The Book of Numbers

Total Page：16

File Type：pdf, Size：1020Kb

springer.com Mathematics : Number Theory Conway, John H., Guy, Richard The Book of Numbers Journey through the world of numbers with the foremost authorities and writers in the field. John Horton Conway and Richard K. Guy are two of the most accomplished, creative, and engaging number theorists any mathematically minded reader could hope to encounter. In this book, Conway and Guy lead the reader on an imaginative, often astonishing tour of the landscape of numbers. The Book of Numbers is just that - an engagingly written, heavily illustrated introduction to the fascinating, sometimes surprising properties of numbers and number patterns. The book opens up a world of topics, theories, and applications, exploring intriguing aspects of real numbers, systems, arrays and sequences, and much more. Readers will be able to use figures to figure out figures, rub elbows with famous families of numbers, prove the primacy of primes, fathom the fruitfulness of fractions, imagine imaginary numbers, investigate the infinite and infinitesimal and more. Order online at springer.com/booksellers Copernicus Springer Nature Customer Service Center LLC 1996, IX, 310 p. 233 Spring Street 1st New York, NY 10013 edition USA T: +1-800-SPRINGER NATURE (777-4643) or 212-460-1500 Printed book [email protected] Hardcover Printed book Hardcover ISBN 978-0-387-97993-9 $ 49,99 Available Discount group Trade Books (1) Product category Popular science Other renditions Softcover ISBN 978-1-4612-8488-8 Prices and other details are subject to change without notice. All errors and omissions excepted. Americas: Tax will be added where applicable. Canadian residents please add PST, QST or GST. Please add $5.00 for shipping one book and $ 1.00 for each additional book. Outside the US and Canada add $ 10.00 for first book, $5.00 for each additional book. If an order cannot be fulfilled within 90 days, payment will be refunded upon request. Prices are payable in US currency or its equivalent. ISBN 978-0-387-97993-9 / BIC: PBH / SPRINGER NATURE: SCM25001 Part of .

Copy Link

Recommended publications

The 'Crisis of Noosphere'
The ‘crisis of noosphere’ as a limiting factor to achieve the point of technological singularity Rafael Lahoz-Beltra Department of Applied Mathematics (Biomathematics). Faculty of Biological Sciences. Complutense University of Madrid. 28040 Madrid, Spain. [email protected] 1. Introduction One of the most significant developments in the history of human being is the invention of a way of keeping records of human knowledge, thoughts and ideas. The storage of knowledge is a sign of civilization, which has its origins in ancient visual languages e.g. in the cuneiform scripts and hieroglyphs until the achievement of phonetic languages with the invention of Gutenberg press. In 1926, the work of several thinkers such as Edouard Le Roy, Vladimir Ver- nadsky and Teilhard de Chardin led to the concept of noosphere, thus the idea that human cognition and knowledge transforms the biosphere coming to be something like the planet’s thinking layer. At present, is commonly accepted by some thinkers that the Internet is the medium that brings life to noosphere. Hereinafter, this essay will assume that the words Internet and noosphere refer to the same concept, analogy which will be justified later. 2 In 2005 Ray Kurzweil published the book The Singularity Is Near: When Humans Transcend Biology predicting an exponential increase of computers and also an exponential progress in different disciplines such as genetics, nanotechnology, robotics and artificial intelligence. The exponential evolution of these technologies is what is called Kurzweil’s “Law of Accelerating Returns”. The result of this rapid progress will lead to human beings what is known as tech- nological singularity.
[Show full text]
Elementary Functions: Towards Automatically Generated, Efficient
Elementary functions : towards automatically generated, efficient, and vectorizable implementations Hugues De Lassus Saint-Genies To cite this version: Hugues De Lassus Saint-Genies. Elementary functions : towards automatically generated, efficient, and vectorizable implementations. Other [cs.OH]. Université de Perpignan, 2018. English. NNT : 2018PERP0010. tel-01841424 HAL Id: tel-01841424 https://tel.archives-ouvertes.fr/tel-01841424 Submitted on 17 Jul 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Délivré par l’Université de Perpignan Via Domitia Préparée au sein de l’école doctorale 305 – Énergie et Environnement Et de l’unité de recherche DALI – LIRMM – CNRS UMR 5506 Spécialité: Informatique Présentée par Hugues de Lassus Saint-Geniès [email protected] Elementary functions: towards automatically generated, efficient, and vectorizable implementations Version soumise aux rapporteurs. Jury composé de : M. Florent de Dinechin Pr. INSA Lyon Rapporteur Mme Fabienne Jézéquel MC, HDR UParis 2 Rapporteur M. Marc Daumas Pr. UPVD Examinateur M. Lionel Lacassagne Pr. UParis 6 Examinateur M. Daniel Menard Pr. INSA Rennes Examinateur M. Éric Petit Ph.D. Intel Examinateur M. David Defour MC, HDR UPVD Directeur M. Guillaume Revy MC UPVD Codirecteur À la mémoire de ma grand-mère Françoise Lapergue et de Jos Perrot, marin-pêcheur bigouden.
