Table Of ContentA Readable Introduction to Real Mathematics
Undergraduate Texts in Mathematics
SeriesEditors:
SheldonAxler
SanFranciscoStateUniversity,SanFrancisco,CA,USA
KennethRibet
UniversityofCalifornia,Berkeley,CA,USA
AdvisoryBoard:
ColinAdams,WilliamsCollege,Williamstown,MA,USA
AlejandroAdem,UniversityofBritishColumbia,Vancouver,BC,Canada
RuthCharney,BrandeisUniversity,Waltham,MA,USA
IreneM.Gamba,TheUniversityofTexasatAustin,Austin,TX,USA
RogerE.Howe,YaleUniversity,NewHaven,CT,USA
DavidJerison,MassachusettsInstituteofTechnology,Cambridge,MA,USA
JeffreyC.Lagarias,UniversityofMichigan,AnnArbor,MI,USA
JillPipher,BrownUniversity,Providence,RI,USA
FadilSantosa,UniversityofMinnesota,Minneapolis,MN,USA
AmieWilkinson,UniversityofChicago,Chicago,IL,USA
Undergraduate Texts in Mathematics are generally aimed at third- and fourth-
year undergraduate mathematics students at North American universities. These
texts strive to provide students and teachers with new perspectives and novel
approaches.Thebooksincludemotivationthatguidesthereadertoanappreciation
ofinterrelationsamongdifferentaspectsofthesubject.Theyfeatureexamplesthat
illustratekeyconceptsaswellasexercisesthatstrengthenunderstanding.
Forfurthervolumes:
http://www.springer.com/series/666
Daniel Rosenthal • David Rosenthal
Peter Rosenthal
A Readable Introduction
to Real Mathematics
123
DanielRosenthal DavidRosenthal
DepartmentofMathematics DepartmentofMathematics
UniversityofToronto andComputerScience
Toronto,ON,Canada St.John’sUniversity
Queens,NY,USA
PeterRosenthal
DepartmentofMathematics
UniversityofToronto
Toronto,ON,Canada
ISSN0172-6056 ISSN2197-5604(electronic)
ISBN978-3-319-05653-1 ISBN978-3-319-05654-8(eBook)
DOI10.1007/978-3-319-05654-8
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014938028
MathematicsSubjectCodes(2010):03E10,11A05,11A07,11A41,11A51,11-01,51-01,97-01,97F30,
97F40,97F50,97F60,97G99,97H99
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Preface
The fundamental purpose of this book is to teach you to understand mathematical
thinking. We have triedto do that in a way that is clear, engaging and emphasizes
thebeautyofmathematics.Youmaybereadingthisbookonyourownorasatext
foracourseyouareenrolledin.Regardlessofyourreasonforreadingthisbook,we
hopethatyouwillfinditunderstandableandinteresting.
Mathematicsisahugeandgrowingbodyofknowledge;noonecanlearnmore
than a fraction of it. But the essence of mathematics is thinking mathematically.
It is our experience that mathematical thinking can be learned by almost anyone
who is willing to make a serious attempt. We invite you to make such an attempt
byreadingthisbook.Itisimportantnottoletyourselfbediscouragedifyoucan’t
easily understand something. Everyone learning mathematics finds some concepts
bafflingatfirst,butusually,withenougheffort,theideasbecomeclear.
Onewayinwhichmathematicsgetsverycomplexisbybuildingonitself;some
mathematical concepts are built on a foundation of many other concepts and thus
requireagreatdealofbackgroundtounderstand.Thatisnotthecaseforthetopics
discussed in this book. Reading this book does not require any background other
thanbasichighschoolalgebraand,forpartsofChapters9and12,somehighschool
trigonometry.
A few questions, among the many, that you will easily be able to answer after
readingthisbookarethefollowing:Is13217(cid:2)3792(cid:2)4115 D19111(cid:2)29145(cid:2)4312(cid:2)475
(see Chapter4)? Is there a largest prime number (i.e., a largest whole number
whose only factors are 1 and itself) (Theorem 1.1.2)? If a store sells one kind of
product for 9 dollars each and another kind for 16 dollars each and receives 143
dollars for the total sale of both, how many products did the store sell at each
price (Example 7.2.7)? How do computers send secret messages to each other
(Chapter6)? Are there more fractions than there are whole numbers? Are there
more real numbers than there are fractions? Is there a smallest infinity? Is there
alargestinfinity(Chapter10)?Whatarecomplexnumbersandwhataretheygood
for(Chapter9)?
The hardest theorem we will prove concerns construction of angles using a
compass and a straightedge. (A straightedge is a ruler-like device but without
vii
viii Preface
measurementsmarkedonit.)Ifyouaregivenanyangle,itiseasytobisectit(i.e.,
divideitintotwoequalsubangles)byusingacompassandastraightedge(wewill
show you how to do that). This and many similar results were discovered by the
AncientGreeks.TheAncientGreekswonderedwhetheranglescouldbe“trisected”
in the sense of being divided into three equal subangles using only a straightedge
andacompass.AgreatdealofmathematicsbeyondthatconceivedofbytheAncient
Greekswasrequiredtosolvethisproblem;itwasnotsolveduntilthe19thcentury.
