The Whole Truth About Whole Numbers: An Elementary Introduction to Number Theory

295 Pages·2015·2.477 MB·English

by  Sylvia Forman, Agnes M. Rash (auth.)

Sylvia Forman
Agnes M. Rash
The Whole Truth
About Whole
Numbers
An Elementary Introduction
to Number Theory
The Whole Truth About Whole Numbers

Sylvia Forman • Agnes M. Rash
The Whole Truth
About Whole Numbers
An Elementary Introduction
to Number Theory
SylviaForman AgnesM.Rash
DepartmentofMathematics DepartmentofMathematics
St.Joseph’sUniversity St.Joseph’sUniversity
Philadelphia,PA,USA Philadelphia,PA,USA
ISBN978-3-319-11034-9 ISBN978-3-319-11035-6(eBook)
DOI10.1007/978-3-319-11035-6
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014951838
MathematicsSubjectClassification(2010):11–XX,12–XX,00–XX
©SylviaFormanandAgnesM.Rash2015
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This work is dedicated to my mother and role
modelforherinspirationandencouragement
and to my husband for his unending support.
—Agnes M. Rash
To Sean, Carl, and Elinore
—Sylvia Forman

Preface to the Instructor
The Whole Truth About Whole Numbers
Thistextbookintroducesthefieldofnumbertheoryatalevelaccessibletononmath
and nonscience majors. The target audience is students in either a liberal arts
mathematics course or a course focused on elementary education majors at any
college or university. The content of this book is similar to what is included in a
standardintroductorynumbertheorycourseofferedformathematicsmajors,butthe
presentationismuchdifferent.Thetextincludesanintroductiontologicandproofs,
atalevelsuitableforliberalartsmajors,aswellasmajorconceptsinnumbertheory
accessibletostudentswith2yearsofhigh-schoolalgebra.Thistextisdesignedto
beusedinaone-semester(15-week)course.
Throughout the book, concepts have been linked and ordered to show connec-
tions between them. For example, the greatest common divisor is first defined in
Chap.2andusedtoidentify primitive Pythagoreantriples inChap.3.InChap.4,
thegreatestcommondivisorisdiscussedinconnectionwithprimefactors,andthen
inChap.5theEuclideanAlgorithmisusedtocalculategreatestcommondivisors.
ProoftechniquesandmethodsareintroducedinChap.2usingthefamiliarconcepts
of even and odd, and then students continue to write proofs about each new topic
theyencounterthroughoutthetext.
InChap.1,theimportanceofprecisecommunicationinmathematicsisstressed,
andsomecommonproblem-solvingtechniquesareintroduced.Examplesaregiven
usingfactsaboutintegers,consideringcasesandlookingforapattern.Termssuch
as even, odd, prime, and divisible are discussed, but these are formally defined in
Chap.2sothatitispossibletoomitChap.1.
Knowinghowtoverifywhetherastatementistrueorfalseisbasictomathemat-
ics.Studentsmustlearnhowtopreciselystateandjustifyideas.Chapter2includesa
briefintroductiontologicincludingthebasiclogicalconnectivesandthetypesof
compound sentences formed using them. Section 2.4 includes formal definitions
ofsomebasicterms(even,odd,prime).Sections2.5and2.6illustratesomecommon
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proof techniques, such as a direct proof, a proof by contradiction, and an indirect
proof(proofbycontrapositive).ProofsbycontradictioncomeupagaininChap.4,in
theproofthatthereareinfinitelymanyprimes.Thecontrapositiveofastatementis
alsousedinChap.4toformulateaprimalitytest;however,indirectproofsarenot
necessarytotheproofsintherestofthetext,sothistopiccouldbeskippedifdesired.
ProvingstatementsfalseusingacounterexampleisalsodiscussedinChap.2.The
last section of Chap. 2 provides a review of divisibility rules which are useful
throughout the text. Chapter 2 should be covered before proceeding to further
