This page contains descriptions of some of my favorite books, mostly of a mathematical nature. I've tried to stick to books that will be enjoyed by (mathematically-inclined) high school and undergraduate students, as well as older readers.
I haven't added links to Amazon or any other online bookstore. Firstly, I'd like you to consider supporting your local real-world bookstore (even if it's a chain, specially if it isn't). Secondly, the same goes for your local library. Thirdly, you could be looking at a $12 postage and handling fee and a two-week wait. Finally, if you do need to buy online (and OK, sometimes it's the only way to get something unusual) you don't really need a link, just a search engine.

• If you haven't read Douglas Hofstadter's book Gödel, Escher, Bach: An Eternal Golden Braid, you really should. It explains Gödel's Incompleteness Theorem, and how it ties in with ideas of self-reference in music, art, and language. Lots about Turing machines, art (Escher and Magritte), music (Bach, Chopin, Cage), and paradoxes.
  I also recommend Hofstadter's very moving Le ton beau de Marot, in which he considers just what is meant by "translation". Both are published by Basic Books and should be available in any good library or large bookstore.

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• While it's not exactly mathematics, I can't resist recommending Robert Pirsig's classic Zen and the Art of Motorcycle Maintenance. (There's a sequel, Lila, that I don't think is quite as good. Your mileage may differ.) It's a sort of philosophical road trip, partly a novel and partly an inquiry into the question "what is Quality?" OK, it's not exactly mathematics, but I know a lot of mathematicians who like it. You should be able to find it in the psychology/philosophy/NewAge/self-help corner of a large bookstore, and there are second-hand copies to be found.

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• Elwyn Berlekamp, John Conway, and Richard Guy's two-volume work Winning Ways (Academic Press) is THE book (or pair thereof (in the latest edition, four)) on the mathematical theory of games. They are, alas, not cheap, but in my opinion should be in every high school and public library.
  You may be familiar with the version of Nim in which each player, in turn, takes one, two, or three toothpicks from a heap; the winner is the person who takes the last one. If so, the chances are that you are also familiar with the winning strategy, and know whether you would rather start with 40 or 41 toothpicks on the table. Here, the use of (natural) numbers to represent game positions is very obvious.
  Slightly less well-known is the strategy for the usual form of Nim, in which there are several heaps and a player can take as many as he or she likes from one heap. Here there is also a strategy, but it requires binary representations and a weird form of addition. John H. Conway showed, some time ago, that any game in which exactly the same set of moves are available to both players can be worked out in this way. Other games, in which some moves are permitted to one player but not the other, can be modelled by a more general class of so-called "surreal" numbers.
  While this book sometimes ventures to the cutting edge of research (at least as of when it was written - and I believe that some of the problems in it are unsolved even now), it is self contained and avoids technicalities, and most of it can be read and understood by an intelligent amateur.

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• A really accessible introduction to the surreal numbers is Donald Knuth's Surreal Numbers: How two ex-students turned on to pure mathematics and found total happiness in which the construction is presented as a short novel. (Addison-Wesley)

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• Many years ago, the British mathematician J.E. Littlewood wrote A Mathematician's Miscellany, a somewhat random but uniformly fascinating collection of mathematical curiosities, anecdotes, and opinions. Some time after his death, his friend and colleague Béla Bollobás re-edited the book as Littlewood's Miscellany (Cambridge University Press). The new book has been somewhat expanded and contains some hilarious "gossip" about Littlewood's academic contemporaries that he may not have wanted to publish during their lifetimes!

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• The most up-to-date source on computer hacker's jargon is the Jargon File. If you want something smaller than a laptop to carry around with you, however, you want the New Hacker's Dictionary or Yellow Book (MIT Press), the print version. Not only are many of the phrases hilarious (as are the explanations), but you may just find that they are in fact the Right Thing when you need to describe something to do with computers, the people who love them, and the sometimes bizarre actions of both species.

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• If you're interested in actually building models of polyhedra, you should consider Magnus Wenninger's book Polyhedron Models (Cambridge University Press). This book, first printed in 1971 in the days before raytracing, is not the best source today for the person who just wants to look at exotic polyhedra; websites such as Vladimir Bulatov's have the advantage of color rendering. However, this book is an ideal reference for the would-be model-maker. There is also a good chance that you will find it in a library.

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• For many years (which many think of as the Golden Age of recreational math) Martin Gardner wrote a column in Scientific American called "Mathematical Games". These columns have been collected in several volumes, often with updates and notes added. Gardner has also written other books, notably several debunking pseudoscience. School, public libraries, and second-hand bookstores are very likely to have several of these. Here's a link to a bibliography .

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• Clifford Stoll's book The Cuckoo's Egg (Doubleday) is a fascinating true story of a computer programmer who starts out trying to track down a tiny discrepancy in the billing for user accounts, and ends up breaking up a spy ring. It's also a warmly moving look at the human side of hacker [counter]culture. Thoroughly recommended.

