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Surreal Numbers 1st Edition
Purchase options and add-ons
- ISBN-100201038129
- ISBN-13978-0201038125
- Edition1st
- PublisherAddison-Wesley Professional
- Publication dateJanuary 1, 1974
- LanguageEnglish
- Dimensions8.18 x 5.36 x 0.35 inches
- Print length128 pages
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Editorial Reviews
From the Back Cover
Nearly 30 years ago, John Horton Conway introduced a new way to construct numbers. Donald E. Knuth, in appreciation of this revolutionary system, took a week off from work on The Art of Computer Programming to write an introduction to Conway's method. Never content with the ordinary, Knuth wrote this introduction as a work of fiction--a novelette. If not a steamy romance, the book nonetheless shows how a young couple turned on to pure mathematics and found total happiness.
The book's primary aim, Knuth explains in a postscript, is not so much to teach Conway's theory as "to teach how one might go about developing such a theory." He continues: "Therefore, as the two characters in this book gradually explore and build up Conway's number system, I have recorded their false starts and frustrations as well as their good ideas. I wanted to give a reasonably faithful portrayal of the important principles, techniques, joys, passions, and philosophy of mathematics, so I wrote the story as I was actually doing the research myself."... It is an astonishing feat of legerdemain. An empty hat rests on a table made of a few axioms of standard set theory. Conway waves two simple rules in the air, then reaches into almost nothing and pulls out an infinitely rich tapestry of numbers that form a real and closed field. Every real number is surrounded by a host of new numbers that lie closer to it than any other "real" value does. The system is truly "surreal." quoted from Martin Gardner, Mathematical Magic Show, pp. 16--19Surreal Numbers, now in its 13th printing, will appeal to anyone who might enjoy an engaging dialogue on abstract mathematical ideas, and who might wish to experience how new mathematics is created.
0201038129B04062001
About the Author
Donald E. Knuth is known throughout the world for his pioneering work on algorithms and programming techniques, for his invention of the Tex and Metafont systems for computer typesetting, and for his prolific and influential writing. Professor Emeritus of The Art of Computer Programming at Stanford University, he currently devotes full time to the completion of these fascicles and the seven volumes to which they belong.
Product details
- Publisher : Addison-Wesley Professional; 1st edition (January 1, 1974)
- Language : English
- Paperback : 128 pages
- ISBN-10 : 0201038129
- ISBN-13 : 978-0201038125
- Item Weight : 6.3 ounces
- Dimensions : 8.18 x 5.36 x 0.35 inches
- Best Sellers Rank: #50,319 in Books (See Top 100 in Books)
- #5 in Number Theory (Books)
- #5 in Technical Writing Reference (Books)
- #518 in Engineering (Books)
- Customer Reviews:
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About the author

Donald E. Knuth was born on January 10, 1938 in Milwaukee, Wisconsin. He studied mathematics as an undergraduate at Case Institute of Technology, where he also wrote software at the Computing Center. The Case faculty took the unprecedented step of awarding him a Master's degree together with the B.S. he received in 1960. After graduate studies at California Institute of Technology, he received a Ph.D. in Mathematics in 1963 and then remained on the mathematics faculty. Throughout this period he continued to be involved with software development, serving as consultant to Burroughs Corporation from 1960-1968 and as editor of Programming Languages for ACM publications from 1964-1967.
He joined Stanford University as Professor of Computer Science in 1968, and was appointed to Stanford's first endowed chair in computer science nine years later. As a university professor he introduced a variety of new courses into the curriculum, notably Data Structures and Concrete Mathematics. In 1993 he became Professor Emeritus of The Art of Computer Programming. He has supervised the dissertations of 28 students.
Knuth began in 1962 to prepare textbooks about programming techniques, and this work evolved into a projected seven-volume series entitled The Art of Computer Programming. Volumes 1-3 first appeared in 1968, 1969, and 1973. Having revised these three in 1997, he is now working full time on the remaining volumes. Volume 4A appeared at the beginning of 2011. More than one million copies have already been printed, including translations into ten languages.
He took ten years off from that project to work on digital typography, developing the TeX system for document preparation and the METAFONT system for alphabet design. Noteworthy by-products of those activities were the WEB and CWEB languages for structured documentation, and the accompanying methodology of Literate Programming. TeX is now used to produce most of the world's scientific literature in physics and mathematics.
His research papers have been instrumental in establishing several subareas of computer science and software engineering: LR(k) parsing; attribute grammars; the Knuth-Bendix algorithm for axiomatic reasoning; empirical studies of user programs and profiles; analysis of algorithms. In general, his works have been directed towards the search for a proper balance between theory and practice.
Professor Knuth received the ACM Turing Award in 1974 and became a Fellow of the British Computer Society in 1980, an Honorary Member of the IEEE in 1982. He is a member of the American Academy of Arts and Sciences, the National Academy of Sciences, and the National Academy of Engineering; he is also a foreign associate of l'Academie des Sciences (Paris), Det Norske Videnskaps-Akademi (Oslo), Bayerische Akademie der Wissenschaften (Munich), the Royal Society (London), and Rossiiskaya Akademia Nauk (Moscow). He holds five patents and has published approximately 160 papers in addition to his 28 books. He received the Medal of Science from President Carter in 1979, the American Mathematical Society's Steele Prize for expository writing in 1986, the New York Academy of Sciences Award in 1987, the J.D. Warnier Prize for software methodology in 1989, the Adelskøld Medal from the Swedish Academy of Sciences in 1994, the Harvey Prize from the Technion in 1995, and the Kyoto Prize for advanced technology in 1996. He was a charter recipient of the IEEE Computer Pioneer Award in 1982, after having received the IEEE Computer Society's W. Wallace McDowell Award in 1980; he received the IEEE's John von Neumann Medal in 1995. He holds honorary doctorates from Oxford University, the University of Paris, St. Petersburg University, and more than a dozen colleges and universities in America.
Professor Knuth lives on the Stanford campus with his wife, Jill. They have two children, John and Jennifer. Music is his main avocation.
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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 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.
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.
For anybody interested in pure mathematics and set theory, this is a book that should sit on anyone's bookshelf. It certainly has its permanent place on mine.
Top reviews from other countries
The book is not easy: it is challenging and it motivated me buy a new notebook and work through the proofs: I'm actually studying it.
Plus, it is very well written.
Wer sich schon einmal Gedanken gemacht hat über "was kommt nach Unendlich" findet hier nicht nur Antworten, sondern wird auch unterhaltsam und mit viel Bedacht auf den Lernprozess hingeführt. Es ist eine spannende Reise, auf die man sich begiebt, und die Entdeckungen unterwegs sind wirklich wunderbar. Gibt es sowas wie "zwei mal Unendlich"? Was kommt heraus, wenn man zwei mal unendlich mit 0 multipliziert?
Surreale Zahlen als ein Zugang zu Non-Standard-Analysis sind eine spannende Beschäftigung.
Ich halte das Buch auch für Jugendliche geeignet, soweit sie ein Grundverständnis für die Konstruktion von Zahlenkörpern haben. Donald Knuth bietet auch Erklärungen und Anregungen für den Unterricht, und weiterführende Fragen, die bald klar machen, welches wunderbares und mächtiges Feld Surreale Zahlen aufmachen.
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 as
0+1 = 1. The system you end up with will blow your mind.
Knuth, computer scientist par excellence, has done an excellent job.






