- Hardcover: 242 pages
- Publisher: A K Peters/CRC Press; 2nd edition (December 11, 2000)
- Language: English
- ISBN-10: 1568811276
- ISBN-13: 978-1568811277
- Product Dimensions: 6.3 x 0.7 x 9.3 inches
- Shipping Weight: 1 pounds (View shipping rates and policies)
- Average Customer Review: 4.2 out of 5 stars See all reviews (4 customer reviews)
- Amazon Best Sellers Rank: #753,395 in Books (See Top 100 in Books)
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On Numbers and Games 2nd Edition
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Conway defines a bunch of mathematical objects. He defines mathematical operations on these objects such as addition and multiplication. The whole work looks suspiciously like a way to define the integers and arithmetic starting from set theory. But we soon see that his construction allows for all sorts of things beyond just integers. We quickly get to fractions and irrationals and we see that he has given us a wonderful new way to construct the real line. Then we discover infinities and all sorts of weird new numbers called nimbers that have fascinating properties.
It all looks a bit abstract until you get to part two (well, he actually starts at part zero so I mean part one). At this point you discover that these objects are in fact positions in games and that the ordinary everyday numbers we know so well are in fact special types of games. Ordinary operations like addition, subtraction and comparison turn out to have interpretations that are game theoretical. So in fact Conway has found a whole new way to think about numbers that is beautiful and completely different to the standard constructions. Even better, you can use this new found knowledge to find ways to win at a whole lot of games.
It's not every day that someone can make a connection like this between two separate branches of mathematics so I consider this book to be nothing less than a work of genius.
BTW This is the Conway who invented (the cellular automaton) the Game of Life and came up with the Monstrous Moonshine Conjectures (whose proof by Borcherds recently won the Fields Medal in mathematics).
John Horton Conway is probably best known as the creator/discoverer of the computer game called "Life," with which he re-founded the entire field of cellular automata. What he does in this book is the _other_ thing he's best known for: he shows how to construct the "surreal numbers" (they were actually named by Donald Knuth).
Conway's method employs something like Dedekind cuts (the objects Richard Dedekind used to construct the real numbers from the rationals), but more general and much more powerful. Conway starts with the empty set and proceeds to construct the entire system of surreals, conjuring them forth from the void using a handful of recursive rules.
The idea is that we imagine numbers created on successive "days". On the first day, there's 0; on the next, -1 and +1; on the next, 2, 1/2, -1/2, and -2; on the next, 3, 3/4, 1/4, -1/4, -3/4, and -3; and so on. In the first countably-infinite round, we get all the numbers that can be written as a fraction whose denominator is a power of two (including, obviously, all the whole numbers). We can get as close to any other real number as we like, but they haven't actually been created yet at this point.
But we're just getting started. Once we get out past the first infinity, things really get weird. By the time we're through, which technically is "never," Conway's method has generated not only all the real numbers but way, way, way more besides (including more infinities than you've ever dreamed of).Read more ›
Conway is the most original mathematician on the planet, as well as a remarkably witty and vivid writer, who combines wordplay and logic better than anyone since Lewis Carroll. The book is far too densely packed to summarize in a short review. All I can say is that it's practically inexhaustible; like all good math books, what you get out is proportional to the effort you make while reading it, but the amount of effort it will repay is a hundred times as much as for an ordinary book.
This is an all-time classic, a "desert island book". Even though this new edition differs from the old one in very minor ways, I bought it immediately because my 1978 copy was falling apart from extreme overuse. (My other "desert island math book" is Cohen's "Set Theory and the Continuum Hypothesis".)