From Natural Numbers to Quaternions

288 Pages·2017·2.81 MB·English

by  Jürg Kramer

Springer Undergraduate Mathematics Series
Jürg Kramer
Anna-Maria von Pippich
From Natural
Numbers to
Quaternions
Springer Undergraduate Mathematics Series
Advisory Board
M. A. J. Chaplain, University of St. Andrews
A. MacIntyre, Queen Mary University of London
S. Scott, King’s College London
N. Snashall, University of Leicester
E. Süli, University of Oxford
M. R. Tehranchi, University of Cambridge
J. F. Toland, University of Cambridge
More information about this series at http://www.springer.com/series/3423
ü
J rg Kramer Anna-Maria von Pippich
(cid:129)
From Natural Numbers
to Quaternions
123
Jürg Kramer Anna-Maria vonPippich
Department ofMathematics Department ofMathematics
Humboldt-Universität zuBerlin Technische UniversitätDarmstadt
Germany Germany
Translation from the German language edition: Von den natürlichen Zahlen zu den
QuaternionenbyJürgKramerandAnna-MariavonPippich,©SpringerSpektrum2013.All
Rights Reserved.
ISSN 1615-2085 ISSN 2197-4144 (electronic)
SpringerUndergraduate MathematicsSeries
ISBN978-3-319-69427-6 ISBN978-3-319-69429-0 (eBook)
https://doi.org/10.1007/978-3-319-69429-0
LibraryofCongressControlNumber:2017958024
MathematicsSubjectClassification(2010): 08–01,11–01,12–01,20–01
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Preface to the English Edition
Thisbookontheconstructionofnumbersystemsfirstappearedin2013ina
Germaneditionwiththesametitle.Itcanbeseenfromthefollowingpreface
to that edition that the goal of this book is to present a basic and compre-
hensive construction of number systems, beginning with the natural num-
bersandendingwithHamilton’squaternions,whileprovidingrelevantal-
gebraicknowledgealongtheway.AsasupplementtotheGermanedition,
anappendixhasbeenaddedtoeachchapterinthisEnglishedition,which
incontrasttotherigorousstyleoftherestofthebook,presentsinthemore
casualformofasurveysomerelatedaspectsofthematerialofthechapter,
includingsomerecentdevelopments.
Wewouldliketoofferourmostheartfeltthankstothetranslator,David
Kramer,forhiscompetentwork,whichhascontributedsignificantlytothis
Englishversionandinmanyplacesledtoamorefelicitouspresentationof
thematerial.
We hope that this book will help students and teachers of mathematics
aswellasallthosewithaninterestinthesubjecttofillinanygapsintheir
mathematicaleducationrelatedtotheconstructionofnumbersystemsand
thattheappendiceswillinspiresomereaderstopursuefurthermathemati-
calstudies.
Berlin,September2017 JürgKramer
Anna-MariavonPippich
Preface to the German Edition
The main topic of this book is an elementary introduction to the construc-
tion of the number systems encountered by mathematics students in their
first semesters of study. Beginning with the natural numbers, we succes-
sivelyconstruct,alongwiththerequisitealgebraicmachinery,allthenum-
berfieldscontainingthenaturalnumbers,includingtherealnumbers,com-
plexnumbers,andHamiltonianquaternions.Ourexperiencehasshownus
that time is frequently lacking in introductory mathematics courses for a
well-founded construction of number systems; this book represents a con-
tributiontowardfillingthatgap.
Theconstructionofnumbersystemsalsorepresentsanimportantcompo-
nentintheprofessionaleducationofmathematicsteachers.Forthisreason,
this book offers a self-contained and compact construction of the number
systemsthatareofrelevancetodifferentgradelevelsfromamathematical
perspectivewithaviewtowardaspectsofpedagogicalcontentknowledge.
Thisbookarosefromacourseinelementaryabstractalgebraandnumber
theorygivenanumberoftimesattheHumboldtUniversityofBerlin.Parts
of the first-named author’s book Zahlen für Einsteiger: Elemente der Algebra
und Zahlentheorie (Vieweg Verlag, Wiesbaden, 2008) have been revised and
expandedforinclusioninthisnewlyconceivedbookontheconstructionof
numbersystems.Numerousexerciseswithextensivesolutionsfacilitatethe
reader’sengagementwiththesubject.
The completion of this book would not have been possible without the
contributionsofmanyindividuals.HerewewishtothankfirstofallChrista
DobersandMatthiasFischmannfortypingthefirstpartsofthemanuscript.
In addition, we wish to thank all the students whose written course notes
contributed to the text. We also wish to thank our colleagues, in particu-
larAndreasFillerandWolfgangSchulz,fortheirnumeroussuggestionsfor
improvingearlyversionsofthemanuscript.Aspecialwordofthanksgoes
to Olaf Teschke for his work on creating the exercises, and we also thank
Barbara Jung and André Henning for their work on writing up solutions
to the exercises. Finally, we offer hearty thanks to Christoph Eyrich for his
expertsupportindesigningthelayoutofthebookandtoUlrikeSchmickler-
Hirzebruchforherencouragementandsupportonbehalfofthepublisher,
SpringerSpektrum.
Berlin,February2013 JürgKramer
Anna-MariavonPippich
Table of Contents
PrefacetotheEnglishEdition v
PrefacetotheGermanEdition vii
Introduction 1
I TheNaturalNumbers 9
