Mathematics and Its History, Third Edition

683 Pages·2011·5.9 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
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