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Surreal Numbers

3.86  ·  Rating details ·  394 ratings  ·  42 reviews
Shows how a young couple turned on to pure mathematics and found total happiness. This title is intended for those who might enjoy an engaging dialogue on abstract mathematical ideas, and those who might wish to experience how new mathematics is created.
Paperback, 128 pages
Published January 1st 1974 by Addison-Wesley Professional
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Feb 19, 2020 rated it really liked it
Also Sprake JHWH

All words in a language, all symbols, all operations that can be performed in the language are contained only within the language itself. That is to say, languages are self-defined; their elements are constituted by other elements of the language not by not-words outside the language. Language is therefore a circular affair. Or, more optimistically, language is helical. It refers to itself endlessly and gets more expressive as it does so as it builds upon itself.

Expressive of wha
Sep 16, 2013 rated it really liked it
Good introduction to the Surreals and overall a good description of the way new mathematical ideas are developed. I like that Knuth decided to stick with Conway's construction of the surreals as cuts between sets, as opposed to other approaches like Gonshor's ordinal-length sequences of +'s and -'s, since this approach makes it easier to build intuition about many important ideas used in later proofs.

The characterization is weak, but that hardly matters. It's a nice attempt to both humanize math
Mar 31, 2017 added it
Shelves: science
Okay, if you don't rack your brain too much with the dialogues of mathematical explanations & derivations, then I guess it's for fun !
The book was about how students should be taught to learn the interesting aspects of the problem rather than to solve it first, how they could train their mind to learn the creative aspects of proofs rather than the proof itself given in textbook.
But after being in a system of 17 year's of examsmanship, I would say I enjoyed his concrete mathematics book better th
Abdullah Ghareeb
May 22, 2017 rated it really liked it
mathematical & mind challenging at the first place which makes it even more exciting and interesting .
actually , I haven't put so much concentration and thought in a book or a novel since Islam between the east and the west which I have not finished so far . but anyway that's not something weird knowing that Donald Knuth of the art of programming is the same gut who wrote it ^_^ it actually adds up :D
Roberto Rigolin F Lopes
Jul 09, 2017 rated it it was amazing
Aww, this is so cute. Knuth is selling us number theory within a romantic plot. This is about a couple having fun in a beach discovering the fundamental laws of numbers. They even discuss why mathematics was profoundly boring at school but so exciting now; specially figuring things out by themselves. Knuth is therefore writing to young mathematicians igniting their curiosity by showing off the pleasures/frustrations of independent work. Be prepared to fire up a great deal of your 100 billion neu ...more
Apr 27, 2021 rated it really liked it
So glad I came across this. Berlekamp & Conway's series on Winning Ways for your Mathematical Plays has been staring at me for years. But Conway works heavily with the concept of surreal numbers there, with hardly any explanation. So, maybe this will be the breakthrough.

Knuth suggests this book for an undergraduate seminar, but I found it above that level. More than once I had to put the book aside, once for over a month, before taking it up again. I worked through every point quite thoroughly,
Aug 19, 2021 rated it liked it
I worked along with it in the beginning and then kinda stopped, but working through the math is obviously the best way to read this book - it's certainly not for the dumb romance! ...more
Paweł Petecki
Mar 15, 2022 rated it really liked it  ·  review of another edition
Great for undergraduate math students.

For someone just interested in mathematics story is nice, but the details of the proofs might be too technical.
Sep 19, 2018 rated it it was amazing
This is one of those books that will blow your mind. Although it is a work of fiction it served as the introduction to the world of John Conway's surreal numbers. This is a small book, barely a novelette, I read it in less than an hour.

The entire book is presented as a dialog between a couple apparently stranded on an island. They find an ancient rock inscribed with:

"In the beginning, everything was void, and J. H. W. H.
Conway began to create numbers. Conway said, "Let
there be two rules which b
Ravindra Pai
May 12, 2018 rated it it was amazing
Knuth explains Number theory with a romantic plot as a conversation between two lovers. The book has many wordplays and very enjoyable read, proving many basic proofs.