1. ThePeanoAxioms ............................................ 9
2. DivisibilityandPrimeNumbers................................ 15
3. TheFundamentalTheoremofArithmetic ....................... 22
4. GreatestCommonDivisor,LeastCommonMultiple ............. 25
5. DivisionwithRemainder ...................................... 29
A. PrimeNumbers:FactsandConjectures ......................... 32
II TheIntegers 45
1. SemigroupsandMonoids...................................... 45
2. GroupsandSubgroups ........................................ 48
3. GroupHomomorphisms ...................................... 54
4. CosetsandNormalSubgroups ................................. 57
5. QuotientGroupsandtheHomomorphismTheorem ............. 63
6. ConstructionofGroupsfromRegularSemigroups............... 68
7. TheIntegers .................................................. 73
B. RSAEncryption:AnApplicationofNumberTheory............. 77
III TheRationalNumbers 93
1. TheIntegersandDivisibilityTheory............................ 93
2. RingsandSubrings ........................................... 97
3. RingHomomorphisms,Ideals,andQuotientRings ..............102
4. FieldsandSkewFields ........................................110
5. ConstructionofFieldsfromIntegralDomains ...................112
6. TheRationalNumbers ........................................117
7. UniqueFactorizationDomains,PrincipalIdealDomains,and
EuclideanDomains ...........................................119
C. RationalSolutionsofEquations:AFirstGlimpse ................129
IV TheRealNumbers 141
1. DecimalRepresentationofRationalNumbers ...................141
2. ConstructionoftheRealNumbers..............................145
3. TheDecimalExpansionofaRealNumber ......................155
x TableofContents
4. EquivalentCharacterizationsofCompleteness ..................159
5. TheRealNumbersandtheRealNumberLine...................164
6. TheAxiomaticPointofView...................................168
D. The p-adicNumbers:AnotherCompletionofQ .................171
V TheComplexNumbers 183
1. TheComplexNumbersasaRealVectorSpace...................183
2. ComplexNumbersofModulus1andtheSpecialOrthogonal
Group........................................................187
3. TheFundamentalTheoremofAlgebra..........................191
4. AlgebraicandTranscendentalNumbers ........................193
5. TheTranscendenceofe ........................................197
E. ZerosofPolynomials:TheSearchforSolutionFormulas..........204
VI Hamilton’sQuaternions 219
1. Hamilton’sQuaternionsasaRealVectorSpace ..................219
2. QuaternionsofModulus1andtheSpecialUnitaryGroup........223
3. QuaternionsofModulus1andtheSpecialOrthogonalGroup ....227
F. Extensions of Number Systems: What Comes after the
Quaternions? .................................................231
SolutionstoExercises 247
SelectedLiterature 279
Index 283
Introduction
TheDevelopmentoftheIntegersandAlgebra
Oneofmankind’searliestintellectualoccupationswascounting.Thedevel-
opmentoftheconceptsofnumbersandtherepresentationofnumbershas
thereforeassumedaplaceofimportanceinthehistoryofeverycivilization.
The enormous effectiveness of our decimal system of numerical represen-
tation is the culmination of centuries—indeed millennia—of earlier efforts
thattogetherrepresentapowerfulculturalattainment.
Theideaofcountingobjects,thatis,ofbringingasetofequivalentobjects
intoaone-to-onecorrespondencewithafixedsetofnumbers,representsa
significantintellectualprocessofabstraction.
Inmoreadvancedcultures,systemsofsymbolicnotationforthesenum-
bers—somemoreeffectivethanothers—weredeveloped.Wementionpar-
ticularlythecuneiformwritingoftheBabylonians,Egyptianhieroglyphics,
Roman numerals, and the system of numerals developed in India. It was
only in the thirteenth and fourteenth centuries that the Indian positional
decimal system finally made its way via the Islamic world to Western Eu-
rope,whichtothisdayuses“Arabic”numerals.
Thedevelopmentofnumbersystemsgoesrelativelycloselyhandinhand
withthedevelopmentofmethodsofcalculation.Inthisregard,theBabylo-
nianandIndiannumbersystems,forexample,werefarsuperiortothoseof
theEgyptiansandRomans.Nevertheless,untillateinthefifteenthcentury,
inboththeancientcivilizationsandWesternEurope,numericalcalculation
wastheprovinceofasmallgroupofspecialistsknownasarithmeticians.It
wasnotuntilthepublicationinthefifteenthcenturyofAdamRies’sbooks
on calculation, which were based on the book Liber Abaci of Leonardo of
Pisa,knownasFibonacci,thattheusualmethodsofcalculationthatweuse
todaybecameaccessibletothe“commonpeople.”Thediffusionofcalcula-
tionaltechniquesislinkedtoasystemizationofarithmeticintheacademic
world, which then led to the development of algebra. At first, algebra was
viewedprimarilyasapracticaltool,butitgraduallytookonalifeofitsown
andeventuallydevelopedintotheindependentdisciplinethatweknowto-
day.Algebrawillthereforeplayasignificantroleineveryrigorousscientifi-
callybasedconstructionofnumbersystems.