23. Infinite series
  • 24. Uniform convergence and power series
  • 25. Complex numbers
  • 26. Complex functions
  • 27. Complex power series
• Part V. Epilogue:
  • 28. Fields
  • 29. Construction of the real numbers
  • 30. Uniqueness of the real numbers
• Suggested reading
• Answers (to selected problems)
• Glossary of symbols
• Index