Preface
Computerscienceisaverylargesubject,andgraduatespursueawidevarietyof careers,includingprogramming,systemsdesign,cryptography,websitedesign,realtimesystems,computeranimation,computergames,datavisualisation,etc.Consequently,itisimpossibletowriteamathematicsbookthatcaterstoallofthesecareer paths.Nevertheless,Ihaveattemptedtodescribearangeofmathematicaltopicsthat Ibelievearerelevant,andhavehelpedmeduringmyowncareerincomputerscience. Thebook’ssubtitle‘AVisualApproach’reflectstheimportanceIplaceoncoloured illustrationsandfunctiongraphs,ofwhichthereareover210and90tables.Many chapterscontainavarietyofworkedexamples.
Thisthirdeditionremainsanintroductorytext,andisaimedatstudentsstudying foranundergraduatedegreeincomputerscience.Therearenownineteenchaptersonnumbers,counting,algebra,logic,combinatorics,probability,modulararithmetic,trigonometry,coordinatesystems,determinants,vectors,complexnumbers, matrices,geometricmatrixtransforms,differentiation,integration,areaandvolume, whichshouldprovidereaderswithasolidfoundation,uponwhichmoreadvanced topicsofmathematicscanbestudied.
Ihavereferencedthekeypeoplebehindthevariousmathematicaldiscoveries covered,whichIhopeaddsahumandimensiontothesubject.Ihavefounditvery interestingandentertainingtodiscoverhowsomemathematiciansridiculedtheir fellowpeers,whentheycouldnotcomprehendthesignificanceofanewinvention.
ThereisnowayIcouldhavewrittenthisbookwithouttheassistanceoftheInternet andmybookspreviouslypublishedbySpringerVerlag.Inparticular,Iwouldliketo acknowledgeWikipediaandRichardElwes’excellentbook Maths1001.Iprepared thisbookonanAppleiMac,using LaTeX 2e ,PagesandtheGrapherpackage, andwouldrecommendthiscombinationtoanyoneconsideringwritingabookon mathematics.Idohopeyouenjoyreadingthisbook,andthatyouaretemptedto studymathematicstoadeeperlevel.
Breinton,Herefordshire,UK October2022
JohnVince
1VisualMathematics
2.5.1TheArithmeticofPositiveandNegative
2.7.9QuaternionsandOctonions.....................19
2.8.5InfinityofPrimes..............................24
2.8.6MersenneNumbers............................25
3.2.2BinaryNumbers...............................32
3.2.3OctalNumbers................................33 3.2.4HexadecimalNumbers.........................33
3.3ConvertingDecimaltoBinary,OctalandHexadecimal.......34
3.3.1ConvertingDecimaltoBinary...................34
3.3.2ConvertingDecimaltoOctal....................35
3.3.3ConvertingDecimaltoHexadecimal.............36
3.4ConvertingBetweenBinaryandOctalNumbers.............38
3.5ConvertingBetweenBinaryandHexadecimalNumbers......39
3.6AddingandSubtractingBinaryNumbers...................41
3.6.1AddingBinaryNumbers........................41
3.6.2SubtractingBinaryNumbersUsingTwo’s Complement..................................42
3.7AddingandSubtractingDecimalNumbers..................42
3.7.1AddingDecimalNumbers......................42
3.7.2SubtractingDecimalNumbersUsingTen’s Complement..................................43
3.8AddingandSubtractingOctalNumbers....................44
3.8.1AddingOctalNumbers.........................44
3.8.2SubtractingOctalNumbersUsingEight’s Complement..................................44 3.9Summary..............................................45
3.10WorkedExamples.......................................45
3.10.1ConvertaDecimalNumberintoBinary...........45
3.10.2ConvertaDecimalNumberintoBinaryUsing anAlgorithm.................................45
3.10.3ConvertaBinaryNumberintoDecimal...........46
3.10.4ConvertaBinaryNumberintoOctal.............46
3.10.5ConvertanOctalNumberintoBinary............46
3.10.6ConvertanOctalNumberintoHexadecimal.......46
3.10.7ConvertaHexadecimalNumberintoOctal........47
3.10.8ConvertaDecimalNumberintoOctal............47
3.10.9ConvertaDecimalNumberintoOctalUsing anAlgorithm.................................47
3.10.10ConvertaDecimalNumberintoHexadecimal.....48
3.10.11AddBinaryNumbers..........................48
3.10.12SubtractBinaryNumbers.......................49
3.10.13AddOctalNumbers............................49
4.7.4FunctionDomainsandRanges..................63
5Logic
5.3.1LogicalConnectives...........................74
5.4LogicalPremises........................................75
5.4.1MaterialEquivalence...........................75
5.4.2Implication...................................76
5.4.3Negation.....................................77
5.4.4Conjunction..................................77
5.4.5InclusiveDisjunction...........................78
5.4.6ExclusiveDisjunction..........................79 5.4.7Idempotence..................................79 5.4.8Commutativity................................79
5.4.9Associativity..................................80
5.4.10Distributivity.................................82
5.4.11deMorgan’sLaws.............................83
5.4.12Simplification.................................83
5.4.13ExcludedMiddle..............................84
5.4.14Contradiction.................................85
5.4.15DoubleNegation..............................85
5.4.16ImplicationandEquivalence....................85
5.4.17Exportation...................................86
5.4.18Contrapositive................................86
5.4.19ReductioAdAbsurdum........................87
5.4.20ModusPonens................................88
5.4.21ProofbyCases................................89
5.5SetTheory.............................................91
5.5.1EmptySet....................................91
5.5.2MembershipandCardinalityofaSet.............91
5.5.3Subsets,SupersetsandtheUniversalSet..........92
5.5.4SetBuilding..................................92
5.5.5Union........................................93