[Show full text]
~Umbers the BOO K O F Umbers
TH E BOOK OF ~umbers THE BOO K o F umbers John H. Conway • Richard K. Guy c COPERNICUS AN IMPRINT OF SPRINGER-VERLAG © 1996 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 1st edition 1996 All rights reserved. No part of this publication may be reproduced, stored in a re trieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Published in the United States by Copernicus, an imprint of Springer-Verlag New York, Inc. Copernicus Springer-Verlag New York, Inc. 175 Fifth Avenue New York, NY lOOlO Library of Congress Cataloging in Publication Data Conway, John Horton. The book of numbers / John Horton Conway, Richard K. Guy. p. cm. Includes bibliographical references and index. ISBN-13: 978-1-4612-8488-8 e-ISBN-13: 978-1-4612-4072-3 DOl: 10.l007/978-1-4612-4072-3 1. Number theory-Popular works. I. Guy, Richard K. II. Title. QA241.C6897 1995 512'.7-dc20 95-32588 Manufactured in the United States of America. Printed on acid-free paper. 9 8 765 4 Preface he Book ofNumbers seems an obvious choice for our title, since T its undoubted success can be followed by Deuteronomy,Joshua, and so on; indeed the only risk is that there may be a demand for the earlier books in the series. More seriously, our aim is to bring to the inquisitive reader without particular mathematical background an ex planation of the multitudinous ways in which the word "number" is used.
[Show full text]
Mitigating Javascript's Overhead with Webassembly
Samuli Ylenius Mitigating JavaScript’s overhead with WebAssembly Faculty of Information Technology and Communication Sciences M. Sc. thesis March 2020 ABSTRACT Samuli Ylenius: Mitigating JavaScript’s overhead with WebAssembly M. Sc. thesis Tampere University Master’s Degree Programme in Software Development March 2020 The web and web development have evolved considerably during its short history. As a result, complex applications aren’t limited to desktop applications anymore, but many of them have found themselves in the web. While JavaScript can meet the requirements of most web applications, its performance has been deemed to be inconsistent in applications that require top performance. There have been multiple attempts to bring native speed to the web, and the most recent promising one has been the open standard, WebAssembly. In this thesis, the target was to examine WebAssembly, its design, features, background, relationship with JavaScript, and evaluate the current status of Web- Assembly, and its future. Furthermore, to evaluate the overhead differences between JavaScript and WebAssembly, a Game of Life sample application was implemented in three splits, fully in JavaScript, mix of JavaScript and WebAssembly, and fully in WebAssembly. This allowed to not only compare the performance differences between JavaScript and WebAssembly but also evaluate the performance differences between different implementation splits. Based on the results, WebAssembly came ahead of JavaScript especially in terms of pure execution times, although, similar benefits were gained from asm.js, a predecessor to WebAssembly. However, WebAssembly outperformed asm.js in size and load times. In addition, there was minimal additional benefit from doing a WebAssembly-only implementation, as just porting bottleneck functions from JavaScript to WebAssembly had similar performance benefits.
