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Surreal Numbers 1st Edition

3.2 out of 5 stars 9 customer reviews
ISBN-13: 978-0201038125
ISBN-10: 0201038129
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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.


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.


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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: 5.4 x 0.4 x 8.2 inches
  • Shipping Weight: 6.4 ounces (View shipping rates and policies)
  • Average Customer Review: 3.2 out of 5 stars  See all reviews (9 customer reviews)
  • Amazon Best Sellers Rank: #49,912 in Books (See Top 100 in Books)

Customer Reviews

Top Customer Reviews

By Shlomo Yona on 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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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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Format: Paperback Verified Purchase
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.
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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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