5.5.6Intersection...................................94
5.5.7RelativeComplement..........................94
5.5.8AbsoluteComplement.........................96
5.5.9PowerSet....................................96
5.6WorkedExamples.......................................97
5.6.1TruthTables..................................97
5.6.2SetBuilding..................................97
5.6.3Sets.........................................99
5.6.4PowerSet....................................99
6Combinatorics ................................................101
6.1Introduction............................................101
6.2Permutations...........................................101
6.3PermutationsofMultisets................................104
6.4Combinations...........................................105
6.5WorkedExamples.......................................107
6.5.1Eight-PermutationsofaMultiset.................107
6.5.2Eight-PermutationsofaMultiset.................108
6.5.3NumberofPermutations........................109
6.5.4NumberofFive-CardHands....................109
6.5.5HandShakeswith100People...................109
6.5.6PermutationsofMISSISSIPPI...................110
7Probability ...................................................111
7.1Introduction............................................111
7.2DefinitionandNotation..................................111
7.2.1IndependentEvents............................113
7.2.2DependentEvents.............................113
7.2.3MutuallyExclusiveEvents......................114
7.2.4InclusiveEvents...............................115
7.2.5ProbabilityUsingCombinations.................115
7.3WorkedExamples.......................................117
7.3.1ProductofProbabilities........................117
7.3.2BookArrangements............................118
7.3.3WinningaLottery.............................118
7.3.4RollingTwoDice..............................118
7.3.5TwoDiceSumto7............................118
7.3.6TwoDiceSumto4............................119
7.3.7DealingaRedAce.............................119
7.3.8SelectingFourAcesinSuccession...............119
7.3.9SelectingCards...............................119
7.3.10SelectingFourBallsfromaBag.................120
7.3.11FormingTeams...............................120
7.3.12DealingFiveCards............................121
8ModularArithmetic
8.1Introduction............................................123
8.2InformalDefinition......................................123
8.3Notation...............................................123
8.4Congruence............................................124
8.5NegativeNumbers.......................................125
8.6ArithmeticOperations...................................125
8.6.1SumsofNumbers.............................126
8.6.2Products.....................................127
8.6.3MultiplyingbyaConstant......................127
8.6.4CongruentPairs...............................128
8.6.5MultiplicativeInverse..........................128
8.6.6ModuloaPrime...............................130
8.6.7Fermat’sLittleTheorem........................133
8.7ApplicationsofModularArithmetic.......................133
8.7.1ISBNParityCheck............................133
8.7.2IBANCheckDigits............................134
8.8WorkedExamples.......................................136
8.8.1NegativeNumbers.............................136
8.8.2SumsofNumbers.............................136
8.8.3RemaindersofProducts........................137
8.8.4MultiplicativeInverse..........................138
8.8.5ProductTableforModulo13....................138
8.8.6ISBNCheckDigit.............................139 Reference.....................................................139
9Trigonometry
9.1Introduction............................................141 9.2Background............................................141
9.3UnitsofAngularMeasurement............................141
9.4TheTrigonometricRatios................................142
9.4.1DomainsandRanges...........................145
9.5InverseTrigonometricRatios.............................145
9.6TrigonometricIdentities..................................147
9.7TheSineRule..........................................148
9.8TheCosineRule........................................148
9.9Compound-AngleIdentities..............................149
9.9.1Double-AngleIdentities........................150
9.9.2Multiple-AngleIdentities.......................151
9.9.3Half-AngleIdentities...........................152
9.10PerimeterRelationships..................................153
9.11WorkedExamples.......................................153
9.11.1DegreestoRadians............................153
9.11.2SineRule.....................................154
9.11.3CosineRule..................................154
9.11.4CompoundAngle..............................155
9.11.5Double-AngleIdentity.........................155
9.11.6PerimeterRelationship.........................156
10CoordinateSystems ...........................................157
10.1Introduction............................................157
10.2Background............................................157
10.3TheCartesianPlane.....................................158
10.4FunctionGraphs........................................158
10.5ShapeRepresentation....................................159
10.5.12DPolygons..................................159
10.5.2AreasofShapes...............................160
10.6TheoremofPythagorasin2D.............................161
10.6.1PythagoreanTriples............................161 10.73DCartesianCoordinates................................162
