The Universal Book of Mathematics

David Darling

Description

Praise for David Darling

The Universal Book of Astronomy

"A first-rate resource for readers and students of popular astronomy and general science. . . . Highly recommended."
-Library Journal

"A comprehensive survey and . . . a rare treat."
-Focus

The Complete Book of Spaceflight

"Darling's content and presentation will have any reader moving from entry to entry."
-The Observatory magazine

Life Everywhere

"This remarkable book exemplifies the best of today's popular science writing: it is lucid, informative, and thoroughly enjoyable."
-Science Books & Films

"An enthralling introduction to the new science of astrobiology."
-Lynn Margulis

Equations of Eternity

"One of the clearest and most eloquent expositions of the quantum conundrum and its philosophical and metaphysical implications that I have read recently."
-The New York Times

Deep Time

"A wonderful book. The perfect overview of the universe."
-Larry Niven
Acknowledgments.

Introduction.

Mathematics Entries A to Z.

References.

Solutions to Puzzles.

Category Index.

Sujets

• Sciences et techniques
• Mathematics
• David Darling

Informations

• Publié par: Turner Publishing Company
• Date de parution: 21 avril 2008
• Nombre de lectures: 0
• EAN13: 9780470307885
• Langue: English
• Poids de l'ouvrage: 4 Mo

Extrait

The Universal Book of Mathematics
The Universal Book of Mathematics
From A bracadabra to Z eno s Paradoxes
David Darling

