Elements of Mathematics: From Euclid to Gödel

439 Pages·2016·4.41 MB·English

by  John Stillwell

Elements of Mathematics

Elements of
Mathematics
FROM EUCLID
TO GÖDEL
John Stillwell
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
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Title:Elementsofmathematics:fromEuclidtoGödel/JohnStillwell.
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Identifiers:LCCN2015045022|ISBN9780691171685(hardcover:alk.paper)
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To Hartley Rogers Jr.
In Memoriam

Contents
Preface xi
1 ElementaryTopics 1
1.1 Arithmetic 2
1.2 Computation 4
1.3 Algebra 7
1.4 Geometry 9
1.5 Calculus 13
1.6 Combinatorics 16
1.7 Probability 20
1.8 Logic 22
1.9 HistoricalRemarks 25
1.10 PhilosophicalRemarks 32
2 Arithmetic 35
2.1 TheEuclideanAlgorithm 36
2.2 ContinuedFractions 38
2.3 PrimeNumbers 40
2.4 FiniteArithmetic 44
2.5 QuadraticIntegers 46
2.6 TheGaussianIntegers 49
2.7 Euler’sProofRevisited 54
√
2.8 2andthePellEquation 57
2.9 HistoricalRemarks 60
2.10 PhilosophicalRemarks 67
3 Computation 73
3.1 Numerals 74
3.2 Addition 77
3.3 Multiplication 79
3.4 Division 82
3.5 Exponentiation 84
viii • Contents
3.6 PandNPProblems 87
3.7 TuringMachines 90
∗
3.8 UnsolvableProblems 94
∗
3.9 UniversalMachines 97
3.10 HistoricalRemarks 98
3.11 PhilosophicalRemarks 103
4 Algebra 106
4.1 ClassicalAlgebra 107
4.2 Rings 112
4.3 Fields 117
4.4 TwoTheoremsInvolvingInverses 120
4.5 VectorSpaces 123
4.6 LinearDependence,Basis,andDimension 126
4.7 RingsofPolynomials 128
4.8 AlgebraicNumberFields 133
4.9 NumberFieldsasVectorSpaces 136
4.10 HistoricalRemarks 139
4.11 PhilosophicalRemarks 143
5 Geometry 148
5.1 NumbersandGeometry 149
5.2 Euclid’sTheoryofAngles 150
5.3 Euclid’sTheoryofArea 153
5.4 StraightedgeandCompassConstructions 159
5.5 GeometricRealizationofAlgebraicOperations 161
5.6 AlgebraicRealizationofGeometric
Constructions 164
5.7 VectorSpaceGeometry 168
5.8 IntroducingLengthviatheInnerProduct 171
5.9 ConstructibleNumberFields 175
5.10 HistoricalRemarks 177
5.11 PhilosophicalRemarks 184
6 Calculus 193
6.1 GeometricSeries 194
6.2 TangentsandDifferentiation 197
Contents • ix
6.3 CalculatingDerivatives 202
6.4 CurvedAreas 208
6.5 TheAreaundery=xn 211
∗
6.6 TheFundamentalTheoremofCalculus 214
6.7 PowerSeriesfortheLogarithm 218
6.8 ∗TheInverseTangentFunctionandπ 226
6.9 ElementaryFunctions 229
6.10 HistoricalRemarks 233
6.11 PhilosophicalRemarks 239
7 Combinatorics 243
7.1 TheInfinitudeofPrimes 244
7.2 BinomialCoefficientsandFermat’sLittle
Theorem 245
7.3 GeneratingFunctions 246
7.4 GraphTheory 250
7.5 Trees 252
7.6 PlanarGraphs 254
7.7 TheEulerPolyhedronFormula 256
7.8 NonplanarGraphs 263
∗
7.9 TheKo˝nigInfinityLemma 264
7.10 Sperner’sLemma 268
7.11 HistoricalRemarks 272
7.12 PhilosophicalRemarks 274
8 Probability 279
8.1 ProbabilityandCombinatorics 280
8.2 Gambler’sRuin 282
8.3 RandomWalk 284
8.4 Mean,Variance,andStandardDeviation 286
∗
8.5 TheBellCurve 290
8.6 HistoricalRemarks 292
8.7 PhilosophicalRemarks 296
9 Logic 298
9.1 PropositionalLogic 299
9.2 Tautologies,Identities,andSatisfiability 302