[Show full text]
John Horton Conway
Obituary John Horton Conway (1937–2020) Playful master of games who transformed mathematics. ohn Horton Conway was one of the Monstrous Moonshine conjectures. These, most versatile mathematicians of the for the first time, seriously connected finite past century, who made influential con- symmetry groups to analysis — and thus dis- tributions to group theory, analysis, crete maths to non-discrete maths. Today, topology, number theory, geometry, the Moonshine conjectures play a key part Jalgebra and combinatorial game theory. His in physics — including in the understanding deep yet accessible work, larger-than-life of black holes in string theory — inspiring a personality, quirky sense of humour and wave of further such discoveries connecting ability to talk about mathematics with any and algebra, analysis, physics and beyond. all who would listen made him the centre of Conway’s discovery of a new knot invariant attention and a pop icon everywhere he went, — used to tell different knots apart — called the among mathematicians and amateurs alike. Conway polynomial became an important His lectures about numbers, games, magic, topic of research in topology. In geometry, he knots, rainbows, tilings, free will and more made key discoveries in the study of symme- captured the public’s imagination. tries, sphere packings, lattices, polyhedra and Conway, who died at the age of 82 from tilings, including properties of quasi-periodic complications related to COVID-19, was a tilings as developed by Roger Penrose. lover of games of all kinds. He spent hours In algebra, Conway discovered another in the common rooms of the University of important system of numbers, the icosians, Cambridge, UK, and Princeton University with his long-time collaborator Neil Sloane.
[Show full text]
Interview with John Horton Conway
Interview with John Horton Conway Dierk Schleicher his is an edited version of an interview with John Horton Conway conducted in July 2011 at the first International Math- ematical Summer School for Students at Jacobs University, Bremen, Germany, Tand slightly extended afterwards. The interviewer, Dierk Schleicher, professor of mathematics at Jacobs University, served on the organizing com- mittee and the scientific committee for the summer school. The second summer school took place in August 2012 at the École Normale Supérieure de Lyon, France, and the next one is planned for July 2013, again at Jacobs University. Further information about the summer school is available at http://www.math.jacobs-university.de/ summerschool. John Horton Conway in August 2012 lecturing John H. Conway is one of the preeminent the- on FRACTRAN at Jacobs University Bremen. orists in the study of finite groups and one of the world’s foremost knot theorists. He has written or co-written more than ten books and more than one- received the Pólya Prize of the London Mathemati- hundred thirty journal articles on a wide variety cal Society and the Frederic Esser Nemmers Prize of mathematical subjects. He has done important in Mathematics of Northwestern University. work in number theory, game theory, coding the- Schleicher: John Conway, welcome to the Interna- ory, tiling, and the creation of new number systems, tional Mathematical Summer School for Students including the “surreal numbers”. He is also widely here at Jacobs University in Bremen. Why did you known as the inventor of the “Game of Life”, a com- accept the invitation to participate? puter simulation of simple cellular “life” governed Conway: I like teaching, and I like talking to young by simple rules that give rise to complex behavior.
[Show full text]
An Aperiodic Convex Space-Filler Is Discovered
\ olume IJ. :"umller 6 The Ne\\ stetter of the :\Iathematical Association of America Den'mller )9l)J An Aperiodic Convex Space-filler is Discovered Earlier this year, Princeton mathematician Conway's discovery was announced at the John Horton Conway discovered a convex Regional Geometry Institute held this past polyhedron that will fill space, but only summer at Smith College. There, the par aperiodically. The space-filler is a biprism, ticipants(over 100,includingundergraduate that is, two slant triangular prisms fused REU participants, graduate students, high together (its faces are four congruent tri school teachers, and research mathemati angles and four congruentparallelograms). ~ cians) assembled cardboard models of the Several years ago, Peter Schmitt, at the ~ biprism and stacked them to witness the University of Vienna, described a non-con- § aperiodic space-filling. One participant, vex aperiodic space-filler, and Conway's ~ Ludwig Danzer, of the University of biprism fills space in a similar manner. First, ~ Dortmund, noted that Conway's tiling was ~ copies of the tile fill a layer (and in this single g not face-to-face, and quickly devised an layer, the tiling is periodic), and layers are ~ "improved" version. The parallelogram stacked to fill space. However, adjacentlay- E: faces ofDanzer's biprismare inscribedwith ers must be rotated with respect to each An example ofConway s tiling built at the Regional congruent triangles (which are considered other by a fixed irrational angle, thereby Geometry Institute at Smith College as faces) and when his polyhedronis stacked ensuring that the tiling is aperiodic. in the same manner as Conway's, the tiling is aperiodic and also face-to-face.