chapters.
Chapter 3 is notessential for understanding the concepts presented later, but it
givesstudentspracticewithelementaryproofsthroughbuildingontheirknowledge
ofafamiliartopic:righttriangles.Whilediscussingthismaterial,theauthorsshow
NOVA’s film The Proof, about Andrew Wiles’ path to proving Fermat’s Last
Theorem. This provides the opportunity to remark that you never know when an
ideawillhaveanewapplication.
Chapter4onprimenumbersiscentraltothestudyofnumbertheory.Section4.3
exploresarithmeticintheevenintegersinordertoshowthattheuniquefactoriza-
tionpropertyoftheintegersdoesnotholdineverynumbersystem,andisoptional.
Section 4.5 discusses some unsolved questions in number theory as well as the
searchforlargerprimesandcouldalsobeomitted.
In Chap. 5, the division algorithm is discussed in detail in order to motivate
the study of the Euclidean Algorithm as well as modular arithmetic described in
Chap.6.Section6.4isanapplicationofcongruencestocheckdigits,sotherestof
thematerialinthebookisnotdependentuponit.TheChineseRemainderTheorem
in Section 6.5 is also not necessary for the remaining material. In Chap. 7,
Section7.5couldalsobeomittedtomaketimeforcoveringcryptography.Euler’s
phifunctionandEuler’stheoreminSections7.4and7.6areneededforChap.8.
Chapter 8 introduces a modern application of number theory to public key
cryptographybutalsoincludesseveralexamplesofprivatekeycodesinSections8.2
and8.3forcontrast.Ifneeded,thesesectionscouldbeomittedtoallowtimeforan
introduction to the application of number theory to the RSA public key code,
containedinSection8.4.
Philadelphia,PA,USA SylviaForman
Philadelphia,PA,USA AgnesM.Rash
Preface to the Student
“ArchimedeswillberememberedwhenAeschylusisforgotten,becauselanguagesdieand
mathematicalideasdonot.”
—G.H.Hardy(1877–1947)
Number theory is the study of whole numbers. Early in recorded history, humans
begantocountusingthenaturalnumbers1,2,3,....Theearliestcountingdidnot
progressveryfarinthissequence,partiallybecausenumberswereassociatedwith
the items that could be counted on fingers (also called digits) and toes. Whole
numbers(orintegers)includenaturalnumbersandtheirnegativesaswellaszero.
Thesenumbersandtheirpropertieshavebeenstudiedextensively,atleastsincethe
timeoftheBabyloniansandthegoldenageofGreece.Manylearnedpeoplewere
fascinatedwiththesenumbersandprovedtheoremsaboutwhole numbersjustfor
thepersonalsatisfactionoffindingtheresults.InthefifthcenturyB.C.,Pythagoras
remarked “All is number” and, like many mathematicians, believed that numbers
hadspecialpropertiesthatcouldbeusedtodescribeorrepresenttheworldaround
them.Forexample,Pythagorasidentified“1”asthenumberofreasonand“10”as
thenumberoftheuniverse.
Thereisacertainsatisfactionthatonegetsbyaccomplishingagoal,performing
adifficulttask,ordiscoveringanewfact.Throughouttheages,thisfascinationwith
wholenumbershasledtomanygreatdiscoveries.Breakthroughs,discoveries,and
newideasoftenoccurasaninspirationoran“aha”moment.WhenArchimedeshad
such an “aha” moment, the story goes that he leapt out of the bathtub and ran
throughthestreetsproclaiming“Eureka,Ihavefoundit!”
There are concepts that were discovered just for the fun of it that eventually
turnedouttobeveryusefulandhavehadprofoundimplicationsformoderntimes.
One such example is the Euclidean Algorithm, discovered by Euclid in the third
centuryB.C.Thisresultisnowwidelyusedincryptographyandthedesignofsecret
codesdiscussedinChap.8.
Wehopethatyouhave“aha”moments,althoughhopefullylessprovocativethan
Archimedes’,aswelearnaboutnumbertheory.
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