It's the idea that counts.

All we have is a bunch of objects ordered neatly in a line,
but we haven't got anything to do with them.

I guess the excitement and the beauty comes
in the discovery, not the hearing.

Rubbish. Wait until you get to infinite sets.

What a miserable night! I kept tossing and turning, and my
mind was racing in circ
Andrew Litfin
Mar 13, 2016 rated it really liked it
From the standpoint of being a mathematical text, this book is awful.
Fortunately, that is not at all the point of this novellette, nor does it pretend in any capacity that it is the point. Knuth states, in no uncertain terms, that the book is designed to give the impression of what it is like to do research-level mathematics, where the answers to questions are totally unknown, and there are no resources to research from. Everything must be tried, and sometimes failure is inevitable. It is in thi
Mar 05, 2018 rated it liked it
There are a lot of numbers in this book. Enough so that I panicked when I opened it, and wondered what I'd just gotten myself into. I am not good with numbers.

I stuck it out.

Knuth - anyone who knows him will attest to this - is good at what he does. Even for someone (me) whose last year of formal math was grade 11, many many years ago, the book was a pleasure. I followed the logic, if not the notation, without too painful an effort (though it was definitely an effort). And the payoff was easil
Dec 27, 2010 rated it liked it
I have to admit that I didn't take the time to follow the math in this book. However, the concept it introduces is interesting. The book introduces the concept of "extraordinal" numbers - a number system in which every number is represented by a set of two sets. The system is interesting not only because it can describe real, rational, and irrational numbers, but because it can give concrete answers to questions like "what is infinity times infinity?" I'll admit that some of it went over my head ...more
Jan 02, 2011 rated it liked it
Shelves: books-i-own
Yes, this is a pretty accurate portrayal of how math and research is done. (minus the beach and kissing.) It does a decent job of portraying how enjoyable it can be to explore and figure things out for yourself, and the author correctly points out that this is sort of "how it should be done," when it comes to education. (If only, if only.) But reading about it just doesn't capture the magic of doing it yourself. So I'm not sure if this book really did what it was supposed to--it was like watchin ...more
Feb 05, 2013 rated it really liked it
I found this book very endearing; a lovely fusion of math and (an admittedly skeletal) story. If you enjoy doing math proofs, I recommend reading this book. I wish there were more math books that attempted a similar structure. If you aren't interested in math at all, unfortunately, I have to recommend that you skip this book. There isn't enough body to the story to make it an entertaining read. It's worth noting that Knuth suggested this book as a text for a course on "mathematical creativity". ...more
Sep 25, 2019 rated it it was amazing
I loved this book. I have worked many hours on the math in it, and could work many more before finishing it. You have to be comfortable with logic and abstraction to do the work suggested by Alice and Bill's conversations.

Surreal numbers (aka hyperreals) are the basis for non-standard analysis, and I get to tell my calculus students about that.

I also love books that combine mathematical work with a story, no matter how simple.
Jan 23, 2014 rated it liked it
If you want to learn the surreal numbers, there are better ways, and if you want to learn to think mathematically, there are better ways. This book does not succeed at the author's goal of inspiring creativity in the reader. However, it was worth my time, for the thing Knuth was not emphasizing: it is a sufficiently complete and mildly entertaining description of the surreal numbers. ...more
Leo Ferres
Aug 05, 2018 rated it liked it
This is by my intellectual crush, Don Knuth. I understand where he's coming from, but wonder if the maieutic method would really work her. In any case, I should go back to it in a more leisurely way, but for now, it's definitely entertaining. I wonder hoe many things can be done in this way to popularize science this century. ...more
Sep 27, 2018 rated it liked it
A beautifully written (using LaTex, of course) novelette that showcases Alice and Bill, a couple in an island vacationing with enough time to exercise the mind in a mathematical “mistery”.

