Mathematics and Its History

681 Pages·2010·5.666 MB·English

by  John Stillwell

Undergraduate Texts in Mathematics
EditorialBoard
S.Axler
K.A.Ribet
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John Stillwell
Mathematics
and Its History
Third Edition
123
JohnStillwell
DepartmentofMathematics
UniversityofSanFrancisco
SanFrancisco,CA94117-1080
USA
[email protected]
EditorialBoard
S.Axler K.A.Ribet
MathematicsDepartment MathematicsDepartment
SanFranciscoStateUniversity UniversityofCaliforniaatBerkeley
SanFrancisco,CA94132 Berkeley,CA94720-3840
USA USA
[email protected] [email protected]
ISSN0172-6056
ISBN978-1-4419-6052-8 e-ISBN978-1-4419-6053-5
DOI10.1007/978-1-4419-6053-5
SpringerNewYorkDordrechtHeidelbergLondon
LibraryofCongressControlNumber:2010931243
MathematicsSubjectClassification(2010):01-xx,01Axx
(cid:2)c SpringerScience+BusinessMedia,LLC2010
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ToElaine,Michael,andRobert
Preface to the Third Edition
The aim of this book, announced in the first edition, is to give a bird’s-
eye view of undergraduate mathematics and a glimpse of wider horizons.
The second edition aimed to broaden this view by including new chapters
on number theory and algebra, and to engage readers better by including
manymoreexercises. Thisthird(andpossiblylast)editionaimstoincrease
breadthanddepth,butalsocohesion, byconnectingtopicsthatwereprevi-
ouslystrangerstoeachother,suchasprojectivegeometryandfinitegroups,
andanalysis andcombinatorics.
Therearetwonew chapters, onsimple groups and combinatorics, and
severalnewsectionsinoldchapters. Thenewsectionsfillgapsandupdate
areas where there has been recent progress, such as the Poincare´ conjec-
ture. The simple groups chapter includes some material on Lie groups,
thus redressing one of the omissions I regretted in the first edition of this
book. Thecoverageofgrouptheory hasnowgrownfrom17pagesand10
exercises inthefirsteditionto61pagesand85exercisesinthisone. Asin
the second edition, exercises often amount toproofs ofbigtheorems, bro-
ken down into small steps. In this way we are able to cover some famous
theorems, such as the Brouwer fixed point theorem and the simplicity of
A ,thatwouldotherwiseconsumetoomuchspace.
5
Eachchapternowbeginswitha“Preview”intendedtoorientthereader
withmotivation,anoutlineofitscontentsand,whererelevant,connections
to chapters that come before and after. I hope this will assist readers who
liketohaveanoverview before plunging intothedetails, andalsoinstruc-
tors looking for a path through the book that is short enough for a one-
semester course. Many different paths exist, at many different levels. Up
to Chapter 10, the level should be comfortable for most junior or senior
undergraduates; afterthat,thetopicsbecomemorechallenging, butalsoof
greater currentinterest.
vii
viii PrefacetotheThirdEdition
Allthefigures havenow beenconverted toelectronic form, whichhas
enabledmetoreducesomethatwereexcessivelylarge,andhencemitigate
thebloating thattendstooccurinneweditions.
Someofthe new material on mechanics in Section 13.2 originally ap-
peared (in Italian) in a chapter I wrote for Volume II of LaMatematica,
edited by Claudio Bartocci and Piergiorgio Odifreddi (Einaudi, Torino,
2008). Likewise, the new Section 8.6 contains material that appeared in
mybookTheFourPillarsofGeometry (Springer, 2005).
Finally,therearemanyimprovementsandcorrections suggestedtome
by readers. Special thanks go to France Dacar, Didier Henrion, David
Kramer, Nat Kuhn, Tristan Needham, Peter Ross, John Snygg, Paul Stan-
ford, Roland van der Veen, and Hung-Hsi Wu for these, and to my son
Robertandmywife,Elaine,fortheirtirelessproofreading.
IalsothanktheUniversityofSanFranciscoforgivingmetheopportu-
nitytoteachthecoursesonwhichmuchofthisbookisbased,andMonash
Universityfortheuseoftheirfacilities whilerevisingit.
JohnStillwell
MonashUniversity andtheUniversityofSanFrancisco
March2010
Preface to the Second Edition
ThiseditionhasbeencompletelyretypedinLATEX,andmanyofthefigures
redoneusingthePSTrickspackage,toimproveaccuracyandmakerevision
easier inthefuture. Intheprocess, severalsubstantial additions havebeen
made.
• Therearethreenewchapters, onChineseandIndiannumbertheory,
