Surreal Numbers

In the early 1970s, mathematician John Conway and computer scientist Donald Knuth had lunch together, during which Conway told Knuth of a way to generate all numbers from a couple of rules.What is a number?Everyone understands what one apple means. We also all understand that if Bill starts with three apples and gives one of those apples to Alice, he will have two apples left. But what are those things we all understand to be "one" or "two" or "three"? They are abstract objects and in modern mathematics we build numbers using set theory. We start with the empty set, then we create a set that contains the empty set, and a set that contains that set, and so on. The empty set is "zero", the set containing the empty set is "one", the set containing one is "two". Each set created this way has a successor set and together they form the Natural numbers. We now have 0,1,2,3,...We create the Integers by giving each Natural number except zero a negative version. We now have 0,1,-1,2,-2,...From the the set of Integers, we create the set of all ordered pairs (a,b) where a is any integer and b is any integer except zero. This gives us all fractions: 1/2, 3/5. We can reduce ordered pairs to simpler ones if they have common factors: 3/3 is the same as 1 while 96/15 is the same as 6 and 2/5. Because they are a ratio of two integers, we call them the Rational numbers.It was a big disappointment for the Greeks to find that these numbers did NOT correspond to every point on the line. All the rational numbers are indeed ON the line but there are points on the line that are NOT fractions--for example the square root of two. This unsatisfactory situation endured until the 19th century when the Real numbers were created from a specific kind of subset of the Rationals called "cuts".So from the empty set, we get the natural numbers, then from those we get the integers, then from those we build the rationals and finally we get the reals. That's four levels of construction.Amazingly John Conway invented a way to get ALL the numbers in one go, in a single level of construction. Conway came up with two rules that yield all the numbers on the real line by starting from the empty set and proceeding by iteration. As a bonus, these two rules also generate infinitesimals and transfinite cardinals. Infinitesimals are numbers greater than zero but smaller than all the non-zero positive real numbers, while transfinite cardinals are numbers that characterize different orders of infinity.Donald Knuth jumped at the chance to use the topic to illustrate how much fun doing mathematics can be. He thought Conway's numbers would make an excellent basis for a story about two students working out how to generate the numbers from Conway's two rules and proving many useful theorems along the way. Knuth came up with the name Surreal Numbers (Conway referred to them just as "numbers") because they are in fact more than the Real numbers and yet they are generated using a simpler set of rules. Surreal!Knuth set his story on an exotic island where the two students, Alice and Bill, discover a stone inscribed with the two rules and a short explanation of how to generate zero, one and minus one. From that starting point, Alice and Bill figure out how to work out all the numbers, and also how to add, subtract and multiply them. (SPOILER ALERT) The experience of working together convinces them that they should get married.As far as dramatic literature goes, this isn't anything impressive. Calling the dialogue silly or corny would be generous. But following the math part of the novelette does effectively convey how it feels to work out mathematical theories for oneself and it will show the interested reader just how much fun he or she can have working out theorems for themselves.Vincent Poirier, Montreal

I found this book to be super interesting. I really enjoy math, although I have come to that late in life and am not very good at it. I just finished reading this fairly quickly and am about to start again at the beginning and take it more slowly this time. I like the emphasis on logical development and proofs and the way Knuth returns to the same topics later to identify the weak points that can be further refined. Knuth is trying to help us develop an intuitive understanding of Conway's amazing discovery/invention but more importantly show us how math is developed rather than just presenting it as a finished product. He makes the material easy to read without even requiring full comprehension which is quite a trick. That is not easy to do!I don't understand the other reviewers negative comments about the "story" or the references to food and sex. Just to be clear, there are no explicit references to sex in this book. There are explicit references to eating but hopefully that won't bother most people. The non-math dialog is very brief, serving as a gentle way to open and exit each small chapter and providing a simple context for a conversation about the mathematical concepts.The purpose of this truncated character and story development is to make the text more accessible to sophomore math students and it works perfectly. I suppose the people who are bothered by this prefer their math straight-up. I can see how a competent mathematician would be annoyed by these brief digressions but this book is not for them. Knuth discusses this in the book's postscript where he points out that the book is targeted to the college sophomore level and he decries the teaching of math concepts in the form of finished products as a major shortcoming of our current education system.I would give this book 6 stars if I could.

Donald Knuth's Surreal Numbers is a small little book telling the story of two people discovering John Horton Conway's surreal numbers. They discover them little by little and through dialog create a mathematical proof for the number system. At times, they go in the wrong direction, at times they revert, but gradually they discover more and more math.The math is interesting, although, towards the end, beyond my basic reading capabilities. But the most interesting aspect of the book (which makes it a 4-star book) is that it tries to express the beauty of math. The people are engaged in a puzzle and are thoroughly enjoying that. As a reader, you feel their excitement. The book is small but managed to express the joy of solving math puzzles. Well done. Recommended for people interested in math.

This was very enjoyable. It is short, but intense. The story is entirely dialogue between two 'characters' and is a very good exploration of how these mathematical concepts work. It is very much a discovery and that is the enjoyable charm of the book. I recommend this for math lovers as well as those who are looking to improve their own learning techniques and approaches. Similar to Feynman, take some time to figure it out for yourself!

Awesome book. Just awesome!

Fantastic !

Best book ever

Conway's system of surreal numbers is one of the most brilliant creations of Mathematics. The system adds to the familiar numbers a vast family of infinite and infinitesimal numbers. It allow you to add, subtract, multiply and divide numbers in this collection, and also to find such things as their seventh roots. The system is amazingly rich.Knuth imagines two young lovers in the future finding the first clues to the system on stone tablets, and then shows how, from these clues, they begin to reconstruct Conway's system. It gives a feel for what research in mathematics is like. Sometimes, our two lovers make errors and have to retrace some of their steps.To be sure, you need to think hard to get the best out of the book. You must not feel discouraged that you have to spend a long time proving such things as0+1 = 1. The system you end up with will blow your mind.Knuth, computer scientist par excellence, has done an excellent job.

I've always been a big fan of Knuth. I think it's partly because of his insistence on the aesthetics of maths and not just formal correctness. I came to this book thinking that I was going to learn some new number theory. I guess I did - but this book is *really* about how fun it is to discover and *prove* mathematical concepts for yourself. The book was apparently written in about a week and developed as Knuth discovered the ideas for himself. The material is not very hard, and is probably worth reading through quickly at first and then going back to later, to try your own hand at proving some of the basic properties. My only complaint was it all finishes too soon!