[Show full text]
John Horton Conway 2013 Book.Pdf
Princeton University Honors Faculty Members Receiving Emeritus Status May 2013 The biographical sketches were written by colleagues in the departments of those honored, except where noted. Copyright © 2013 by The Trustees of Princeton University 350509-13 Contents Faculty Members Receiving Emeritus Status Leonard Harvey Babby 1 Mark Robert Cohen 4 Martin C. Collcutt 6 John Horton Conway 10 Edward Charles Cox 14 Frederick Lewis Dryer 16 Thomas Jeffrey Espenshade 19 Jacques Robert Fresco 22 Charles Gordon Gross 24 András Peter Hámori 28 Marie-Hélène Huet 30 Morton Daniel Kostin 32 Heath W. Lowry 34 Richard Bryant Miles 36 Chiara Rosanna Nappi 39 Susan Naquin 42 Edward Nelson 44 John Abel Pinto 47 Albert Jordy Raboteau 49 François P. Rigolot 54 Daniel T. Rodgers 57 Gilbert Friedell Rozman 61 Peter Schäfer 64 José A. Scheinkman 68 Anne-Marie Slaughter 71 Robert Harry Socolow 74 Zoltán G. Soos 78 Eric Hector Vanmarcke 81 Maurizio Viroli 83 Frank Niels von Hippel 85 Andrew John Wiles 87 Michael George Wood 89 John Horton Conway John Conway is a mathematician whose interests run broad and deep, ranging from classical geometry to the 196,884-dimensional Monster group to infinity and beyond. Perhaps his greatest achievement (certainly his proudest achievement) is the invention of new system of numbers, the surreal numbers—a continuum of numbers that include not only real numbers (integers, fractions, and irrationals such as pi, which in his heyday he could recite from memory to more than 1,100 digits), but also the infinitesimal and the infinite numbers. When he discovered them in 1970, the surreals had John wandering around in a white-hot daydream for weeks.
[Show full text]
Genius at Play: the Curious Mind of John Horton Conway a Review by Elizabeth T
BOOK REVIEW Genius at Play: The Curious Mind of John Horton Conway A review by Elizabeth T. Milic´evic´ Communicated by Thomas Garrity Genius at Play: The Curious Mind of John Horton Conway By Siobhan Roberts the general public, having a rich fascination with the Bloomsbury USA, 2015 notion of a crazed mathematician as portrayed in a wide 480 pages variety of media, will cling to the promise on the book ISBN-13: 978-1620405932 cover of exploring a mathematical “genius,” and thus be drawn to read Roberts’ honest and detailed account of In her biography of John H. Conway titled Genius at Play, this intriguing character. There is sufficient support pro- Siobhan Roberts explains that the only medium through which Conway is capable of reaching out to other humans vided throughout this biography that Conway is indeed a is through “a giant prosthetic carapace of mathematical genius, often in the form of direct quotations from other knowledge and mathematical appetite” (p. 296). If there prominent research mathematicians who are in a position were a reliable way to document such mathematical en- to evaluate his work. However, Conway’s world view of counters with Conway, and to apply a reasonable metric mathematics, and his for the distance between encounters, I would wager that a own position therein, significant number of us in the mathematical community is considerably more would achieve a somewhat lower “Conway number” than inclusive than the The biography’s we have “Erdo˝s number.” On their first trip to the United jacket cover of this title plays States from Belgrade, Serbia even my non-mathematician biography reveals.
[Show full text]
George Pólya Lecturer Series
2020 IMPACT REPORT About Us 2 Contents Board of Directors 4 Letter from Executive Director 4 Supporting Our Community 6 in the Midst of a Pandemic Community 8 Our Members 10 Sections 12 SIGMAAs 13 MAA Connect 14 Shaping the Future 16 MAA Competitions 17 M-Powered 18 Mathematics on the Global Stage 21 International Mathematical Olympiad 21 European Girls’ Mathematical Olympiad 22 Romanian Masters of Mathematics 23 Cyberspace Mathematical Competition 24 MAA AMC Awards and Certificate Program 26 Edyth May Sliffe Award 26 Putnam Competition 30 Programs 31 MAA Project NExT 33 Diversity, Equity, and Inclusion 37 MAA 2020 Award Recipients 42 Communication 43 MAA Press & Publications 44 Communications Highlights 50 Math Values 54 Newsroom 55 Financials 56 Financial Snapshot 57 Why Your Gift Matters 58 Message from the MAA President 61 MATHEMATICAL ASSOCIATION OF AMERICA 2020 IMPACT REPORT COMMUNICATION About Us The Mathematical Association of America (MAA) is the world’s largest community of mathematicians, students, and enthusiasts. We accelerate the understanding of our world through mathematics because mathematics drives society and shapes our lives. OUR VISION We envision a society that values the INCLUSIVITY power and beauty of mathematics OUR and fully realizes its potential to CORE VALUES promote human flourishing. COMMUNITY OUR MISSION The mission of the MAA is to advance the understanding of mathematics and its impact on our world. TEACHING & LEARNING 2 3 MATHEMATICAL ASSOCIATION OF AMERICA 2020 IMPACT REPORT versus understanding, and what authentic assessment supports our goals for our students might look like. While these questions were Letter From lurking in the background pre-pandemic, I fully anticipate that we will 2020 Board continue these discussions to accelerate the work already underway, which is represented by the Committee on the Undergraduate of Directors Leadership Program in Mathematics (CUPM).
