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?
A good book for learning mathematical trickery
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4"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](http://www.amazon.com/dp/0883854503)* and *[Real Computing Made Real](http://www.amazon.com/dp/0486442217)*. – 2011-05-16
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2You might also be interested in *[The Art and Craft of Problem Solving](http://www.amazon.com/dp/0471789011)*. – 2011-05-16
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0Wait until you see set theoretic proofs :-) – 2011-05-16
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0Thanks, I'll definitely check them out. All of them look great. You should post these as answers, I'll upvote = ) – 2011-05-16
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0@Asaf Looking forward to it, one day = ) – 2011-05-16
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2@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." – 2011-05-16
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0@Qiaochu I have to look at some books and type up several examples. – 2011-05-17
3 Answers
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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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0This looks very good. Much closer to what I'm looking for. – 2011-05-16
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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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0This is great! = ) Thanks! Not exactly what I'm looking for, but indeed very promising. – 2011-05-16