Surreal Numbers. Get Books. Nearly 30 years ago, John Horton Conway introduced a new way to construct numbers. Never content with the ordinary, Knuth wrote this. An Introduction to the Theory of Surreal Numbers. These notes provide a formal introduction to the theory of surreal numbers in a clear and lucid style. Foundations of Analysis over Surreal Number Fields.
In this volume, a tower of surreal number fields is defined, each being a real-closed field having a canonical formal power series structure and many other higher order properties. Skip to content. Toggle navigation. 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 guess the excitement and the beauty comes in the discovery, not the hearing. Wait until you get to infinite sets. What a miserable night! I kept tossing and turning, and my mind was racing in circ Knuth explains Number theory with a romantic plot as a conversation between two lovers.
I kept tossing and turning, and my mind was racing in circles. I dreamed I was proving things and making logical deductions, but when I woke up they were all foolishness. Maybe this mathematics isn't good for us after all. We were so happy yesterday. Well, when we were going around in circles like this before, how did we break out? The main thing was to use induction, I mean to show that the proof in one case depended on the truth in a previous case, which depended on a still previous case, and so on, where the chain must eventually come to an end.
Mar 13, Andrew Litfin 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 From the standpoint of being a mathematical text, this book is awful. It is in this context that the book shines. Despite being short enough to be read in an afternoon, one comes away thinking that they could indeed be Alice or Bill, if only they were brave enough to play around with the rules, just to see what pops out.
If you are looking for a more formal, complete introduction to the theory of surreal numbers, read "On Numbers and Games," by John Conway yes, the same "J. Conway" named in this book! Mar 05, Clove 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 There are a lot of numbers in this book.
And the payoff was easily worth it. If I'd had teachers like him back then, I might've taken my math pursuits further! Hit the spot. Dec 27, Alex 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?
Jan 02, J. Yes, this is a pretty accurate portrayal of how math and research is done. 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 Yes, this is a pretty accurate portrayal of how math and research is done.
So I'm not sure if this book really did what it was supposed to--it was like watching people have fun. It made me want to have the fun, but it wasn't particularly fun in itself. View 2 comments. Feb 05, Julian rated it really liked it Shelves: mathematics , allegory , inspirational. 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".
Sep 25, Sue 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, George 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. Aug 05, Leo Ferres 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. Sep 27, Manuel rated it liked it. 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, Dmk 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. Yet even with such groundbreaking findings and a wealth of popular-level.
From Antiphilosophy to Worlds and from Beckett to Wittgenstein, the entries in this dictionary provide detailed explanations and engagements with Badious's key concepts and major interlocutors. The 18 revised full papers and 19 invited papers and invited extended abstracts were carefully reviewed and selected from 40 submissions. The conference CiE has six special sessions — two sessions, cryptography. This book offers an introduction to modern ideas about infinity and their implications for mathematics.
It unifies ideas from set theory and mathematical logic, and traces their effects on mainstream mathematical topics of today, such as number theory and combinatorics. An Introduction to the Theory of Surreal Numbers. Surreal Numbers by Donald Ervin Knuth. On Numbers and Games by John H.
Number and Numbers by Alain Badiou. Hypernumbers and Extrafunctions by Mark Burgin.
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