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.

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    I've seen this book, and it's quite nice... – J. M. is not a mathematician Oct 1 '11 at 17:36
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    @J. M:Aha I have seen that book too,but I guess one need a moderate understanding of at-least high school mathematics to comprehend that books isn't? – Quixotic Oct 1 '11 at 17:42
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    Once you get past the very basics, you might try What is Mathematics? – Nick Strehlke Oct 1 '11 at 18:37
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    i hate books with only text and no images... – Muhammad Umer Jul 24 '13 at 9:19
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    Ian Stewart's various books are nice. – Akiva Weinberger Jan 6 '15 at 17:39

15 Answers 15

up vote 42 down vote accepted

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.

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    That's as good a recommendation as you're going to find from any of us. – Mathemagician1234 Oct 1 '11 at 18:17

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:


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:

  • 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

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.

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    Haha, innovative solution, thanks :) – Aditya M P Sep 30 '13 at 9:07

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.

I strongly recommend NO BULLSHIT guide to Math and Physics. Impressive work, excellent for people scared of maths, too.

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.

(1) How to Prove It by Daniel J. Velleman

(2) Fundamentals of Algebra and Trigonometry by Swokowski&Cole

(3) Calculus vol.1&2 by Tom M. Apostol

(4) Introduction to Linear Algebra by Gilbert Strang

(5) Mathematical Analysis by Tom M. Apostol

(6) Introductory Functional Analysis with Applications by Kreyszig

  • Calculus by Apostol is not a beginner book in my opinion, a good book but not for beginners. – Prathik Rajendran M Jul 24 '16 at 12:16

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.

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    I also recommend looking at the Gelfand books, even if you think you already know the material. (For example, can you factor $a^3 + b^3 + c^3 - 3abc?$ Think about what the Factor Theorem from precalculus tells you for $x^3 + b^3 + c^3 - 3xbc$ when $x = -b - c.$) See also the (translations of the) various books in the Russian "Popular Lectures in Mathematics Series", which I listed (via a URL) here: math.stackexchange.com/questions/55353 – Dave L. Renfro Oct 3 '11 at 15:07

I think the following books would be great to start learning to think in mathematics

  1. How to read and do proofs by Daniel Solow.
  2. How to solve it by George Polya.
  3. Mathematics and Plausible reasoning by George Polya.

It depends what your level is and what you're interested in. I think a book that's not about maths but uses maths is probably more interesting for most people. I've noticed this in undergraduates as well: give someone a course using the exact same maths but with the particulars of their subject area subbed in, and they'll like it much better. Examples being speech therapy, economics, criminal justice, meterorology, kinesiology, ecology, philosophy, audio engineering….

tensor penrose manifold cohomology of the tribar http://www.jstor.org/discover/pgs/index?id=10.2307/1575844&img=dtc.17.tif.gif&uid=3739664&uid=2&uid=4&uid=3739256&sid=21104710897987&orig=/discover/10.2307/1575844?uid=3739664&uid=2&uid=4&uid=3739256&sid=21104710897987

That said … for me a great introduction was Penrose's The Road to Reality. It pulls no punches unlike many popular physics books. I've always been interested in "the deep structure of the universe/reality" so ... that was a topic in line with my advice from above. But also Penrose is an excellent writer and takes the time to draw pictures.

Nicolas Bourbaki's Éléments de mathématique (there are English versions) assumes "no special knowledge of mathematics, it tries to take up mathematics from the very beginning, proceed axiomatically and give complete proofs." (quoted from Wikipedia)

In practice, I found it very difficult to read. Everything is really explained there from bottom to up, but I could understand it just once I knew what was being explained.

However, take a lot at it. You may like to have it at hand while reading another book.

Mathematics: From the Birth of Numbers by Gullberg is a great overview at about the 1st year university level. It also has problems in the text you can solve while having a cup of tea. Very nice.

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