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I've seen several question here on what book to read to learn writing and reading proofs. This question is not about that. I've been doing that for a while, and I'm quite comfortable with proofs. I am looking for resources (books, ideally) that can teach not the concept of proofs, but rather some of the specific mathematical tricks that are commonly employed in proofs: those that mostly include clever number manipulation, ad-hoc integration techniques, numerical methods and other thing you are likely never to learn in theory-oriented books. I come mainly from applied math and engineering, and when I look at proofs from Stochastic Processes, Digital Signal Processing, Non-Linear Systems and other applied subjects, I feel like I need to learn a new method to understand every proof I read. Is there any good literature on such mathematical tricks?

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"numerical methods" - you'll pick up a whole lot of practical numerical advice/tricks of the trade from Acton's two books: Numerical Methods That (usually) Work and Real Computing Made Real. – J. M. May 16 '11 at 15:30
You might also be interested in The Art and Craft of Problem Solving. – J. M. May 16 '11 at 15:32
Wait until you see set theoretic proofs :-) – Asaf Karagila May 16 '11 at 15:36
Thanks, I'll definitely check them out. All of them look great. You should post these as answers, I'll upvote = ) – Phonon May 16 '11 at 15:36
@Phonon: can you be more specific about these examples you describe? I'm not really sure what "clever number manipulation" or "ad-hoc integration techniques" could be referring to. More generally, beyond a handful of very general things the "tricks" you're going to see will depend on the field (although not necessarily in an obvious way), so I wouldn't say that there are "proof tricks" so much as "tricks for certain kinds of proofs." – Qiaochu Yuan May 16 '11 at 16:35

I don't know if you're interested in inequalities, but a very nice book which teaches lots of tricks is Steele's The Cauchy–Schwarz Master Class.

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This looks very good. Much closer to what I'm looking for. – Phonon May 16 '11 at 20:17

I enjoyed Mahajan's Street-Fighting Mathematics. It has a strongly "applied" bent, and is freely available.

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The Tricki ("Trick Wiki") is an attempt to catalogue such things, although it is somewhat less successful than was initially hoped.

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This is great! = ) Thanks! Not exactly what I'm looking for, but indeed very promising. – Phonon May 16 '11 at 17:59

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