Number Systems: A Path into Rigorous Mathematics

317 Pages·2021·10.436 MB·English

by  Anthony Kay

Number Systems

Number Systems
A Path into Rigorous Mathematics
Anthony Kay
First edition published 2022
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Contents
Preface and Acknowledgments ix
1 Introduction: The Purpose of This Book 1
1.1 A Very Brief Historical Context . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Axiomatic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 The Place of Number Systems within Mathematics . . . . . . . . . . . . . 2
1.4 Mathematical Writing, Notation, and Terminology . . . . . . . . . . . . . . 3
1.5 Logic and Methods of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Sets and Relations 7
2.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.2 Subsets and Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.3 Union, Intersection, and Complement . . . . . . . . . . . . . . . . . 9
2.1.4 Ordered Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Relations between Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Relations in General . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Relations on a Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.2 Order Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.3 Transitivity and Proofs . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Binary Operations and Algebraic Structures . . . . . . . . . . . . . . . . . 17
3 Natural Numbers, N 19
3.1 Peano’s Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Addition of Natural Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Multiplication of Natural Numbers . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 Exponentiation (Powers) of Natural Numbers . . . . . . . . . . . . . . . . . 27
3.5 Order in the Natural Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.6 Bounded Sets in N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.7 Cardinality, Finite and Infinite Sets . . . . . . . . . . . . . . . . . . . . . . 36
3.7.1 Some Useful Notations . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.7.2 Finite Sets, Their Subsets and Injections. . . . . . . . . . . . . . . . 39
3.7.3 Finiteness and Boundedness of Sets . . . . . . . . . . . . . . . . . . 41
3.7.4 Infinite Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.8 Subtraction: The Inverse of Addition . . . . . . . . . . . . . . . . . . . . . 44
v
vi Contents
4 Integers, Z 49
4.1 Definition of the Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Arithmetic on Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3 Algebraic Structure of Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3.1 An Abelian Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3.2 A Commutative Ring . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4 Order in Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4.1 How to Solve Inequalities . . . . . . . . . . . . . . . . . . . . . . . . 66
4.5 Finite, Infinite, and Bounded Sets in Z . . . . . . . . . . . . . . . . . . . . 70
5 Foundations of Number Theory 73
5.1 Integer Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Expressing Integers in Any Base . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3 Prime Numbers and Prime Factorisation . . . . . . . . . . . . . . . . . . . 80
5.3.1 Prime Numbers and Prime Factorisation in N . . . . . . . . . . . . . 80
5.3.2 Primes in Z and Other Number Systems . . . . . . . . . . . . . . . . 86
5.4 Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.5 Modular Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.6 Z as an Algebraic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 97
d
6 Rational Numbers, Q 107
6.1 Definition of the Rationals . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.2 Addition and Multiplication on Q . . . . . . . . . . . . . . . . . . . . . . . 109
6.3 Countability of Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.4 Exponentiation and Its Inverse(s) on Q . . . . . . . . . . . . . . . . . . . . 117
6.4.1 Integer Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.4.2 Roots and Fractional Powers . . . . . . . . . . . . . . . . . . . . . . 118
6.4.3 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.5 Order in Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.6 Bounded Sets in Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.7 Expressing Rational Numbers in Any Base . . . . . . . . . . . . . . . . . . 133
6.7.1 Terminating Base-b Representations . . . . . . . . . . . . . . . . . . 135
6.7.2 Repeating Base-b Representations . . . . . . . . . . . . . . . . . . . 137
6.7.3 Fractions from Repeating Base-b Representations . . . . . . . . . . . 141
6.8 Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7 Real Numbers, R 149
7.1 The Requirements for Our Next Number System . . . . . . . . . . . . . . . 149
7.2 Dedekind Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.3 Order and Bounded Sets in R . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.4 Addition in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.5 Multiplication in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.6 Exponentiation in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
7.7 Expressing Real Numbers in Any Base . . . . . . . . . . . . . . . . . . . . 169
7.8 Cardinality of R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.9 Algebraic and Transcendental Numbers . . . . . . . . . . . . . . . . . . . . 177
Contents vii
8 Quadratic Extensions I: General Concepts and Extensions of Z and Q 183
8.1 General Concepts of Quadratic Extensions . . . . . . . . . . . . . . . . . . 183
8.2 Introduction to Quadratic Rings: Extensions of Z . . . . . . . . . . . . . . 190
√
8.3 Units in Z[ k] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
√
8.4 Primes in Z[ k] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
8.4.1 Basic Theorems about Primes . . . . . . . . . . . . . . . . . . . . . . 195
8.4.2 Associates Classes and Conjugates of Primes . . . . . . . . . . . . . 196
8.4.3 How to Search for Primes . . . . . . . . . . . . . . . . . . . . . . . . 199
√
8.5 Prime Factorisation in Z[ k] . . . . . . . . . . . . . . . . . . . . . . . . . . 202
8.6 Quadratic Fields: Extensions of Q . . . . . . . . . . . . . . . . . . . . . . . 205
8.6.1 Algebraic Numbers in Quadratic Fields . . . . . . . . . . . . . . . . 206
8.6.2 Quadratic Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
8.7 Norm-Euclidean Rings and Unique Prime Factorisation . . . . . . . . . . . 212
9 Quadratic Extensions II: Complex Numbers, C 221
9.1 Complex Numbers as a Quadratic Extension . . . . . . . . . . . . . . . . . 221
9.2 Exponentiation by Real Powers in C: A First Approach . . . . . . . . . . . 224
9.3 Geometry of C; the Principal Value of the Argument, and the Number π . 227
9.3.1 The Unit Circle and the Principal Value of the Argument of a
Complex Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
9.3.2 The Number π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
9.4 Use of the Argument to Define Real Powers in C . . . . . . . . . . . . . . . 238
9.4.1 The PVA of a Product . . . . . . . . . . . . . . . . . . . . . . . . . . 238
9.4.2 The Multiple-Valued Argument and the Definition of Real Powers . 243
9.4.3 Evaluating Rational Powers of Complex Numbers. . . . . . . . . . . 247
9.5 Exponentiation by Complex Powers; the Number e . . . . . . . . . . . . . 250
9.5.1 The Number e and Its Powers. . . . . . . . . . . . . . . . . . . . . . 252
9.5.2 General Exponentiation and Logarithms in C . . . . . . . . . . . . . 259
9.5.3 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . 260
9.6 The Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . . . . 263
9.6.1 Factorisation of Polynomials . . . . . . . . . . . . . . . . . . . . . . 269
9.7 Cardinality of C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
10 Yet More Number Systems 275
10.1 Constructible Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
10.2 Hypercomplex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
11 Where Do We Go from Here? 289
11.1 Number Theory and Abstract Algebra . . . . . . . . . . . . . . . . . . . . . 289
11.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
A How to Read Proofs: The “Self-Explanation” Strategy 291
Bibliography 297
Index 299

