In pursuit of the unknown. 17 equations that changed the world

353 Pages·2012·3.125 MB·English

by  Stewart, Ian

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mathematical building blocks that form the foundations of great equations: Pythagoras’s theorem,
modern life, In Pursuit of the Unknown is also a penetrating 1 7 E q uations That for instance, or Newton’s Law of Gravity,
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exploration of how we have long used equations to make —New ScieNtiSt C h a n ged the World or Einstein’s theory of relativity. But the way these
sense of, and in turn influence, our world. mathematical breakthroughs have contributed to human
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interpersonal relationships. And the Black-Scholes model, SPOT GLOSS ON
EQUATIONS ON
$26.99 US / $30.00 CAN used by bankers to track the prices of financial derivatives FRONT COVER
ISBN 978-0-465-02973-0 over time, led to massive growth in the financial sector,
52699 Ian Stewa r t
Jacket design by Jennifer Carrow A Member of the Perseus Books Group thereby contributing to the banking crisis of 2008—the
03/12 www.basicbooks.com
effects of which we are still feeling today.
9 780465 029730 (continued on back flap)
IN PURSUIT OF THE UNKNOWN
Also by Ian Stewart:
Concepts of Modern Mathematics
Game, Set, and Math
The Problems of Mathematics
Does God Play Dice?
Another Fine Math You’ve Got Me into
Fearful Symmetry(with Martin Golubitsky)
Nature’s Numbers
From Here to Infinity
The Magical Maze
Life’s Other Secret
Flatterland
What Shape Is a Snowflake?
The Annotated Flatland
Math Hysteria
The Mayor of Uglyville’s Dilemma
Letters to a Young Mathematician
Why Beauty Is Truth
How to Cut a Cake
Taming the Infinite/The Story of Mathematics
Professor Stewart’s Cabinet of Mathematical Curiosities
Professor Stewart’s Hoard of Mathematical Treasures
Cows in the Maze
Mathematics of Life
with Terry Pratchett and Jack Cohen
The Science of Discworld
The Science of Discworld II: the Globe
The Science of Discworld III: Darwin’s Watch
with Jack Cohen
The Collapse of Chaos
Figments of Reality
Evolving the Alien/What Does a Martian Look Like?
Wheelers(science fiction)
Heaven(science fiction)
IN PURSUIT OF THE
UNKNOWN
17 Equations
That Changed the World
✺
IAN STEWART
A Member of the Perseus Books Group
New York
Copyright © 2012 by Ian Stewart
Published in the United States in 2012 by Basic Books,
A Member of the Perseus Books Group
Published in Great Britain in 2012 by Profile Books
All rights reserved. Printed in the United States of America. No part of this book may be
reproduced in any manner whatsoever without written permission except in the case of
brief quotations embodied in critical articles and reviews. For information, address Basic
Books, 387 Park Avenue South, New York, NY 10016-8810.
Books published by Basic Books are available at special discounts for bulk purchases in the
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please contact the Special Markets Department at the Perseus Books Group, 2300 Chestnut
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[email protected].
A CIP catalog record for this book is available from the Library of Congress.
LCCN: 2011944850
ISBN: 978-0-465-02973-0
10 9 8 7 6 5 4 3 2 1
Contents
WhyEquations?/viii
1
Thesquawonthehippopotamus/1
Pythagoras’sTheorem
2
Shorteningtheproceedings/21
Logarithms
3
Ghostsofdepartedquantities/35
Calculus
4
Thesystemoftheworld /53
Newton’sLawofGravity
5
Portentoftheidealworld/73
TheSquareRootofMinusOne
6
Muchadoaboutknotting/89
Euler’sFormulaforPolyhedra
7
Patternsofchance/107
NormalDistribution
8
Goodvibrations/131
WaveEquation
9
Ripplesandblips/149
FourierTransform
10
Theascentofhumanity/165
Navier–StokesEquation
11
Wavesintheether/179
Maxwell’sEquations
12
Lawanddisorder/195
SecondLawofThermodynamics
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vi Contents
13
Onethingisabsolute/217
Relativity
14
Quantumweirdness /245
Schro¨dinger’sEquation
15
Codes,communications,andcomputers/265
InformationTheory
16
Theimbalanceofnature/283
ChaosTheory
17
TheMidasformula/295
Black–ScholesEquation
WhereNext?/317
Notes/321
IllustrationCredits/330
Index/331
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To avoide the tediouse repetition of these woordes: is
equalle to: I will sette as I doe often in woorke use, a paire of
paralleles, or gemowe lines of one lengthe:=======, bicause
noe .2. thynges, can be moare equalle.
Robert Recorde, The Whetstone of Witte, 1557
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Why Equations?
E
quations are the lifeblood of mathematics, science, and technology.
Without them, our world would not exist in its present form.
