# A good book for learning mathematical trickery

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?

-
"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.

-
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.

-

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

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