Preface and Acknowledgments
For me, being a mathematician is not a profession; it is a genetic condition. One of my
earliest memories, from age 3 or 4, is that having realised that the decimal system enabled
me to continue counting indefinitely, I would silently count to myself while getting on with
whatever I was doing; on one occasion I got to 1,112 before being interrupted, and this
number has stayed in my memory ever since that day. So from an early age I appreciated
mathematical concepts as abstractions; but later, during my last two years at school, a
brilliantPhysicsteacher,BarryJackson,showedmehowexcitingitcanbetoapplymathe-
maticstoproblemsinphysics.Thissetmeonacareerasanappliedmathematician,mainly
working in fluid mechanics.
However,Ineverlostmyloveofnumbers,ormyappreciationofmathematicalbeautyfor
itsownsake.MycolleaguesatLoughboroughUniversityunderstoodthis:whenanewfirst-
year module on “Numbers” was introduced to the syllabus in 2001 by Andrew Osbaldestin
in his capacity as Teaching Coordinator, he decided that I was the best person to teach it.
He also very kindly provided a rather vague module specification; this left me free to read
aroundthesubjectandthendecideformyselfwhatIwasgoingtoteach.Chapters3to7of
thisbook,ontherigoroustheoryofnumbersystemsfromtheNaturalNumberstotheReal
Numbers, contain all the material that I taught in this module, and a substantial amount
of further material that could not be included in a module of 22 lectures. Chapters 8 to 10
contain material on further number systems, in particular the Complex Numbers, which I
would also have included if more time had been available.
Unlike every other module that I have taught (and there have been many!), I did not
feelthatanyoftheexistingbooksonthissubjectsatisfiedalltherequirementsforstudents:
coveringallthematerialinthemodule,setattherightlevelofmathematicalsophistication,
and clearly presented. So after some years, I conceived the idea of writing a book myself: I
wanted to cover the theory of number systems rigorously, assuming no other mathematical
knowledge apart from Na¨ıve Set Theory; and the book should be suitable for students with
nopreviousexperienceoftherigorousdevelopmentofmathematicaltheoryfromaxioms.So
therigourwouldneedtobetemperedwithexplanationsthatappealedtostudents’intuition.
JustwhenIeventuallyfeltreadytostartwritingmybook,thereappearedIanStewartand
David Tall’s The Foundations of Mathematics [Second edition], which does a brilliant job
of introducing students to the rigorous theory of number systems and much else. I almost
wantedtoabandonmyproject;butStewartandTallhaveabroadercanvasthanIintended
to cover, while I wanted to go into more detail on many aspects. I can only hope that
my work approaches the standard of clarity provided by those eminent authors. Whereas
Edmund Landau, whose Foundations of Analysis is one of the earliest books to cover the
theoryofnumbersystems,describeshisbookasbeingwrittenin“mercilesstelegraphstyle”,
mypursuitofclaritytendstotakemetotheoppositeextreme.Itrustthatmyexplanations
will be regarded as thorough and unambiguous, rather than simply verbose!
Althoughthebookgrewoutofamoduleforfirst-yearuniversitystudents,itdoesinclude
some more advanced material. Here is a brief guide to what you will find in each chapter.
Following the introductory notes in Chapter 1, the next chapter covers all the Set Theory
that readers will be assumed to know in the remainder of the book. This is Na¨ıve Set
ix