10.7.1TheoremofPythagorasin3D...................163
10.8PolarCoordinates.......................................163
10.9SphericalPolarCoordinates..............................164
10.10CylindricalCoordinates..................................165
10.11BarycentricCoordinates..................................166
10.12HomogeneousCoordinates...............................167
10.13WorkedExamples.......................................167
10.13.1AreaofaShape...............................167
10.13.2DistanceBetweenTwoPoints...................168
10.13.3PolarCoordinates.............................168
10.13.4SphericalPolarCoordinates.....................169
10.13.5CylindricalCoordinates........................169
10.13.6BarycentricCoordinates........................170 Reference.....................................................171
11Determinants
11.3LinearEquationswithTwoVariables.......................174 11.4LinearEquationswithThreeVariables.....................178
11.4.1Sarrus’sRule.................................184
11.5MathematicalNotation...................................184
11.5.1Matrix.......................................185
11.5.2OrderofaDeterminant.........................185
11.5.3ValueofaDeterminant.........................185
11.5.4PropertiesofDeterminants......................187
11.6WorkedExamples.......................................188
11.6.1DeterminantExpansion.........................188
11.6.2ComplexDeterminant..........................188
11.6.3SimpleExpansion.............................189
11.6.4SimultaneousEquations........................189
12Vectors
12.32DVectors.............................................192
12.3.1VectorNotation...............................192
12.3.2GraphicalRepresentationofVectors..............193
12.3.3MagnitudeofaVector..........................194
12.43DVectors.............................................195
12.4.1VectorManipulation...........................196
12.4.2ScalingaVector...............................196
12.4.3VectorAdditionandSubtraction.................197
12.4.4PositionVectors...............................198
12.4.5UnitVectors..................................199
12.4.6CartesianVectors..............................199
12.4.7Products.....................................200
12.4.8ScalarProduct................................200
12.4.9TheDotProductinLightingCalculations.........202
12.4.10TheScalarProductinBack-FaceDetection........203
12.4.11TheVectorProduct............................204
12.4.12TheRight-HandRule..........................209
12.5DerivingaUnitNormalVectorforaTriangle...............209
12.6SurfaceAreas...........................................210
12.6.1Calculating2DAreas..........................211 12.7Summary..............................................212
12.8WorkedExamples.......................................212
12.8.1PositionVector................................212
12.8.2UnitVector...................................212
12.8.3VectorMagnitude.............................213
12.8.4AngleBetweenTwoVectors....................213
12.8.5VectorProduct................................213
13.2RepresentingComplexNumbers..........................215
13.2.1ComplexNumbers.............................215
13.2.2RealandImaginaryParts.......................216
13.2.3TheComplexPlane............................216
13.3ComplexAlgebra.......................................216
13.3.1AlgebraicLaws...............................216
13.3.2ComplexConjugate............................218
13.3.3ComplexDivision.............................220
13.3.4Powersof i ...................................221
13.3.5RotationalQualitiesof i ........................222
13.3.6ModulusandArgument........................224
13.3.7ComplexNorm................................226
13.3.8ComplexInverse..............................227
13.3.9ComplexExponentials.........................228
13.3.10deMoivre’sTheorem..........................232
13.3.11 n thRootofUnity..............................234
13.3.12 n thRootsofaComplexNumber.................235
13.3.13LogarithmofaComplexNumber................236
13.3.14RaisingaComplexNumbertoaComplex Power........................................237
13.3.15VisualisingSimpleComplexFunctions...........239
13.3.16TheHyperbolicFunctions......................243
13.4Summary..............................................244
13.5WorkedExamples.......................................244
13.5.1ComplexAddition.............................244
13.5.2ComplexProducts.............................245
13.5.3ComplexDivision.............................245
13.5.4ComplexRotation.............................245
13.5.5PolarNotation................................246
13.5.6RealandImaginaryParts.......................246
13.5.7MagnitudeofaComplexNumber................247
13.5.8ComplexNorm................................247
13.5.9ComplexInverse..............................248
13.5.10deMoivre’sTheorem..........................248 13.5.11 n thRootofUnity..............................249
13.5.12RootsofaComplexNumber....................250
13.5.13LogarithmofaComplexNumber................251
13.5.14RaisingaNumbertoaComplexPower...........251 References....................................................251
14Matrices
14.4MatrixNotation.........................................258
14.4.1MatrixDimensionorOrder.....................259
14.4.2SquareMatrix.................................259
14.4.3ColumnVector................................259
14.4.4RowVector...................................259
14.4.5NullMatrix...................................260
