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Surreal Numbers [Paperback]

by Donald E. Knuth
3.0 out of 5 stars  See all reviews (8 customer reviews)

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Book Description

January 11, 1974 0201038129 978-0201038125 1
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--19 Surreal 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

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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--19

Surreal 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

  • Paperback: 119 pages
  • Publisher: Addison-Wesley Professional; 1 edition (January 11, 1974)
  • Language: English
  • ISBN-10: 0201038129
  • ISBN-13: 978-0201038125
  • Product Dimensions: 8.2 x 5.4 x 0.4 inches
  • Shipping Weight: 6.4 ounces (View shipping rates and policies)
  • Average Customer Review: 3.0 out of 5 stars  See all reviews (8 customer reviews)
  • Amazon Best Sellers Rank: #540,465 in Books (See Top 100 in Books)

More 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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33 of 36 people found the following review helpful
5.0 out of 5 stars Couldn't put it down. November 12, 2001
Format:Paperback
This little book, written as a "novel", actually tries to show us that each of us is actually able to be an amature mathematician, and that "pure mathematics" is not that complicated once you get down to the rules.
For readers familiar with group theory notations, this is an easy and fun read.
Byeond the superlatives given all over to the nice and simple manner in which the number system is built in front of our eyes, I would also like to add I have noticed some ideas Knuth wanted the readers to absorb by reading this book of his:
* People too much into civilization need time off to "rest".
* After a long while of "resting" people need brain stimulations.
* The joy and interest in mathematics comes with the discovery, or at least after trying the best you can. Only then can you appreciate what others did in mathematics.
* Teachers in schools would rather tell you about math, and make you takes exams, and will not encourage creativity. This results in that only in graduate school are people allowed (and demanded) to start creating things of their own.
* Solving good math puzzles or solving any problem, is satisfying, and makes you horny!
* definitions proofs to theorems and ideas should be expressed as simple as possible, and they can always be expressed in a simple way.
I could go on with more ideas Knuth wanted to pass to the readers...
I read the book in one time, not putting it down for a minute. The flow of ideas and progress in building the number system (up to the pseudo-numbers) is clear and fun. I actually felt as if I was discovering things myself.
There is a lot which can be "further probed" after readnig the book, and Knuth appeals to teachers to gives seminars based on this text, and guides them how he would want those seminars to be like.
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8 of 8 people found the following review helpful
Format:Paperback
First of all, a word of advice for the future readers of this book. Do not read it for its story. From the literary point of view, it's bad. Perhaps the only type of reader that will benefit or enjoy this book is the mathematical one.

In this book, you will find an exposition of a construction of a special number system (formally, a proper class of number systems). However, this exposition does not follow the formal or even traditional method employed in most mathematics books. It is told in form of a story. Two characters find a stone inscribed with the axioms of the construction of some "surreal numbers" and spend the whole book thinking what these axioms mean in some intuitive way.

In a mathematician's perspective (rather, my own), it is very entertaining. The characters' point of view is just as that of two mathematicians talking about some problem. And the construction is very interesting from a mathematician's point of view. So, yet again, for a mathematician, it will be like listening to two colleagues talking about some problem. It also has another element cooked for mathematicians: it has a small discussion about the fact that mathematical thinking is not taught until graduate school.

In conclusion, this book is a book about advanced mathematics written in a funny style. Do not expect the story to be good in a literary sense.
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14 of 18 people found the following review helpful
Format:Paperback
This book, written in Knuth's classic style, employs a unique dialog to guide the reader through the derivation of the fascinating mathematical topic of surreal numbers. Its short length and humor makes it a must for any math fan interested in the methods used for deriving new concepts in math, and the exercises included make it a useful book for math teachers interested in giving something new to their students. All said, a lovely book.
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41 of 59 people found the following review helpful
Format:Paperback
Stanford mathematician D. E. Knuth, in his slim volume Surreal Numbers, attempts to impart to the reader the notion of surreal numbers by way of a very unusual tactic: the dialogue. The book has two characters-a man and a woman-that, as a couple, have left Western society to find peace and their inner selves on a beach in India. After several months of swimming and picking berries, their intellectual needs begin to weigh heavy on them. In other words, they become bored. As luck would have it, their boredom is halted by the discovery of a stone tablet on the beach near their camp. The tablet reads, in Hebrew, "In the beginning, everything was void, and J. H. W. H. Conway began to create numbers." The tablet continues, giving the basic axioms that serve as basis for the creation of surreal numbers. The rest of the dialogue consists of our lovely couple discovering theorems and properties of surreal numbers using the axioms from the stone tablet. We see them take many wrong paths in their journey, only to realize and correct their errors in moments of sudden and poorly explained revelation. ....
To the math buff, I recommend studying up on these magnificent numbers. However, I do not recommend you use the present text to do so. It does little justice to the beauty of surreal numbers, and does even less in its explanation of their properties. The intention of the author, as he states in the postscript, is to present a math text in such a way that the reader not only learns of the topic discussed, but participates in its development. He sees the common math text as a dry conveyer of theorems and proofs that hides the intriguing and moving path of discovery that resulted in these theorems and proofs. He seeks, in Surreal Numbers, to write a sort of antithesis to this type of book.
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