However, equations have a reputation for being scary: Stephen
Hawking’s publishers told him that every equation would halve the sales
of A Brief History of Time, but then they ignored their own advice and
allowed him to include E = mc2 when cutting it out would allegedly have
sold another 10 million copies. I’m on Hawking’s side. Equations are too
importanttobehiddenaway.Buthispublishershadapointtoo:equations
are formal and austere, they look complicated, and even those of us who
love equations can be put off if we are bombarded with them.
Inthisbook,Ihaveanexcuse.Sinceit’saboutequations,Icannomore
avoid including them than I could write a book about mountaineering
withoutusingtheword‘mountain’.Iwanttoconvinceyouthatequations
have played a vital part in creating today’s world, from mapmaking to
satnav, from music to television, from discovering America to exploring
themoonsofJupiter.Fortunately,youdon’tneedtobearocketscientistto
appreciate the poetry and beauty of a good, significant equation.
Therearetwokindsofequationsinmathematics,whichonthesurface
look very similar. One kind presents relations between various
mathematical quantities: the task is to prove the equation is true. The
other kind provides information about an unknown quantity, and the
mathematician’s task is to solve it – to make the unknown known. The
distinction is not clear-cut, because sometimes the same equation can be
usedinbothways,butit’sausefulguideline.Youwillfindbothkindshere.
Equations in pure mathematics are generally of the first kind: they
revealdeepandbeautifulpatternsandregularities.Theyarevalidbecause,
given our basic assumptions about the logical structure of mathematics,
there is no alternative. Pythagoras’s theorem, which is an equation
expressed in the language of geometry, is an example. If you accept
Euclid’s basic assumptions about geometry, then Pythagoras’s theorem is
true.
Equations in applied mathematics and mathematical physics are
usually of the second kind. They encode information about the real
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WhyEquations? ix
world;theyexpresspropertiesoftheuniversethatcouldinprinciplehave
been very different. Newton’s law of gravity is a good example. It tells us
howtheattractiveforcebetweentwobodiesdependsontheirmasses,and
how far apart they are. Solving the resulting equations tells us how the
planets orbit the Sun, or how to design a trajectory for a space probe. But
Newton’slawisn’tamathematicaltheorem;it’strueforphysicalreasons,it
fitsobservations.Thelawofgravitymighthavebeendifferent.Indeed,itis
different: Einstein’s general theory of relativity improves on Newton by
fitting some observations better, while not messing up those where we
already know Newton’s law does a good job.
Thecourseofhumanhistoryhasbeenredirected,timeandtimeagain,
byanequation.Equationshavehiddenpowers.Theyrevealtheinnermost
secrets of nature. This is not the traditional way for historians to organise
the rise and fall of civilisations. Kings and queens and wars and natural
disasters abound in the history books, but equations are thin on the
ground. This is unfair. In Victorian times, Michael Faraday was
demonstrating connections between magnetism and electricity to
audiences at the Royal Institution in London. Allegedly, Prime Minister
William Gladstone asked whether anything of practical consequence
wouldcomefromit.Itissaid(onthebasisofverylittleactualevidence,but
whyruinanicestory?)thatFaradayreplied:‘Yes,sir.Onedayyouwilltax
it.’Ifhedidsaythat,hewasright.JamesClerkMaxwelltransformedearly
experimental observations and empirical laws about magnetism and
electricity into a system of equations for electromagnetism. Among the
many consequences were radio, radar, and television.
Anequationderivesitspowerfromasimplesource.Ittellsusthattwo
calculations, which appear different, have the same answer. The key
symbolistheequalssign,¼.Theoriginsofmostmathematicalsymbolsare
eitherlostinthemistsofantiquity,oraresorecentthatthereisnodoubt
where they came from. The equals sign is unusual because it dates back
morethan450years,yetwenotonlyknowwhoinventedit,weevenknow
why.TheinventorwasRobertRecorde,in1557,inTheWhetstoneofWitte.
He used two parallel lines (he used an obsolete word gemowe, meaning
‘twin’)toavoidtediousrepetitionofthewords‘isequalto’.Hechosethat
symbolbecause‘notwothingscanbemoreequal’.Recordechosewell.His
symbol has remained in use for 450 years.
The power of equations lies in the philosophically difficult
correspondence between mathematics, a collective creation of human
minds,andanexternalphysicalreality.Equationsmodeldeeppatternsin
the outside world. By learning to value equations, and to read the stories
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