14.4.6UnitMatrix...................................260
14.4.7Trace........................................261
14.4.8DeterminantofaMatrix........................262
14.4.9Transpose....................................262
14.4.10SymmetricMatrix.............................263
14.4.11AntisymmetricMatrix..........................265
14.5MatrixAdditionandSubtraction..........................267
14.5.1ScalarMultiplication...........................268 14.6MatrixProducts.........................................268
14.6.1RowandColumnVectors.......................268
14.6.2RowVectorandaMatrix.......................269
14.6.3MatrixandaColumnVector....................270
14.6.4SquareMatrices...............................271
14.6.5RectangularMatrices..........................272
14.7InverseMatrix..........................................273
14.7.1InvertingaPairofMatrices.....................280
14.8OrthogonalMatrix......................................281
14.9DiagonalMatrix........................................282
14.10WorkedExamples.......................................282
14.10.1MatrixInversion...............................282
14.10.2IdentityMatrix................................283
14.10.3SolvingTwoEquationsUsingMatrices...........284
14.10.4SolvingThreeEquationsUsingMatrices..........285
14.10.5SolvingTwoComplexEquations................286
14.10.6SolvingThreeComplexEquations...............287
14.10.7SolvingTwoComplexEquations................288
14.10.8SolvingThreeComplexEquations...............289
15.2.22DScaling...................................295
15.2.32DReflections................................297
15.2.42DShearing..................................299
15.2.52DRotation..................................300
15.2.62DScaling...................................303
15.2.72DReflection.................................304
15.2.82DRotationAboutanArbitraryPoint............305 15.33DTransforms..........................................306
15.3.13DTranslation................................306
15.3.23DScaling...................................306
15.3.33DRotation..................................307
15.3.4RotatingAboutanAxis........................311
15.3.53DReflections................................312
15.4RotatingaPointAboutanArbitraryAxis...................313
15.4.1Matrices.....................................313
15.5DeterminantofaTransform..............................316
15.6PerspectiveProjection...................................317
15.7WorkedExamples.......................................320
15.7.12DScaleandTranslate.........................320
15.7.22DRotation..................................321
15.7.3DeterminantoftheRotateTransform.............322
15.7.4DeterminantoftheShearTransform..............322
15.7.5Yaw,PitchandRollTransforms.................322
15.7.6RotationAboutanArbitraryAxis................323
15.7.73DRotationTransformMatrix..................324
15.7.8PerspectiveProjection..........................325
16.4.4GraphicalInterpretationoftheDerivative.........334
16.7DifferentiatingImplicitFunctions.........................349
16.8DifferentiatingExponentialandLogarithmicFunctions.......352
16.9.3Differentiatingsec.............................359 16.9.4Differentiatingcot.............................360
16.9.5Differentiatingarcsin,arccosandarctan..........361
16.9.6Differentiatingarccsc,arcsecandarccot..........362
16.10DifferentiatingHyperbolicFunctions.......................362
16.10.1Differentiatingsinh,coshandtanh...............364
16.11HigherDerivatives......................................366
16.12HigherDerivativesofaPolynomial........................366
16.13IdentifyingaLocalMaximumorMinimum.................369
16.14PartialDerivatives.......................................370
16.14.1VisualisingPartialDerivatives...................373 16.14.2MixedPartialDerivatives.......................375
16.15ChainRule.............................................376
16.16TotalDerivative.........................................378
16.17PowerSeries...........................................379
16.18WorkedExamples.......................................382
16.18.1Antiderivative1...............................382
16.18.2Antiderivative2...............................382
16.18.3DifferentiatingSumsofFunctions...............383
16.18.4DifferentiatingaFunctionProduct...............383
16.18.5DifferentiatinganImplicitFunction..............383
16.18.6DifferentiatingaGeneralImplicitFunction........384
16.18.7LocalMaximumorMinimum...................385
16.18.8PartialDerivatives.............................386
16.18.9MixedPartialDerivative1......................386
16.18.10MixedPartialDerivative2......................387
16.18.11TotalDerivative...............................387
17Calculus:Integration ..........................................389
17.1Introduction............................................389
17.2IndefiniteIntegral.......................................389 17.3IntegrationTechniques...................................390
17.3.1ContinuousFunctions..........................390
17.3.2DifficultFunctions.............................391 17.4TrigonometricIdentities..................................392
17.4.1ExponentNotation.............................395
17.4.2CompletingtheSquare.........................396
17.4.3TheIntegrandContainsaDerivative..............398
17.4.4ConvertingtheIntegrandintoaSeries ofFractions...................................400
17.4.5IntegrationbyParts............................401