[Show full text]
MOVES 2015 - Abstracts
MOVES 2015 - Abstracts Plenary Talks Elwyn Berlekamp, University of California, Berkeley The Mathematics of Amazons John H. Conway, Princeton University How turning triangles inside out helps us understand their geometry Richard Guy, University of Calgary A Triangle has Eight Vertices Research Talks Aviv Adler, MIT The Hardness of Clickomania Clickomania is a popular computer puzzle game where the player is tasked with clearing a board of colored tiles by removing contiguous same-colored groups. Can it be quickly determined whether a given game is winnable? No; even in very restricted cases, it's NP-complete (and sometimes it's w[1]-hard). Max Alekseyev, George Washington University From Silent Circles to Graph Circuits We present an elegant solution to the question posed by Philip Brocoum, who described the following game as a preliminary event in a drama class he once attended at MIT. This game was played repeatedly by all the students until silence was achieved. An even number of people stand in a circle with their heads lowered. On cue, everyone looks up and stares either at one of their two immediate neighbors (left or right) or at the person diametrically opposed. If two people make eye contact, both will scream! What is the probability that everyone will be silent? Hossein Behforooz, Utica College On the Celebration of the 500th Birthday of Melancholia I and the D¨urer Magic Square Magic squares are a major component and the most important part of Recreational Mathematics. Although it is a very old subject, it is still alive and it is growing every day.
[Show full text]
Numbers 1 to 100
Numbers 1 to 100 PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Tue, 30 Nov 2010 02:36:24 UTC Contents Articles −1 (number) 1 0 (number) 3 1 (number) 12 2 (number) 17 3 (number) 23 4 (number) 32 5 (number) 42 6 (number) 50 7 (number) 58 8 (number) 73 9 (number) 77 10 (number) 82 11 (number) 88 12 (number) 94 13 (number) 102 14 (number) 107 15 (number) 111 16 (number) 114 17 (number) 118 18 (number) 124 19 (number) 127 20 (number) 132 21 (number) 136 22 (number) 140 23 (number) 144 24 (number) 148 25 (number) 152 26 (number) 155 27 (number) 158 28 (number) 162 29 (number) 165 30 (number) 168 31 (number) 172 32 (number) 175 33 (number) 179 34 (number) 182 35 (number) 185 36 (number) 188 37 (number) 191 38 (number) 193 39 (number) 196 40 (number) 199 41 (number) 204 42 (number) 207 43 (number) 214 44 (number) 217 45 (number) 220 46 (number) 222 47 (number) 225 48 (number) 229 49 (number) 232 50 (number) 235 51 (number) 238 52 (number) 241 53 (number) 243 54 (number) 246 55 (number) 248 56 (number) 251 57 (number) 255 58 (number) 258 59 (number) 260 60 (number) 263 61 (number) 267 62 (number) 270 63 (number) 272 64 (number) 274 66 (number) 277 67 (number) 280 68 (number) 282 69 (number) 284 70 (number) 286 71 (number) 289 72 (number) 292 73 (number) 296 74 (number) 298 75 (number) 301 77 (number) 302 78 (number) 305 79 (number) 307 80 (number) 309 81 (number) 311 82 (number) 313 83 (number) 315 84 (number) 318 85 (number) 320 86 (number) 323 87 (number) 326 88 (number)
[Show full text]