It is playful and invites readers to follow the steps of Alice and Bill and play with mathematical proofs from scratch, creating a whole universe day by day.
Jun 25, 2015 rated it it was amazing
Amazing story, it's hard to understand it all, I fail to understand proofs since second half of it. But I highly reccomend to read it. Narrative styl is funny and unique and it gives you many interesting information about Conaways amazing way how to define numbers. ...more
Joshua Lin
Aug 25, 2016 rated it liked it
The math was ok
But the story and background dialogue of A and B's romantic relationship was passable at best
Unless there's like some hidden message in their quick romantic quips, it wasn't really endearing to me
Arttu Ojanperä
Aug 05, 2018 rated it liked it
An entertaining little novella about theoretical mathematics and the joy of discovery. Recommended for mqth students or people otherwise familiar with the concepts of set theory, algebra etc. Others are bound to find this book a bit challenging.
Nov 13, 2018 rated it it was ok
Just the Maths would have helped a lot more than injecting it with a fictitious plot. It makes it way too verbose and I lost all my interest by the third chapter. It didn’t work for me, may be other night like it.
Apr 30, 2015 rated it really liked it
One of the most amazing abstract algebra childrens book I've ever read! ...more
Ian Schiffman
Jun 14, 2021 rated it really liked it
"I have prepared 'Surreal Numbers' as a mathematical dialogue of the 1970's, emphasizing the nature of creative mathematical explorations. Of course, I wrote this mostly for fun, and I hope that it will transmit some pleasure to its readers, but I must also admit that I also had a serious purpose in the back of my mind. Namely, I wanted to provide some material which would help to overcome one of the most serious shortcomings in our present education system, the lack of training for research wor ...more
Mark West
May 28, 2022 rated it liked it
Shelves: physical, nonfiction
While short, this is great insight into the process of discovery, mathematical or otherwise. While not something I've experienced much myself, this process I feel a deep connection to, and I such I feel that connection to this book. That said, there are many details glossed over as the characters intuit their findings, which got a little old after a while. This is meant to encourage the reader to take the book slowly as to follow the characters' deductions, but it mostly just made me not pay att ...more
Jake McGuffie
Dec 26, 2020 rated it really liked it
really great take on the process of developing theory, including all the frustrations and annoyances that made my university life hell at times. pieces of indecipherable maths are ironically included (usually followed by one of the characters remarking how obtuse it all is) which was fun and made me feel less alone in my experience as a maths student, but still, some parts of the theory remained unilluminated by the end in my opinion. will return to this lots and lots but not enough to discover ...more
Sep 01, 2020 rated it really liked it
A great read for recreational mathematics nuts out there. ;) You'll definitely like this theory and the numbers that come out of it better than the reals or complex numbers you learned in school. It's relatively concise and definitely consistent and lets you handle infinities and infinitesimals easily and in a way that makes sense. And Knuth is right about it being necessary to teach creativity in math from the beginning.. why wait for the fun until the last years of graduate school?! ...more
Jul 06, 2022 rated it really liked it
Great style and such an interesting topic. Not only writes about the numbers, but the process of discovery and exploration. Starts from scratch, so no prior knowledge is needed, just some curiosity. I'd highly suggest working through each proof by yourself especially those that aren't derived in the book. If I was to read it again, I would try to only read a chapter a day, and really try to predict what will follow in the next day. ...more
Anthony O'Connor
Jul 13, 2020 rated it it was amazing
A masterly introduction to one of the non standard number systems providing some real insight into what they are and how they work.
Written in the form of a dialogue between two enthusiasts who just happen to be stranded on an island. Full of acerbic comments on the nature of forced education and solid thoughts on how best to learn and do math in general.
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Donald Ervin Knuth, born January 10th 1938, is a renowned computer scientist and Professor Emeritus of the Art of Computer Programming at Stanford University.

Author of the seminal multi-volume work The Art of Computer Programming ("TAOCP"), Knuth has been called the "father" of the analysis of algorithms, contributing to the development of, and systematizing formal mathematical techniques for, the

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