什么是数学：对思想和方法的基本研究

PREFACETOSECONDEDITION

PREFACETOREVISEDEDITIONS

PREFACETOFIRSTEDITION

HowTOUSETHEBOOK

WHATISMATHEMATICS?

CHAPTERⅠ.THENATURALNUMBERS

Introduction

1.CalculationwithIntegers

1.LawsofArithraetic.2.TheRepresentationofIntegers.3.ComputationinSystemsOtherthantheDecimal.

2.TheInfinitudeoftheNumberSystem,MathematicalInduction

1.ThePrincipleofMathematical.Induction.2.TheArithmeticalProgression.3.TheGeometricalProgression.4.TheSumoftheFirstnSquares.5.AnImportantInequality.6.TheBinomialTheorem.7.FurtherRemarksonMathematicalInduction.

SUPPLEMENTTOCHAPTERI.THETHEORYOFNUMBERS

Introduction

1.ThePrimeNumbers

1.FundamentalFacts.2.TheDistributionofthePrimes.3.FormulasProducingPrimes.b.PrimesinAritluneticalProgressions.c.ThePrimeNumberTheorem.d.TwoUnsolvedProblemsConcerningPrimeNumbers.

2.Congruences

1.GeneralConcepts.2.FermatsTheorem.3.QuadraticResidues.

3.PythagoreanNumbersandFermatsLastTheorem

4.TheEuclideanAlgorithm

1.GeneralTheory.2.ApplicationtotheFundamentalTheoremofArithmetic.3.EulersFunction.FermatsTheoremAgain.4.ContinuedFractions.DiophantineEquations.

CHAPTERⅡ.THENUMBERSYSTEMOFMATHEMATICS

Introduction

1.TheRationalNumbers

1.RationalNumbersasaDeviceforMeasuring.2.IntrinsicNeedfortheRationalNumbers.PrincipalofGeneration.3.GeometricalInterpretationofRationalNumbers.

2.IncommensurableSegments,IrrationalNumbers,andtheConceptofLimit

1.Introduction.2.DecimalFractions.InfiniteDecimals.3.Limits.InfiniteGeometricalSeries.4.RationalNumbersandPeriodicDeci-maiN.5.GeneralDefinitionofIrrationalNumbersbyNested

Intervals6.AlternativeMethodsofDefiningIrrationalNumbers.DedekindCuts.

3.RemarksonAnalyticGeometry

1.TheBasicPrinciple.2.EquationsofLinesandCurves.

4.TheMathematicalAnalysisofInfinity

1.FundamentalConcepts.2.TheDenumerabilityoftheRationalNumbersandtheNon-DenumerabilityoftheContinuum.3.Cantors"CardinalNumbers."4.TheIndirectMethodofProof.5.TheParadoxesoftheInfinite.6.TheFoundationsofMathematics.

5.ComplexNumbers

1.TheOriginofComplexNumbers.2.TheGeometricalInterpretationofComplexNumbers.3.DeMoivresFormulaandtheRootsofUnity.4.TheFundamentalTheoremofAlgebra.

6.AlgebraicandTranscendentalNumbers

1.DefinitionandExistence.2.LiouvillesTheoremandtheConstructionofTranscendentalNumbers.

SUPPLEMENTTOCHAPTERII.THEALGEBRAOFSETS

1.GeneralTheory.2.ApplicationtoMathematicalLogic.3.AnApplicationtotheTheoryofProbability.

CHAPTERⅠ.GEOMETRICALCONSTRUCTIONS.THEALGEBRAOFNUMBERFIELDS

Introduction

PartⅠ.ImpossibilityProofsandAlgebra

1.FundamentalGeometricalConstructions

1.ConstructionofFieldsandSquareRootExtraction.2.RegularPolygons.3.ApolloniusProblem.

2.ConstructibleNumbersandNumberFields

1.GeneralTheory.2.AllConstructibleNumbersareAlgebraic.

3.TheUnsolvabilityoftheThreeGreekProblems

1.DoublingtheCube.2.ATheoremonCubicEquations.3.TrisectingtheAngle.4.TheRegularHeptagon.5.RemarksontheProblemofSquaringtheCircle.

PartⅡ.VariousMethodsforPerformingConstructions

4.GeometricalTransformations.Inversion

1.GeneralRemarks.2.PropertiesofInversion.3.GeometricalConstrnctionofInversePoints.4.HowtoBisectaSegmentandFindtheCenterofaCirclewiththeCompassAlone.

5.ConstructionswithOtherTools.MascheroniConstructionswithCompassAlone

1.AClassicalConstructionforDoublingtheCube.2.RestrictiontotheUseoftheCompassAlone.3.DrawingwithMechanicalInstruments.MechanicalCurves.Cycloids.4.Linkages.PeauceUiersandHartsInversors.

6.MoreAboutInversionsanditsApplications

1.InvarianceofAngles.FamiliesofCircles.2.ApplicationtotheProblemofApollonius.3.RepeatedReflections.

CHAPTERⅣ.PROJECTIVEGEOMETRY.AXIOMATICS.NON-EucLIDEANGEOMETRIES.

1.Introduction

……

CHAPTERⅤTOPOLOGY

CHAPTERⅥFUNCTIONSANDLIMITS

CHAPTERⅦMAXIMAANDMINIMA

CHAPTERⅧTHECALCULUS

CHAPTERⅨRECENTDEVELOPMENTS

APPENDIX:SUPPLEMENTARYREMARKS,PROBLEMS,ANDEXERCISES

SUGGESTIONSFORFURTHERREADING

SUGGESTIONSFORADDITIONALREADING

INDEX

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