17.4.6IntegrationbySubstitution......................405 17.4.7PartialFractions...............................408 17.5Summary..............................................410 17.6WorkedExamples.......................................410
17.6.1IntegratingaFunctionContainingitsOwn Derivative....................................410
17.6.2DividinganIntegralintoSeveralIntegrals.........412
17.6.3IntegratingbyParts1..........................412
17.6.4IntegratingbyParts2..........................413
17.6.5IntegratingbySubstitution1....................415
17.6.6IntegratingbySubstitution2....................416
17.6.7IntegratingbySubstitution3....................417
17.6.8IntegratingwithPartialFractions................417
18Area .........................................................419
18.1Introduction............................................419
18.2AreaUnderaGraph.....................................419
18.3CalculatingAreas.......................................419
18.4PositiveandNegativeAreas..............................428
18.5AreaBetweenTwoFunctions.............................430
18.6Areaswiththey-Axis....................................432
18.7AreawithParametricFunctions...........................433
18.8TheRiemannSum......................................435
18.9SurfaceofRevolution....................................437
18.9.1SurfaceAreaofaCylinder......................438
18.9.2SurfaceAreaofaRightCone...................439
18.9.3SurfaceAreaofaSphere.......................442
18.9.4SurfaceAreaofaParaboloid....................443
18.10SurfaceAreaUsingParametricFunctions...................445
18.11DoubleIntegrals........................................447
18.12Jacobians..............................................448
18.12.11DJacobian..................................449
18.12.22DJacobian..................................450
18.12.33DJacobian..................................455
18.13DoubleIntegralsforCalculatingArea......................458 18.14Summary..............................................463 18.14.1SummaryofFormulae.........................463 19Volume .......................................................465 19.1Introduction............................................465
19.2SolidofRevolution:Disks................................465
19.2.1VolumeofaCylinder..........................467
19.2.2VolumeofaRightCone........................467
19.2.3VolumeofaRightConicalFrustum..............470 19.2.4VolumeofaSphere............................471 19.2.5VolumeofanEllipsoid.........................472
19.2.6VolumeofaParaboloid.........................474
19.3SolidofRevolution:Shells...............................475
19.3.1VolumeofaCylinder..........................476
19.3.2VolumeofaRightCone........................477
19.3.3VolumeofaSphere............................478
19.3.4VolumeofaParaboloid.........................479
19.4VolumeswithDoubleIntegrals............................481
19.4.1ObjectswithaRectangularBase.................482
19.4.2RectangularBox..............................482
19.4.3RectangularPrism.............................483
19.4.4CurvedTop...................................484
19.4.5ObjectswithaCircularBase....................485
19.4.6Cylinder.....................................485
19.4.7TruncatedCylinder............................486
19.5VolumeswithTripleIntegrals.............................488
19.5.1RectangularBox..............................489
19.5.2VolumeofaCylinder..........................490
19.5.3VolumeofaSphere............................492
19.5.4VolumeofaCone.............................493
1.2VisualBrainsVersusAnalyticBrains
Iconsidermyselfa visual person,aspictureshelpmeunderstandcomplexproblems. Ialsodon’tfindittoodifficulttovisualiseobjectsfromdifferentviewpoints.I rememberlearningaboutelectrons,neutronsandprotonsforthefirsttime,whereour planetarysystemprovidedasimplemodeltovisualisethehiddenstructureofmatter. Mymentalimageofelectronswasoneofsmallorangespheres,spinningarounda small,centralnucleuscontainingblueprotonsandgreyneutrons.Andalthoughthis visualmodelisseriouslyflawed,itprovidedafirststeptowardsunderstandingthe structureofmatter.
Asmyknowledgeofmathematicsgrew,this,too,wasimagebased.Equations werecurvesandsurfaces,simultaneousequationswereintersectingorparallellines, etc.,andwhenIembarkeduponcomputerscience,Ifoundanaturalapplication formathematics.Forme,mathematicsisavisualscience,althoughIdoappreciate thatmanyprofessionalmathematiciansneedonlyaformal,symbolicnotationfor constructingtheirworld.Suchpeopledonotrequirevisualscaffolding—theyseem tobeabletomanipulateabstractmathematicalconceptsatasymboliclevel.Their booksdonotrequireillustrationsordiagrams—Greeksymbols,upside-downand back-to-frontLatinfontsaresufficienttoannotatetheirideas.
Today,whenreadingpopularsciencebooksonquantumtheory,Istilltrytoform imagesof3Dfieldsofenergyandprobabilityoscillatinginspace—tonoavail—andI haveacceptedthathumanknowledgeofsuchphenomenaisbestlefttoamathematical description.Nevertheless,mathematicians,suchasSirRogerPenrose,knowthe importanceofvisualmodelsincommunicatingcomplexmathematicalideas.His
©SpringerNatureSwitzerlandAG2023
J.Vince, FoundationMathematicsforComputerScience, https://doi.org/10.1007/978-3-031-17411-7_1
book TheRoadtoReality (Penrose 2004)isdecoratedwithbeautiful,informative, hand-drawnillustrations,whichhelpreadersunderstandthemathematicsofscience. InthisbookIrelyheavilyonimagestocommunicateanidea.Theyaresimpleand arethefirststeponaladdertowardsunderstandingadifficultidea.Eventually,when that Eureka momentarrives,thatmomentwhenyousaytoyourself,‘Iunderstand whatyouaresaying,’theimagebecomescloselyassociatedwiththemathematical notation.
