Surreal Numbers

How two ex-students turned on to pure mathematics and found total happiness

by Donald E. Knuth (Reading, Massachusetts: Addison-Wesley, 1974), vi+119pp.
ISBN 0-201-03812-9
Illustrated by Jill C. Knuth

Czech translation by Helena Nesetrilová, Nadreálná císla, in Pokroky Matematiky, Fyziky a Astronomie 23 (1978), 66--76, 130--139, 187--196, 246--261.
German translation by Brigitte and Karl Kunisch, Insel der Zahlen, (Braunschweig: Friedrich Vieweg & Sohn, 1979), 124pp.
Japanese translation by Junji Koda, Chogen Jis Su (Tokyo: Kaimei Sha Ltd., 1978), 179pp.
Another Japanese translation by Junji Koda, Chogen Jis Su, in Basic Sugaku 1978, no. 8, pages 31--38; 1978, no. 9, 70--74; 1978, no. 10, 44--47; 1978, no. 11, 45--49; 1978, no. 12, 50--54; 1979, no. 1, 61--64; 1979, no. 2, 51--56; 1979, no. 3, 54--59.
Japanese translation by Shunsuke Matsuura, Shifuku no Chogen Jis Su, illustrated by Saito Yusuke (Tokyo: Kashiwa Shobo, 2004), 174pp.
Spanish translation by Lluc Garriga, Números Surreales (Barcelona: Reverté, 1979), 101pp.
Hungarian translation by Jáanos Virágh and Zoltán Ésik, Száamok valóoson innen és túl (Budapest: Gondolat, 1987), 136+ii pp.
Portuguese translation by Jorge Nuno Silva, Números Surreais (Lisbon: Gradiva, 2002), 113pp.
Chinese translation by Bo Gao, Yan Jiu Zhi Mei (Beijing: Publishing House of Electronics Industry, 2011), xii+189 pp.
Russian translation by M. S. Strigunova, Сюрреальные числа (Moscow: Binom, 2014), 110pp.
Italian translation, by Francesco Oliveri, Numeri Surreali (Milan: Franco Angeli, 2016), 110 pp.

A few years ago John Horton Conway of the University of Cambridge hit on a remarkable new way to construct numbers ... Conway explained his new system to Donald E. Knuth, a computer scientist at Stanford University, when they happened to meet at lunch one day in 1972. Knuth was immediately fascinated by its possibilities and its revolutionary content. In 1973 during a week of relaxation in Oslo, Knuth wrote an introduction to Conway's method in the form of a novelette. ... I believe it is the only time a major mathematical discovery has been published first in a work of fiction. ... 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]

The usual numbers are very familiar, but at root they have a very complicated structure. Surreals are in every logical, mathematical and aesthetic sense better. --- Martin Kruskal

[quoted by Robert Mathews in New Scientist, 2 September 1995]

The First Prize in the 1996 Westinghouse Science Talent Search went to Jacob Lurie for his exciting project on computability of surreal numbers. [See Ivars Peterson's MathLand, March 1996; Journal of Symbolic Logic 63 (1998), 337--371.] Lurie read this book!

I am very grateful to Knuth for inventing this name [surreal numbers]---the original of ONAG said “Because of the generality of this Class, we shall simply describe its members as numbers, without adding any restricting adjective.” “Surreal Numbers” is much better! -- John H. Conway

[from the Prologue to the second edition of On Numbers and Games (ONAG)]

Available from the publisher, Addison-Wesley Publishing Company. (Don't believe anybody who tells you that the publishers are out of stock! The twenty-fifth printing was made in February 2020, and the twenty-sixth printing four months later.)

It was surreal.
I don't know what surreal means.
But I heard someone say it once on TV, and it sounded smart.
So this was surreal. -- Charles Barkley

[quoted from the sports page of the International Herald Tribune (17 May 1997)]

Related Books

The story of how Surreal Numbers came to be written is told in Mathematical People by Donald J. Albers and G. L. Alexanderson (Birkhauser Boston, 1985), pages 200--202. John Conway's official presentation of the theory appears in his incredible book On Numbers and Games; the second edition has been published by A K Peters, Ltd (2001). See also the second edition of Winning Ways for your Mathematical Plays by Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy, Volume 1 (2001), Volume 2 (2003), Volume 3 (2003), Volume 4 (2004).

Many other excellent books and webpages about surreal numbers have also been prepared, easily findable by your favorite search engine.


For a list of corrections to all known errors in the first fourteen printings of this book, you may download either the errata file in plain TeX format (2338 bytes) or the errata file in DVI format (3456 bytes) or the errata file in compressed PostScript format (21801 bytes); the latter files were generated by the TeX file.

The 15th printing (November 2003), the 16th printing (April 2005), the 17th (April 2006), the 18th (March 2007), the 19th (January 2011), and subsequent printings up to the 25th (February 2020) contain only the errors listed below, as far as I know:

page 11, line 1 (23 August 2008)
change "number to work with" to "numbers to work with"
page 11, line 8 from the bottom (17 April 2020)
change "0 is not greater than" to "0 is not greater than or equal to"
page 12, line 2 from the bottom (27 May 2020)
change "his notation" to "that notation"
page 69, lines 5 and 6 (19 January 2021)
change "which contains" to "the set of all negatives of $X_L$'s elements. So $-(X_L)$ contains
page 73, line 4 (25 June 2009)
change "What are your thinking" to "What are you thinking"
page 87, line 5 (18 September 2010)
change "Look IV" to "Look, IV"

I will gratefully deposit 0x$1.00 ($2.56) to the account of the first person who finds and reports anything that remains technically, historically, typographically, or politically incorrect. Please send suggested corrections to, or send snail mail to Prof. D. Knuth, Computer Science Department, Gates Building 4B, Stanford University, Stanford, CA 94305-9045 USA. I may not be able to read your message until many months have gone by, because I'm working intensively on The Art of Computer Programming. However, I promise to reply in due time.

DO NOT SEND EMAIL TO KNUTH-BUG EXCEPT TO REPORT ERRORS IN BOOKS! And if you do report an error via email, please do not include attachments of any kind; your message should be readable on brand-X operating systems for all values of X.

Answer Sheet

Page 118 of Surreal Numbers promises that “I will send hints to the solutions of exercises 9, 19, and 22 to any bona fide teachers who request them by writing to me at Stanford University.”

That promise is still valid, but anyone can now also download either the hint file in plain TeX format (1144 bytes) or the hint file in DVI format (2176 bytes) or the hint file in compressed PostScript format (13571 bytes) via the World Wide Web.

Don Knuth's home page

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