John Wiley Sons, Inc.
This book is printed on acid-free paper.
Copyright 2004 by David Darling. All rights reserved
Published by John Wiley Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, [phone], fax [phone], or on the web at www.copyright.com . Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley Sons, Inc., 111 River Street, Hoboken, NJ 07030, [phone], fax [phone].
Limit of Liability/Disclaimer of Warranty: While the publisher and the author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor the author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.
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Library of Congress Cataloging-in-Publication Data:
Darling, David J.
The universal book of mathematics : from abracadabra to Zeno s paradoxes / David Darling.
p. cm. Includes bibliographical references and index.
ISBN 0-471-27047-4 (cloth: alk. paper)
1. Mathematics-Encyclopedias. I. Title. QA5 .D27 2004 510 .3-dc22 2003024670 Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigor should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere .
-William S. Anglin
But leaving those of the Body, I shall proceed to such Recreation as adorn the Mind; of which those of the Mathematicks are inferior to none .
-William Leybourn (1626-1700)
The last thing one knows when writing a book is what to put first .
-Blaise Pascal (1623-1662)
Contents
Acknowledgments
Introduction
Mathematics Entries A to Z
References
Solutions to Puzzles
Category Index
Acknowledgments
Many people have helped me enormously in assembling this collection of mathematical oddities, delights, whimsies, and profundities. Thanks especially go to Jan Wassenaar ( www.2dcurves.com ) for drawing many of the plane curves that are featured in the book; Robert Webb ( www.software3d.com ) for numerous photos of his wonderful, homemade polyhedra; Jos Leys ( www.josleys.com ) for his mesmerizing fractal artwork; Xah Lee ( www.xahlee.org ) for a variety of ingenious digital imagery; Sue and Brian Young at Mr. Puzzle Australia ( www.mrpuzzle.com.au ) and Kate and Dick Jones at Kadon Enterprises ( www.gamepuzzles.com ) for their advice and photos of puzzles from their product lines and personal collections; Gideon Weisz ( www.gideonweisz.com ) and Istvan Orosz for stunning recursive and anamorphic art images; my good friend Andrew Dogs Barker for stimulating discussions and the solution to one of the problems; William Waite for pictures from his antique math collection; and Peter Cromwell, Lord Lady Dunsany, Peter Knoppers, John Lienhard, John Mainstone, David Nicholls, Paul and Colin Roberts, Anders Sandberg, John Sullivan, and others for their valuable contributions.
I m greatly indebted to Stephen Power, senior editor, and to Lisa Burstiner, senior production editor, at John Wiley Sons, for their encouragement and unfailing attention to detail, and even proffering of alternative, clever solutions to some of the problems in the book. Any errors that remain are entirely my own responsibility. Thanks also to my marvelous agent, Patricia Van der Leun. And last but most of all, thanks to my family for letting me pursue a career that is really a fantasy.
Introduction
You are lost in a maze: How do you find your way out? You want to build a time machine, but is time travel logically possible? How can one infinity be bigger than another? Why can t you drink from a Klein bottle? What is the biggest number in the world to have a proper name, and how can you write it? Who claimed he could see in the fourth dimension? And what does iteration mean? And what does iteration mean?
Mathematics was never my strong point in school, but because I wanted to become an astronomer, I was told to stick with it. Fortunately, in my last two years before heading off to university, I had a wonderful oldfashioned, eccentric teacher (he actually wore a black gown when teaching), called Mr. Kay (known to one and all as Danny ), who would suddenly divert from the chalk and blackboard to ask, But how did the universe come to be asymmetric -that s what I want to know, or These imaginary numbers are very interesting; in part, because they are so remarkably real. During lunchbreak, Danny and the senior chemistry teacher, Mr. Erp (whose nickname I need hardly spell out), would always meet in the chemistry prep room for a game of chess. They looked and acted very much like characters from a Wellsian science fiction tale, and I sometimes imagined them musing on formulas for invisibility or doorways to higher dimensions. At any rate, though I was never a shining student, I realize what a profound effect those two deeply imaginative, thoughtful men had on my future career. I did become an astronomer. I did persevere with math to a certain level of competence. But, much more than that, my curiosity was fired by the wonderful and weird possibilities of these subjects: curved space, M bius bands, parallel universes, patterns in the heart of chaos, alternative realities. These strange possibilities, and a thousand others, make up the stuffing of this book. If you want a comprehensive, academic dictionary of mathematics, look elsewhere. If you want rigor and proof, try the next shelf. Herein you will find only the unusual and the outrageous, the fanciful and the fantastic: a compendium of the mathematics they didn t teach you in school.
Entries range from short definitions to lengthy articles on topics of major importance or unusual interest. These are arranged alphabetically according to the first word of the entry name and are extensively cross-referenced. Terms that appear in bold type have their own entries. A number of puzzles are included for the reader to try; the answers to these can be found at the back of the book. Also at the back are a comprehensive list of references and a category index. Readers are invited to visit the author s Web site at www.daviddarling.info for the latest news in mathematics and related subjects.
A
abacus
A counting frame that started out, several thousand years ago, as rows of pebbles in the desert sands of the Middle East. The word appears to come from the Hebrew b q (dust) or the Phoenician abak (sand) via the Greek abax , which refers to a small tray covered with sand to hold the pebbles steady. The familiar frame-supporting rods or wires, threaded with smoothly running beads, gradually emerged in a variety of places and mathematical forms.
In Europe, there was a strange state of affairs for more than 1,500 years. The Greeks and the Romans, and then the medieval Europeans, calculated on devices with a place-value system in which zero was represented by an empty line or wire. Yet the written notations didn t have a symbol for zero until it was introduced in Europe in 1202 by Fibonacci , via the Arabs and the Hindus.

abacus A special form of the Chinese abacus (c. 1958) consisting of two abaci stacked one on top of the other. Luis Fernandes

The Chinese suan pan differs from the European abacus in that the board is split into two decks, with two beads on each rod in the upper deck and five beads, representing the digits 0 through 4, on each rod in the bottom. When all five beads on a rod in the lower deck are moved up, they re reset to the original position, and one bead in the top deck is moved down as a carry. When both beads in the upper deck are moved down, they re reset and a bead on the adjacent rod on the left is moved up as a carry. The result of the computation is read off from the beads clustered near the separator beam between the upper and lower decks. In a sense, the abacus works as a 5-2-5-2-5-2 . . . -based number system in which carries and shifts are similar to those in the decimal system. Since each rod represents a digit in a decimal number, the capacity of the abacus is limited only by the number of rods on the abacus. When a user runs out of rods, she simply adds another abacus to the left of the row.
The Japanese soroban does away with the dual representations of fives and tens by having only four counters in the lower portion, known as earth, and only one counter in the upper portion, known as heaven. The world s largest abacus is in the Science Museum in London and measures 4.7 meters by 2.2 meters.
Abbott, Edwin Abbott (1838-1926)
An English clergyman and author