1.3LearningMathematics
IwasfortunateinmystudiesinthatIwastaughtbypeopleinterestedinmathematics, andtheirinterestrubbedoffonme.Ifeelsorryforchildrenwhohavegivenup onmathematics,simplybecausetheyarebeingtaughtbyteacherswhoseprimary subjectisnotmathematics.Iwasnevertooconcernedabouttheusesofmathematics, althoughappliedmathematicsisofspecialinterest.
Oneoftheproblemswithmathematicsisitsincrediblebreadthanddepth.It embraceseverythingfrom2Dgeometry,calculus,topology,statistics,complexfunctionstonumbertheoryandpropositionalcalculus.Allofthesesubjectscanbestudiedsuperficiallyortoamind-numbingcomplexity.Fortunately,nooneisrequired tounderstandeverything,whichiswhymathematicianstendtospecialiseinoneor twoareasanddevelopaspecialistknowledge.
1.4WhatMakesMathematicsDifficult?
‘Whatmakesmathematicsdifficult?’isalsoadifficultquestiontoanswer,butone thathastobeaskedandanswered.Therearemanyanswerstothisquestion,and Ibelievethatproblemsbeginwithmathematicalnotationandhowtoreadit;how toanalyseaproblemandexpressasolutionusingmathematicalstatements.Unlike learningaforeignlanguage—whichIfindverydifficult—mathematicsisalanguage thatneedstobelearnedbydiscoveringfactsandbuildinguponthemtodiscovernew facts.Consequently,agoodmemoryisalwaysanadvantage,aswellasasenseof logic.
Mathematicscanbedifficultforanyone,includingmathematicians.Forexample, whentheideaof √ 1wasoriginallyproposed,itwascriticisedandlookeddown uponbymathematicians,mainlybecauseitspurposewasnotfullyunderstood.Eventually,ittransformedtheentiremathematicallandscape,includingphysics.Similarly, whentheGermanmathematicianGeorgCantor(1845–1919),publishedhispapers onsettheoryandtransfinitesets,somemathematicianshoundedhiminadisgraceful manner.TheGermanmathematicianLeopoldKronecker(1823–1891),calledCantora‘scientificcharlatan’,a‘renegade’,anda‘corrupterofyouth’,anddideverythingtohinderCantor’sacademiccareer.Similarly,theFrenchmathematicianand
physicistHenriPoincaré(1854–1912),calledCantor’sideasa‘gravedisease’,whilst theAustrian-BritishphilosopherandlogicianLudwigWittgenstein(1889–1951) complainedthatmathematicsis‘riddenthroughandthroughwiththepernicious idiomsofsettheory.’Howwrongtheyallwere.Today,settheoryisamajorbranchof mathematicsandhasfounditswayintoeverymathcurriculum.Sodon’tbesurprised todiscoverthatsomemathematicalideasareinitiallydifficulttounderstand—you areingoodcompany.
1.5DoesMathematicsExistOutsideOurBrains?
Manypeoplehaveconsideredthequestion‘Whatismathematics?’Somemathematiciansandphilosophersarguethatnumbersandmathematicalformulaehave somesortofexternalexistenceandarewaitingtobediscoveredbyus.Personally, Idon’tacceptthisidea.Ibelievethatweenjoysearchingforpatternsandstructure inanythingthatfindsitswayintoourbrains,whichiswhywelovepoetry,music, storytelling,art,singing,architecture,science,aswellasmathematics.Thepiano, forexample,isaninstrumentforplayingmusicusingdifferentpatternsofnotes. Whenthepianowasinvented—afewhundredyearsago—themusicofChopin, LisztandRachmaninoffdidnotexistinanyform—ithadtobecomposedbythem. Similarly,bybuildingasystemforcountingusingnumbers,wehaveanamazingtool forcomposingmathematicalsystemsthathelpusmeasurequantity,structure,space andchange.Suchsystemshavebeenappliedtotopicssuchasfluiddynamics,optimisation,statistics,cryptography,gametheoryprobabilitytheory,andmanymore. Iwillattempttodevelopthissameideabyshowinghowtheconceptofnumber, andthevisualrepresentationofnumberrevealsallsortsofpatterns,thatgiveriseto numbersystems,algebra,trigonometry,geometry,analyticgeometryandcalculus. Theuniversedoesnotneedanyofthesemathematicalideastorunitsmachinery,but weneedtheseideastounderstanditsoperation.
