On Numbers and Games: A Mathematical Masterpiece by John H. Conway
If you are looking for a book that will challenge your mind, delight your imagination, and introduce you to a new world of numbers and games, then you should definitely check out On Numbers and Games by John H. Conway. This book is not only a brilliant work of mathematics, but also a fascinating journey into the mind of one of the most creative and charismatic mathematicians of the 20th century.
What is On Numbers and Games?
On Numbers and Games, or ONAG for short, is a book that was written by John Horton Conway in 1969, and published in 1976. It is one of those rare publications that sprang to life in a moment of creative energy and has remained influential for over half a century.
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The main idea: surreal numbers and games
The book's main idea is to define a new class of numbers, called surreal numbers, that includes both the real numbers and the infinite ordinal numbers of Georg Cantor. These surreal numbers are obtained by defining numbers as the strengths of positions in certain games, where a game is a situation where two players take turns making moves according to some rules, until one player wins or the game ends in a draw.
For example, Conway defines zero as the game where neither player has any moves, so it is always a draw. He defines one as the game where only the left player has a move, which is to go to zero, so the left player always wins. He defines minus one as the game where only the right player has a move, which is to go to zero, so the right player always wins. He defines half as the game where both players have one move, which is to go to zero, so whoever moves first loses. And so on.
Using this idea, Conway constructs an amazing system of numbers that contains not only all the familiar numbers (integers, fractions, irrational numbers, etc.), but also new infinitely large numbers, infinitesimally small numbers, and all sorts of new numbers in between. He also develops an algebra and an analysis of these numbers, showing how they can be added, subtracted, multiplied, divided, compared, etc.
The structure: two parts, zero and one
The book is divided into two parts: part zero and part one. Part zero deals with numbers only, while part one deals with games in general.
In part zero, Conway introduces his definition of numbers as games, and shows how to construct the surreal numbers using a recursive process based on their "birthdays". He then explores the properties of these numbers, such as their order structure, their field structure, their number theory, their infinitesimals, their asymptotics, etc.
In part one, Conway extends his theory to more general games that are not necessarily numbers. He introduces concepts such as simplifying games, impartial games (where both players have the same moves), nimbers (the values of impartial games), partizan games (where both players have different moves), sums of games (where players can choose which game to play on each turn), temperatures (a measure of how much a game favors one player over another), etc.
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The style: witty, playful, and profound
and more. He was also known for his charisma, humor, inventiveness, and passion for mathematics. He died in 2020, at the age of 82, from complications of COVID-19.
ONAG is one of Conway's most famous and influential books, and it showcases his genius and legacy. It is a book that he wrote for himself, for his own enjoyment and satisfaction, but also for the benefit and delight of others. It is a book that reveals his vision and imagination, his style and elegance, his depth and breadth, his playfulness and seriousness, his curiosity and wisdom.
How can you download On Numbers and Games?
If you are interested in reading ONAG, you have several options to download it. Here are some of them:
The original edition (1976)
The original edition of ONAG was published in 1976 by Academic Press. It is a hardcover book with 242 pages. It is out of print, but you can still find some copies online from various sellers. You can also download a scanned PDF version of the book from this link: [On Numbers and Games (1976)].
The second edition (2000)
The second edition of ONAG was published in 2000 by A K Peters. It is a paperback book with 256 pages. It is an updated and revised version of the original edition, with some corrections, additions, and changes. It also includes a foreword by Richard Guy and an afterword by Donald Knuth. You can buy the book online from various retailers, or download a PDF version of the book from this link: [On Numbers and Games (2000)].
The online resources (PDF, videos, etc.)
There are also some online resources that you can use to learn more about ONAG and its topics. For example, you can watch some videos of Conway himself explaining some of the concepts and results from the book, such as this one: [John Conway: Surreal Numbers - How playing games led to more numbers than anybody ever thought of]. You can also find some lecture notes, slides, articles, and exercises on surreal numbers and games from various sources, such as this one: [Surreal Numbers].
Conclusion
In conclusion, ONAG is a mathematical masterpiece that explores the fascinating world of numbers and games. It is a book that combines rigor and creativity, logic and intuition, theory and practice, fun and beauty. It is a book that will challenge your mind, delight your imagination, and introduce you to a new world of numbers and games.
If you are interested in reading ONAG, you can download it from various sources, such as the original edition (1976), the second edition (2000), or the online resources (PDF, videos, etc.). You can also find more information about ONAG and its author John H. Conway from various websites, books, articles, etc.
We hope that this article has given you a brief overview of ONAG and its topics. We encourage you to download and enjoy the book yourself. We are sure that you will find it rewarding and enlightening.
FAQs
Here are some frequently asked questions about ONAG:
What are surreal numbers?
Surreal numbers are a class of numbers that includes both the real numbers and the infinite ordinal numbers of Georg Cantor. They are defined as the strengths of positions in certain games.
What are games?
Games are situations where two players take turns making moves according to some rules, until one player wins or the game ends in a draw.
Who is John H. Conway?
John H. Conway was one of the most prolific and influential mathematicians of the 20th century. He wrote ONAG in 1969, and published it in 1976.
Why should I read ONAG?
You should read ONAG because it is a classic of recreational mathematics, a source of inspiration and insight for many fields, and a tribute to Conway's genius and legacy.
How can I download ONAG?
You can download ONAG from various sources, such as the original edition (1976), the second edition (2000), or the online resources (PDF, videos, etc.).
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