on hypercomplex numbers, and on algebraic number theory. These
fill some gaps in the first edition and give more insight into later
developments.
• Therearemanymoreexercises. This,Ihope,correctsaweaknessof
thefirstedition,whichhadtoofewexercises,andsomethatweretoo
hard. Some of the monster exercises in the first edition, such as the
one in Section 2.2 comparing volume and surface area of the icosa-
hedron and dodecahedron, have now been broken into manageable
parts. Nevertheless, there are still a few challenging questions for
thosewhowantthem.
• Commentary has been added to the exercises to explain how they
relate to the preceding section, and also (when relevant) how they
foreshadow latertopics.
• The index has been given extra structure to make searching easier.
Tofind Euler’s work on Fermat’s last theorem, for example, one no
longer has to look at 41 different pages under “Euler.” Instead, one
canfindtheentry“Euler,andFermat’slasttheorem”intheindex.
• The bibliography has been redone, giving more complete publica-
tiondataformanyworkspreviously listedwithlittleornone. Ihave
found the online catalogue of the Burndy Library of the Dibner In-
stitute at MIT helpful in finding this information, particularly for
ix
x PrefacetotheSecondEdition
early printed works. Forrecent works I have made extensive use of
MathSciNet,theonlineversionofMathematical Reviews.
There are also many small changes, some prompted by recent mathe-
matical events, such as the proof of Fermat’s last theorem. (Fortunately,
this one did not force a major rewrite, because the background theory of
ellipticcurveswascoveredinthefirstedition.)
I thank the many friends, colleagues, and reviewers who drew my at-
tentiontofaultsinthefirstedition,andhelpedmeintheprocessofrevision.
Specialthanks gotothefollowingpeople.
• My sons, Michael and Robert, who did most of the typing, and my
wife,Elaine,whodidagreatdealoftheproofreading.
• My students in Math 310 at the University of San Francisco, who
triedoutmanyoftheexercises,andtoTristanNeedham,whoinvited
metoUSFinthefirstplace.
• MarkAarons,DavidCox,DuaneDeTemple,WesHughes,Christine
Muldoon, Martin Muldoon, and Abe Shenitzer, for corrections and
suggestions.
JohnStillwell
MonashUniversity
Victoria, Australia
2001
Preface to the First Edition
One of the disappointments experienced by most mathematics students is
thattheynevergetacourseonmathematics. Theygetcourses incalculus,
algebra, topology, andsoon,butthedivisionoflaborinteaching seemsto
prevent these different topics from being combined into a whole. In fact,
some of the most important and natural questions are stifled because they
fall on the wrong side of topic boundary lines. Algebraists do not discuss
thefundamental theorem ofalgebra because “that’s analysis” andanalysts
do not discuss Riemann surfaces because “that’s topology,” for example.
Thusifstudents aretofeeltheyreally knowmathematics bythetimethey
graduate, thereisaneedtounifythesubject.
Thisbookaimstogiveaunifiedviewofundergraduatemathematicsby
approaching thesubject through itshistory. Sincereaders should havehad
somemathematical experience, certain basics areassumed andthemathe-
matics is not developed formally as in a standard text. Onthe other hand,
themathematicsispursuedmorethoroughly thaninmostgeneralhistories
of mathematics, because mathematics is our main goal and history only
the means of approaching it. Readers are assumed to know basic calcu-
lus,algebra,andgeometry,tounderstandthelanguageofsettheory,andto
have met some more advanced topics such as group theory, topology, and
differential equations. Ihave tried to pick outthe dominant themes ofthis
body of mathematics, and to weave them together as strongly as possible
bytracingtheirhistorical development.
Indoingso, Ihavealsotried totieupsometraditional loose ends. For
example, undergraduat√es can solve quadratic equations. Why not√cubics?
They can integrate 1/ 1− x2 but are told not to worry about 1/ 1− x4.
Why? Pursuing thehistory ofthese questions turnsout tobeveryfruitful,
leading to a deeper understanding of complex analysis and algebraic ge-
ometry, among other things. Thus I hope that the book will be not only a
xi