It can be proven that many angles, including angles of 60 degrees, cannot be so
trisected.Wepresentacompleteproofofthisasanillustrationofcomplicatedbut
beautifulmathematicalreasoning.
The most important question you’ll be able to answer after reading this book,
although you would have difficulty formulating the answer in words, is: what is
mathematicalthinkingreallylike?Ifyoureadandunderstandmostofthisbookand
do a fair number of the problems that are provided, you will certainly have a real
feelingformathematicalthinking.
We hope that you read this book carefully. Reading mathematics is not like
reading a novel, a newspaper, or anything else. As you go along, you have to
reallyreflectonthemathematicalreasoningthatwearepresenting.Afterreadinga
descriptionofanidea,thinkaboutit.Whenreadingmathematicsyoushouldalways
haveapencilandpaperathandandreworkwhatyouread.
Mathematics consists of theorems, which are statements proven to be true. We
willprove anumber oftheorems.Whenyoubegin readingabout atheorem,think
about why it may be true before you read our proof. In fact, at some points you
may be able to prove the theorem we state without looking at our proof at all. In
anyevent,youshouldmakeatleastasmallattemptbeforereadingtheproofinthe
book.Itisoftenusefultocontinuesuchattemptswhileinthemiddleofreadingthe
proofthatwepresent;oncewehavegottenyouacertainwaytowardstheresult,see
ifyoucancontinueonyourown.
If you adopt such an approach and are patient, we believe that you will learn
tothinkmathematically.Wearealsoconvincedthatyouwillfeelthatmuchofthe
mathematicsthatyoulearnisbeautiful,inthesensethatyouwillfindthatthelogical
argumentthatestablishesthetheoremiswhatmathematicianscall“elegant.”
We chose the material for this book based on the following criteria: the mathe-
matics is beautiful, it is “real” in the sense that it is useful in many mathematical
contextsanditisaccessiblewithoutagreatdealofmathematicalbackground.The
theoremsthatweprovehaveapplicationstomathematicsandtoproblemsinother
subjects.Someoftheseapplicationswillbepresentedinwhatfollows.
Eachchapterendswithasectionentitled“Problems.”Theproblemssectionsare
divided into three subsections. The first, “Basic Exercises,” consists of problems
whosesolutionyoushoulddotoassureyourselfthatyouhaveanunderstandingof
the fundamentals of the material. The subsections entitled “Interesting Problems”
contain problems whose solutions depend upon the material of the chapter and
seemtohavemathematicalorotherinterest.Thesubsectionslabeled“Challenging
Problems” contain problems that we expect you will, indeed, find to be quite
challenging. You should not be discouraged if you cannot solve some of the
Preface ix
problems. However, if you do solve problems that you find difficult at first,
especially those that we have labeled “challenging,” we hope and expect that
you will feel some of the pleasure and satisfaction that mathematicians feel upon
discoveringnewmathematics.
Each chapter is divided into sections. Important items, such as definitions and
theorems, are numbered in a way that locates them within a chapter and a section
of that chapter. We put the chapter number, then the section number, and then the
numberoftheitemwithinthatsection.Forexample,7.2.4referstothefourthitem
insectiontwoofchapterseven.
Sincetheonlyprerequisiteforunderstandingthisbookishighschoolalgebra,it
is suitable as a textbook for a wide variety of courses. In particular, it is our view
thatitwouldbeappropriateforcoursesforgeneralartsandsciencesstudentswho
want to get an appreciation of mathematics, for courses for prospective teachers,
and for an introductory course for mathematics majors. Instructors can vary the
levelofthecoursebythepaceatwhichtheyproceed,thedifficultyoftheproblems
that they assign, and the material they omit. The book is also written so as to be
useableforindependentstudybyanyonewhoisinterestedinlearningmathematics.
In particular, high school students who like mathematics might be directed to this
book.
Instructorsandreaderswhowishtoomitsomeofthematerial(perhapsonlyat
first)shouldbeawareofthefollowing.Chapters1through7eachdepend,atleastto
someextent,ontheirpredecessors.Chapter8usessomeofthematerialinChapter4.
Chapters 9–11 are essentially independent of each other and of all other chapters.
Chapter 12 depends basically only on Chapter11 and on the concepts of rational
andirrationalnumbersasdiscussedinChapter8.
This book was developed from lecture notes for a course that was given at the
University of Toronto over a period of 15years. It has been greatly improved by
suggestionsfromstudentsandcolleagues.WeareparticularlygratefultoProfessor
Heydar Radjavi of the University of Waterloo for his assistance and to two
anonymousreviewersfortheircomments.Inspiteofallthesuggestions,wearesure
that further improvements could be made. We would appreciate your sending any
comments,corrections,orsuggestionstoanyoftheauthorsattheire-mailaddresses
givenbelow.
DanielRosenthal:[email protected]
DavidRosenthal:[email protected]
PeterRosenthal:[email protected]
Toronto,ON,Canada DanielRosenthal
Queens,NY,USA DavidRosenthal
Toronto,ON,Canada PeterRosenthal
