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Surreal Numbers epub download

by Donald E. Knuth


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.

In stock on December 31, 2018. 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.

Knuth Surreal Numbers Addison-Wesley Publishing Company Inc. 1974 Acrobat 7 Pdf 1. Mb. Scanned by artmisa using Canon DR2580C + flatbed option. Knuth Surreal Numbers Addison-Wesley Publishing Company Inc.

Поиск книг BookFi BookSee - Download books for free Ronald L. Graham, Donald E. Knuth, Oren Patashnik.

Поиск книг BookFi BookSee - Download books for free. Surreal numbers: how two ex-students turned on to pure mathematics and found total happiness : a mathematical novelette. Ronald L.

Nearly 30 years ago, John Horton Conway introduced a new way to construct numbers

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.

Surreal Numbers book.

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.

Donald E. Knuth - Surreal Numbers. Knuth - Surreal numbers: how two ex-students turned on to pure mathematics and found total happiness : a mathematical novelette.

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.

Surreal Numbers epub download

ISBN13: 978-0201038125

ISBN: 0201038129

Author: Donald E. Knuth

Category: Reference

Subcategory: Writing Research & Publishing Guides

Language: English

Publisher: Addison-Wesley Professional; 1 edition (January 11, 1974)

Pages: 119 pages

ePUB size: 1483 kb

FB2 size: 1546 kb

Rating: 4.9

Votes: 825

Other Formats: txt lrf docx lit

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I ℓ٥ﻻ ﻉ√٥υ
In the early 1970s, mathematician John Conway and computer scientist Donald Knuth had lunch together, during which Conway told Knuth of a way to generate all numbers from a couple of rules.

What is a number?

Everyone understands what one apple means. We also all understand that if Bill starts with three apples and gives one of those apples to Alice, he will have two apples left. But what are those things we all understand to be "one" or "two" or "three"? They are abstract objects and in modern mathematics we build numbers using set theory. We start with the empty set, then we create a set that contains the empty set, and a set that contains that set, and so on. The empty set is "zero", the set containing the empty set is "one", the set containing one is "two". Each set created this way has a successor set and together they form the Natural numbers. We now have 0,1,2,3,...

We create the Integers by giving each Natural number except zero a negative version. We now have 0,1,-1,2,-2,...

From the the set of Integers, we create the set of all ordered pairs (a,b) where a is any integer and b is any integer except zero. This gives us all fractions: 1/2, 3/5. We can reduce ordered pairs to simpler ones if they have common factors: 3/3 is the same as 1 while 96/15 is the same as 6 and 2/5. Because they are a ratio of two integers, we call them the Rational numbers.

It was a big disappointment for the Greeks to find that these numbers did NOT correspond to every point on the line. All the rational numbers are indeed ON the line but there are points on the line that are NOT fractions--for example the square root of two. This unsatisfactory situation endured until the 19th century when the Real numbers were created from a specific kind of subset of the Rationals called "cuts".

So from the empty set, we get the natural numbers, then from those we get the integers, then from those we build the rationals and finally we get the reals. That's four levels of construction.

Amazingly John Conway invented a way to get ALL the numbers in one go, in a single level of construction. Conway came up with two rules that yield all the numbers on the real line by starting from the empty set and proceeding by iteration. As a bonus, these two rules also generate infinitesimals and transfinite cardinals. Infinitesimals are numbers greater than zero but smaller than all the non-zero positive real numbers, while transfinite cardinals are numbers that characterize different orders of infinity.

Donald Knuth jumped at the chance to use the topic to illustrate how much fun doing mathematics can be. He thought Conway's numbers would make an excellent basis for a story about two students working out how to generate the numbers from Conway's two rules and proving many useful theorems along the way. Knuth came up with the name Surreal Numbers (Conway referred to them just as "numbers") because they are in fact more than the Real numbers and yet they are generated using a simpler set of rules. Surreal!

Knuth set his story on an exotic island where the two students, Alice and Bill, discover a stone inscribed with the two rules and a short explanation of how to generate zero, one and minus one. From that starting point, Alice and Bill figure out how to work out all the numbers, and also how to add, subtract and multiply them. (SPOILER ALERT) The experience of working together convinces them that they should get married.