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• And, again by Clifford Stoll, Silicon Snake Oil (Doubleday). This book is a reaction against the Cult of the Virtual. A lost cause? I mean, here am I pounding away on a keyboard and here are you reading it on the Internet- and I'm telling you to go read a book about why we ought to get back to the Real World?
  You bet. First off, this is not a technoLuddite rant about why Computers Should Be Smashed and Done Away With. It's written by a man who has spent more time in the trenches of deep system-level hacking than most of us ever will. His argument is that just because something can be done with a computer doesn't mean that's always the best way of doing it. If you've read The Cuckoo's Egg already, you'll recognize that this is not a recantation of Stoll's earlier lifestyle - it's stressing the need for balance and perspective that was, at a personal level, a theme throughout the first book.
  There's also some good stuff about caving, and about how to build your own railroad engine (which I want to make clear I am not advising you to try yourself.)

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• Some time ago (around 1960) the late Clifton Fadiman edited Fantasia Mathematica and The Mathematical Magpie . These two little anthologies of short stories of a mathematical nature - some definitely in the science fiction vein, some more mainstream - contain many classics, none requiring a great deal of mathematical knowledge to read. Some of the stories are a bit dated, but most age well. These books appear to be back in print now, though a second-hand bookstore or library may be a good place to look too.

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• The Book of Numbers by John Conway and Richard Guy (Springer) is a wonderful introduction to many different sorts of numbers; it should definitely be in school libraries. It has sections on figurate numbers (square numbers, triangular numbers, cubic numbers, etc), prime numbers, constructible numbers, complex numbers, infinite numbers (several kinds!), continued fractions, imaginary numbers, quaternions... Even the sections on advanced topics are very readable, much of it at a junior-high level.

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• Charles Petzold is known to computer programmers as the author of some of the authoritative books on programming. His excellent book Code (Microsoft Press,1999) is aimed at the nonprogrammer who wants, not necessarily to become a programmer, but to really understand how their computer works. Programmers will probably know most of it already, if they happen to be very well up in the history of their craft. But the nostalgia trip might be worthwhile - remember the 4004 microprocessor?
  The organization of this book is both historical and structural. It begins, with Braille and Morse code, and progresses through telegraphs, switching circuits, relays, transistors, and simple integrated circuits before reaching the microprocessor. Petzold shows how, while simple computation can be done using only simple switches, more complicated tasks require switches that can be controlled by other switches. Such switches can be combined into logic gates, which can be combined into counters, adders, and memory units; and these in turn can be combined into a computer.

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• Want great math books (science books, art books, clipart CD's) cheap? Dover Books have been selling paperback reprints of classics for half a century now. Since the beginning they have used excellent paper and bindings. The covers of the oldest Dovers (printed around 1950) on my bookshelf are getting a little frail, the pages and spine are still in first-class shape. You can find these at any good bookstore (this may be the test for a good bookstore), or order direct from the publisher. Many of their books are priced around $20 Canadian. They have recently added a line of hardbacks aimed at the professional market the Dover Phoenix series.

  Most Dover books are reprints - they have much better coverage of subjects that were well-established thirty years ago or more than of very modern subjects. On the other hand, most accessible math was established then.

  A few of my favorites:

  • The entire popular mathematics section is recommended - and I've read a good number of these. In particular, the USSR Olympiad Problem Book and Yaglom and Yaglom's Challenging Mathematical Problems with Elementary Solutions should be on the shelf of anybody training seriously for math competitions.

  • Benjamin Bold's Famous Problems of Geometry and How to Solve Them is an excellent, short, elementary-but-rigorous introduction to the questions of trisecting the angle, duplicating the cube, and squaring the circle, accessible to high school students. (A more advanced book on the same subject, still under C$50, is Felix Klein's Famous Problems of Elementary Geometry )

  • Rouse Ball's Mathematical Recreations and Essays was first written in 1892. It has been a classic ever since, passing through 12 editions. In 1938 the late H.S.M. Coxeter revised and updated it. It's a little old-fashioned - the style is pre-Gardner, and illustrations date, largely, to a period before photos were easily reproduced - but it is very readable.

  • Coxeter's Regular Polytopes , first published in 1963, is accessible to undergraduate students and worth an attempt by high school students. As with any book on polyhedra written before raytracing and the Web, its illustrations should be supplemented with those on a good website such as Vladimir Bulatov's site.

  • Newman's The World of Mathematics" is a classic 4-volume anthology of essays. Excellent resource for teachers, student, and people who want to know what makes mathematicians tick. The four volumes will cost about $100 Canadian, but guarantee a great read as well as an easy introduction to the main topics of math up to about 1950.

  • And of course Abramowitz & Stegun's massive Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables . This is not quite as essential to any scientist's bookshelf as it was before the advent of programs like MAPLE - the actual values of many functions are better found using a calculator these days - but it still has a huge amount of information. It's the place I go if I want to write a C or Java program that uses (say) a normal distribution. And it's under $50 Canadian.

  Every student starting their professional library, every high school trying to keep a good elementary-recreational math section going on a tight budget, and every book addict should check this site out. (I know I said I wouldn't be recommending online bookstores, but this isn't exactly a bookstore.)

  There is also a Dover Thrift series, which (like the offerings of the Wordsworth Press) consists of shorter, long-out-of-copyright, classics priced very cheaply. These are not printed on the famous acid-free paper, and may not last more than a decade; but at two or three dollars, the price is right. Titles I've seen include the Analects of Confucius, Bierce's Devil's Dictionary, and Bellamy's Looking Backward . Not a lot of mathematics, though.

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