Dual Review: On Numbers and Games, by John H. Conway and Winning Ways, volume 1, by Elwyn R. Berlekamp, John Horton Conway, and Richard K. Guy.
I first heard of John Conway's "Surreal Numbers" from Martin Gardner. In one of his "Mathematical Games" columns, Gardner explained Conway's method for "creating numbers out of nothing," obtaining, in the process, a bewildering zoo of infinite and infinitesimal numbers in addition to the usual real numbers. When, a short time later (in 1977, it was), I saw On Numbers and Games (originally published in 1976) for sale at a local science bookstore, I couldn't resist buying a copy.
What a marvelous book it turned out to be! First of all, it was fun to read. (Just look at the names of things: "contorted fractions", "hackenbush unrestrained", "col" and "snort"...) Second, the theory it developed was fascinating. I was only a fledgling mathematician at the time, so many of the details were beyond me, but I could still appreciate and enjoy the creativity and insight on display here.
Here, 25 years later, is a new edition of the book, which has long been out of print. So let's begin by saying that even an unchanged new printing would be a great thing to have: people who missed the chance of buying the book then can now get a copy. The new edition does not include a great number of changes: some corrections have been made, and an Epilogue discusses what progress has been made since 1976 in studying the Surreal Numbers. Coming from Conway, it contains several interesting ideas and even suggested questions for further research.
One of the more interesting things I learned when I read On Numbers and Games was that Conway had discovered a connection between numbers and combinatorial games. In fact, a number turned out to be a certain kind of game. This theory is developed in the "first" part of ONAG (the number theory in the zeroth part), but further development was promised in a forthcoming book. This was Winning Ways, a two-volume book written by Elwyn Berlekamp, John Conway, and Richard Guy. This was another marvelous book. It developed an elaborate and powerful theory of combinatorial games and then applied it to a wide array of different (and fascinating) games.
The theory introduced in WW has continued to develop, of course. The results of a recent workshop on the subject were published in a book called Games of No Chance, reviewed on MAA Online some years ago. This is to have a sequel in the near future. This book, then, is the entry point to a living mathematical theory.
The original WW appeared in two large volumes. The new edition splits each of these volumes in two, so that what we have on hand is the first of four volumes. The new edition is only lightly revised. The authors have added "Extras" at the ends of the chapters and inserted many references to more recent work. They also say that they have "corrected some of the one hundred and sixty-three mistakes." The book is beautifully produced, with color images where they are helpful. It's great to see this book back in print.
Fernando Q. Gouvêa is the Secret Master of MAA Reviews.