As far as dramatic literature goes, this isn't anything impressive. Calling the dialogue silly or corny would be generous. But following the math part of the novelette does effectively convey how it feels to work out mathematical theories for oneself and it will show the interested reader just how much fun he or she can have working out theorems for themselves.

Vincent Poirier, Montreal
I ℓ٥ﻻ ﻉ√٥υ
In the early 1970s, mathematician John Conway and computer scientist Donald Knuth had lunch together, during which Conway told Knuth of a way to generate all numbers from a couple of rules.

What is a number?

Everyone understands what one apple means. We also all understand that if Bill starts with three apples and gives one of those apples to Alice, he will have two apples left. But what are those things we all understand to be "one" or "two" or "three"? They are abstract objects and in modern mathematics we build numbers using set theory. We start with the empty set, then we create a set that contains the empty set, and a set that contains that set, and so on. The empty set is "zero", the set containing the empty set is "one", the set containing one is "two". Each set created this way has a successor set and together they form the Natural numbers. We now have 0,1,2,3,...

We create the Integers by giving each Natural number except zero a negative version. We now have 0,1,-1,2,-2,...

From the the set of Integers, we create the set of all ordered pairs (a,b) where a is any integer and b is any integer except zero. This gives us all fractions: 1/2, 3/5. We can reduce ordered pairs to simpler ones if they have common factors: 3/3 is the same as 1 while 96/15 is the same as 6 and 2/5. Because they are a ratio of two integers, we call them the Rational numbers.

It was a big disappointment for the Greeks to find that these numbers did NOT correspond to every point on the line. All the rational numbers are indeed ON the line but there are points on the line that are NOT fractions--for example the square root of two. This unsatisfactory situation endured until the 19th century when the Real numbers were created from a specific kind of subset of the Rationals called "cuts".

So from the empty set, we get the natural numbers, then from those we get the integers, then from those we build the rationals and finally we get the reals. That's four levels of construction.

Amazingly John Conway invented a way to get ALL the numbers in one go, in a single level of construction. Conway came up with two rules that yield all the numbers on the real line by starting from the empty set and proceeding by iteration. As a bonus, these two rules also generate infinitesimals and transfinite cardinals. Infinitesimals are numbers greater than zero but smaller than all the non-zero positive real numbers, while transfinite cardinals are numbers that characterize different orders of infinity.

Donald Knuth jumped at the chance to use the topic to illustrate how much fun doing mathematics can be. He thought Conway's numbers would make an excellent basis for a story about two students working out how to generate the numbers from Conway's two rules and proving many useful theorems along the way. Knuth came up with the name Surreal Numbers (Conway referred to them just as "numbers") because they are in fact more than the Real numbers and yet they are generated using a simpler set of rules. Surreal!

Knuth set his story on an exotic island where the two students, Alice and Bill, discover a stone inscribed with the two rules and a short explanation of how to generate zero, one and minus one. From that starting point, Alice and Bill figure out how to work out all the numbers, and also how to add, subtract and multiply them. (SPOILER ALERT) The experience of working together convinces them that they should get married.

As far as dramatic literature goes, this isn't anything impressive. Calling the dialogue silly or corny would be generous. But following the math part of the novelette does effectively convey how it feels to work out mathematical theories for oneself and it will show the interested reader just how much fun he or she can have working out theorems for themselves.

Vincent Poirier, Montreal
Cordantrius
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.
Cordantrius
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.
LoboThommy
A wonderful, accessible and lightweight introduction to abstract math. If you want to understand what numbers really are, read this.
LoboThommy
A wonderful, accessible and lightweight introduction to abstract math. If you want to understand what numbers really are, read this.
Vizuru
Awesome book, just wish there was an extended version though.
Vizuru
Awesome book, just wish there was an extended version though.
Owomed
A fun read if you enjoy maths. The book even encouraged me to tinker with the problem within.
Owomed
A fun read if you enjoy maths. The book even encouraged me to tinker with the problem within.
Natety
It's great even as a book by itself
Natety
It's great even as a book by itself