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I'm looking for an understandable book about mathematical proofs.

As an engineer/computer scientist major I'm used to do higher math on a daily base, but whenever I'm asked in an exercise to "show" something I get the "where should I start? and what exactly do they want me to do?" syndrome.

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A classic reference is Polya's How to Solve It (, which has been brought up on math.SE before. You might try searching for it here to see past threads. – Qiaochu Yuan May 13 '11 at 9:36
up vote 7 down vote accepted

There are books devoted to your question, e.g., Daniel Solow's book, How to Read and Do Proofs: An Introduction to Mathematical Thought Processes, but I'm not sure that's the way to go. It might be better to just pick up a good textbook in Number Theory (like Niven, Zuckerman, and Montgomery) or Abstract Algebra (like Fraleigh) or Discrete Math (maybe Grimaldi, or Brualdi), to see how people actually do proofs when they need them.

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Another book devoted to the question is Daniel Velleman, How To Prove It, – Gerry Myerson May 17 '11 at 3:59

I just came across this physics forum with a thread on how to write mathematical proofs. At first I was going to copy the list of texts there here, sorting out possible duplicates. But there are so many resources listed: books, tutorials, university-sponsored guides, etc. that I'll simply steer you to the site.

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Read Pólya's How to Solve It. See also these slides.

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You might also like this: How to Solve it: Modern Heuristics. I've read the first few pages and it's not bad at all.

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It's a very nice book but it's more advanced than Pólya's. – lhf May 13 '11 at 10:50
Yes it is a bit advanced. But I think glancing through it it worthwhile. The preface of the book has the following quote. "Whoever reads and understands the content of this book will be equipped with the most powerful tools for solving problems currently known". That seems tempting. :) – kodyv May 13 '11 at 11:09
Thank you, that looks like a fantastic book, especially for someone interested in optimization. – littleO Nov 19 '12 at 1:39

There are many mathematics texts, ranging from the middle school level to the undergraduate level, that are designed, at least in part, to serve as an introduction to proof. I would recommend that you select a text of this nature about a mathematical or allied field that you find interesting.

Three examples, off the top of my head:

  • Tom Apostol: Calculus, Volume I
  • James Munkres: Topology
  • Uhh … I can't find it right now, but there's an introductory real analysis text called something like Analysis that I used as a college freshman which is designed that way.

If you're into computer science, you could pick up a lot of proof techniques (and exercises) from Donald Knuth's The Art of Computer Programming, but that might be a bit on the intimidating side.

My introduction to proof came from my seventh grade geometry teacher, Darlyn Counihan, whose homework and tests consisted of nothing but proofs. Sadly, I hear that most geometry classes in the U.S. these days don't require students to write a single proof, which makes me wonder just what exactly such classes are for.

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Perhaps you might like: Solving Mathematical Problems: A Personal Perspective. By Terence Tao.

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You can search for "thoughts - Alpha" this is a free downloadable online PDF book for mathematical proofs. the book is a compilation of proofs for basic mathematics (Trigonometric Identities, logarithms, basic series, volumes and surfaces, basic calculus)

Book's webpage:

Author's scribd page:

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Writing proofs is essentially a sequence of statements and their justifications. We all learn some form of proof writing in geometry when we write two column proofs. With that being said, there are many techniques used in proof oriented problems.

There are many good books which have already been mentioned. I advise you not to touch Munkres unless you understand real analysis at the level of Baby Rudin.

Start with a simple intro: Appendix of Hungerford's undergrad Abstract Algebra.

Move onto first 10-15 pages of Baby Rudin to build your math background. Cover the beginning of chapter 2 in Baby Rudin.

Munkres Topology first 7 sections. The rest of the material requires much greater mathematical maturity.

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