1.6SymbolsandNotation
Oneofthereasonswhymanypeoplefindmathematicsinaccessibleisduetoits symbolsandnotation.Let’slookatsymbolsfirst.TheEnglishalphabetpossessesa reasonablerangeoffamiliarcharactershapes:
a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z
whichfindtheirwayintoeverybranchofmathematicsandphysics,andpermit ustowriteequationssuchas E = mc 2
Itisimportantthatwhenweseeanequation,weareabletoreaditaspartofthetext. Inthecaseof E = mc 2 ,thisisreadas‘ E equals m , c squared’,where E standsfor energy, m formass,and c thespeedoflight.Inthecaseof A = π r 2 ,thisisreadas‘ A equalspi, r squared’,where A standsforarea, π theratioofacircle’scircumference toitsdiameter,and r thecircle’sradius.Greeksymbols,whichhappentolooknice andimpressive,havealsofoundtheirwayintomanyequations,andoftendisruptthe flowofreading,simplybecausewedon’tknowtheirEnglishnames.Forexample, theEnglishtheoreticalphysicistPaulDirac(1902–1984)derivedanequationfora movingelectronusingthesymbols αi and β ,whichare4 × 4matrices,where
andisreadas
‘thesumoftheproductsalpha-i beta,andbetaalpha-i ,equalszero.’
Althoughwewillnotcomeacrossmovingelectronsinthisbook,wewillhave tobefamiliarwiththefollowingGreeksymbols:
alpha
beta
xi
gamma o omicron
delta
pi epsilon
zeta
eta
theta
iota
kappa
lambda
mu
rho
sigma
tau
upsilon
phi
chi
psi
omega andsomeupper-casesymbols:
Gamma
Delta
Theta
Lambda
Xi
Pi.
Sigma
Upsilon
Phi
Psi
Omega
Beingabletoreadanequationdoesnotmeanthatweunderstandit—butwearea littlecloserthanjustbeingabletostareatajumbleofsymbols!Therefore,infuture, whenIintroduceanewmathematicalobject,Iwilltellyouhowitshouldberead.
Reference
PenroseR(2004)Theroadtoreality:acompleteguidetothelawsoftheuniverse.Cape
Chapter2
Numbers
2.1Introduction
Thischapterrevisesthesetsofnumbersemployedinmathematicssuchasnatural,integer,rational,irrational,real,algebraic,transcendental,imaginary,complex, quaternionsandoctonions.Italsodescribeshowthesenumbersbehaveinthecontext ofthreelaws:commutative,associativeandthedistributivelaw.
Asprimenumbersfindtheirwayintoallaspectsofcryptography,thechapter introducesthefundamentaltheoremofarithmetic,primenumberdistribution,perfect numbersandMersennenumbers.Thechapterconcludeswiththeconceptofinfinity andsomeworkedexamples.
2.2Counting
Ourbrain’svisualcortexpossessessomeincredibleimageprocessingfeatures.For example,childrenknowinstinctivelywhentheyaregivenlesssweetsthananother child,andadultsknowinstinctivelywhentheyareshort-changedbyaParisiantaxi driver,ordrivenaroundtheArcdeTriumphseveraltimes,onthewaytotheairport!
Intuitively,wecanassesshowmanydonkeysareinafieldwithoutcountingthem, andgenerally,weseemtoknowwithinasecondortwo,whethertherearejustafew, dozens,orhundredsofsomething.Butwhenaccuracyisrequired,onecan’tbeat counting.Butwhatiscounting?
Wellnormally,wearetaughttocountbyourparentsbyfirst,memorisingthe countingwords one,two,three,four,five,six,seven,eight,nine,ten,etc.,andsecond, associatingthemwithourfingers,sothatwhenaskedtocountthenumberofdonkeys inapicturebook,eachdonkeyisassociatedwithacountingword.Wheneach donkeyhasbeenidentified,thenumberofdonkeysequalsthelastwordmentioned. However,thisstillassumesthatweknowthemeaningof one,two,three,four, etc.Memorisingthesecountingwordsisonlypartoftheproblem—gettingthem
©SpringerNatureSwitzerlandAG2023
J.Vince, FoundationMathematicsforComputerScience, https://doi.org/10.1007/978-3-031-17411-7_2
inthecorrectsequenceistherealchallenge.Theincorrectsequence one,two,five, three,nine,four,etc.,introducesanelementofrandomnessintoanycalculation,but practicemakesperfect,andit’susefultomasterthecorrectsequencebeforegoing touniversity!
2.3SetsofNumbers
A set isacollectionofdistinctobjectscalledits elements or members.Forexample, eachsystemofnumberbelongstoasetwithgivenaname,suchas N forthenatural numbers, R forrealnumbers,and Q forrationalnumbers.Whenwewanttoindicate thatsomethingiswhole,realorrational,etc.,weusethenotation
∈ N
whichreads‘n isamemberof(∈)theset N’,i.e. n isawholenumber.Similarly,
∈ R
standsfor‘ x isarealnumber.’
A well-orderedset possessesauniqueorder,suchasthenaturalnumbers N. Therefore,if P isthewell-orderedsetofprimenumbersand N isthewell-ordered setofnaturalnumbers,wecanwrite P ={2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,
Bypairingtheprimenumbersin P withthenumbersin N,wehave {2, 1}, {3,
,...
andwecanreasonthat2isthe1stprime,and3isthe2ndprime,etc.However,we stillhavetodeclarewhatwemeanby1, 2, 3, 4, 5,etc.,andwithoutgettingtoo philosophical,Iliketheideaofdefiningthemasfollows.Theword one,represented by1,standsforone-nessofanything:onefinger,onehouse,onetree,onedonkey,etc. Theword two,representedby2,is‘onemorethanone.’Theword three,represented by3,is‘onemorethantwo,’andsoon.
