On Numbers and Games

by John H. Conway

ONAG, as the book is known, is one of those rare publications that sprang to life in a moment of creative energy and has remained influential for over a quarter of a century. Still in high demand, it is being republished with some adjustments and corrections. The original motivation for writing the book was an attempt to understand the relation between the theories of transfinite numbers and mathematical games. By defining numbers as the strengths of positions in certain games, the author arrives at a new class, the surreal numbers (so named by Donald Knuth) that includes at the same time the real numbers and the ordinal numbers.

Genres: Mathematics, Nonfiction, Science, Games, Number Theory, Computer Science, Puzzles

238 pages, Hardcover

First published June 1, 1976

About the author

John Horton Conway, often credited as John H. Conway, was a Professor of Mathematics at Princeton University, known for inventing the "Game of Life."

Reviews

[Douglas Summers-Stay · Rating 5 out of 5]

I haven't just finished this book (I read it in 2002) which breaks my usual rule for posting on this site, but hearing of Conway's death I wanted to recommend this to anyone looking for a fun, easy math-related book with lots of interesting ideas. Most of what I remember is the material about about surreal numbers.

[meadow · Rating 4 out of 5]

I was initially interested in this book for surreal numbers, but I ended up reading a lot about games too. It's a very exciting book, full of new ideas. The tone is a pretty informal, and I think it could do with more rigor and exercises, but as a light introduction, it does a great job.

[Ronny · Rating 1 out of 5]

I was really excited to read this book and I started off with the first section which presented surreal numbers. I even broke out the pen and paper to follow along. Unfortunately, I was lost amid the imprecise descriptions and possible errors.

Fortunately, the author had written earlier that the second part was readable without the first, and since my interest was in the applicability of surreal numbers to games, I tried that.

Unfortunately, that too did not work. The author seemed to make mistakes in his dominoes example and at no point did I feel any relevance for surreal numbers to games.

I skimmed a bit more before finally giving up.

It's a shame. I really wanted to like this book, and the subject matter might be as good as others say, but the presentation was just poor. Perhaps a solid proofing and beefing up the anemic sections might make this worthwhile.

For now, it's a poor introduction to a tantalizing subject.

[Miktro]

Absolutely brilliant. And it has a chapter zero!

Breathtaking volumes of ideas packed into such short paragraphs that deciphering one of them can take you a week.

[George · Rating 5 out of 5]

Freakin' surreal numbers.

[Stuart Neil]

It’s a kind of magic! For the curious person with a bit of a maths background, it’s a step by step derivation of a newly defined number system. Let’s you see how a pro mathematician goes about getting something quite general and advanced from the simple easy to understand stuff. I’ve yet to fully appreciate the games connections but loved working at the theory. Hey some people waste time on computer games, and “brain training puzzles”; better to pick up some mathematical weight training and run with it. This book facilitates that.

[David Hildebrand · Rating 5 out of 5]

Conway's approach to foundations is frustratingly poorly ordered and I urge readers to look at the appendix to section 0 first. After skimming the book once it becomes extremely lucid and insightful if not haphazard. Truly a mathematical classic.

[Shayne · Rating 3 out of 5]

Don't know as I can say that I fully understood this book, but I am certainly done with it. I can see how it has earned such a high position in the canon of game literature.