Wearenowinapositiontoassociatesomemathematicalnotationwithournumbersbyintroducingthe + and = signs.Weknowthat + means add,butitalsocan standfor more.Wealsoknowthat = means equal,anditcanalsostandfor isthe sameas.Thusthestatement
2 = 1 + 1 isreadas‘twoisthesameasonemorethanone.’
whichisreadas‘threeisthesameasonemorethantwo.’Butaswealreadyhavea definitionfor2,wecanwrite
Developingthisidea,andincludingsomeextracombinations,wehave
andcanbecontinuedwithoutlimit.Thenumbers,1,2,3,4,5,6,etc.,arecalled naturalnumbers,andaretheset N.
2.4Zero
Theconceptofzerohasawell-documentedhistory,whichshowsthatithasbeenused bydifferentculturesoveraperiodoftwo-thousandyearsormore.ItwastheIndian mathematicianandastronomerBrahmagupta(598-c.–670)whoarguedthatzero wasjustasvalidasanynaturalnumber,withthedefinition: theresultofsubtracting anynumberfromitself.However,eventoday,thereisnouniversalagreementasto whetherzerobelongstotheset N,consequently,theset N0 standsforthesetof naturalnumbersincludingzero.
Intoday’spositionaldecimalsystem,whichisa placevaluesystem,thedigit 0isaplaceholder.Forexample,203standsfor:twohundreds,notensandthree units.Although0 ∈ N0 ,itdoeshavespecialpropertiesthatdistinguishitfromother membersoftheset,andBrahmaguptaalsogaverulesshowingthisinteraction.
If x ∈ N0 ,thenthefollowingrulesapply:
addition: x + 0 = x
subtraction: x 0 = x
multiplication: x × 0 = 0 × x = 0
division:0/ x = 0
undefineddivision: x /0
Theexpression0/0iscalledan indeterminateform,asitispossibletoshowthat underdifferentconditions,especiallylimitingconditions,itcanequalanything.So forthemoment,wewillavoidusingituntilwecovercalculus.
2.5NegativeNumbers
Whennegativenumberswerefirstproposed,theywerenotacceptedwithopenarms, asitwasdifficulttovisualise 5ofsomething.Forinstance,ifthereare5donkeys inafield,andtheyareallstolen,thefieldisnowempty,andthereisnothingwe candointhearithmeticofdonkeystocreateafieldof 5donkeys.However,in appliedmathematics,numbershavetorepresentallsortsofquantitiessuchastemperature,displacement,angularrotation,speed,acceleration,etc.,andwealsoneed toincorporateideassuchasleftandright,upanddown,beforeandafter,forwards andbackwards,etc.Fortunately,negativenumbersareperfectforrepresentingallof theabovequantitiesandideas.
Considertheexpression4 x ,where x ∈ N0 .When x takesoncertainvalues, wehave
andunlessweintroducenegativenumbers,weareunabletoexpresstheresultof 4 5.Consequently,negativenumbersarevisualisedasshowninFig. 2.1,where the numberline showsnegativenumberstotheleftofthenaturalnumbers,which are positive,althoughthe + signisomittedforclarity. -6-5-4-3-2-10123456
Fig.2.1 Thenumberline
Movingfromlefttoright,thenumberlineprovidesanumericalcontinuumfrom largenegativenumbers,throughzero,towardslargepositivenumbers.Inanycalculationwecouldagreethatanglesabovethehorizonarepositive,andanglesbelow thehorizon,negative.Similarly,amovementforwardsispositive,andamovement backwardsisnegative.Sonowweareabletowrite
withoutworryingaboutcreatingimpossibleconditions.
2.5.1TheArithmeticofPositiveandNegativeNumbers
Onceagain,Brahmaguptacompiledalltherules,Tables 2.1 and 2.2,supportingthe addition,subtraction,multiplicationanddivisionofpositiveandnegativenumbers. Therealflyintheointment,beingnegativenumbers,whichcauseproblemsfor children,mathteachersandoccasionalaccidentsformathematicians.Perhaps,the oneruleweallrememberfromourschooldaysisthat‘twonegativesmakeapositive.’
Anotherproblemwithnegativenumbersariseswhenweemploythesquare-root function.Astheproductoftwopositiveornegativenumbersresultsinapositive result,thesquare-rootofapositivenumbergivesrisetoapositive and anegativeanswer.Forexample, √4 =±2. Thismeansthatthesquare-rootfunctiononly appliestopositivenumbers.Nevertheless,itdidnotstoptheinventionofthe imaginary unit i ,where i 2 =−1.However, i isnotanumber,butanoperator,andis describedlater.
Table2.1 Rulesforaddingandsubtractingpositiveandnegativenumbers
