Table Of ContentJohn Vince
Foundation
Mathematics
for Computer
Science
A Visual Approach
Third Edition
Foundation Mathematics for Computer Science
John Vince
Foundation Mathematics
for Computer Science
A Visual Approach
Third Edition
JohnVince
BournemouthUniversity
Poole,UK
ISBN 978-3-031-17410-0 ISBN 978-3-031-17411-7 (eBook)
https://doi.org/10.1007/978-3-031-17411-7
1stedition:©SpringerInternationalPublishingSwitzerland2015
2nd&3rdeditions:©SpringerNatureSwitzerlandAG2020,2023
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Thisbookisdedicatedtomywifeandbest
friend,Heidi.
Preface
Computer science is a very large subject, and graduates pursue a wide variety of
careers,includingprogramming,systemsdesign,cryptography,websitedesign,real-
timesystems,computeranimation,computergames,datavisualisation,etc.Conse-
quently,itisimpossibletowriteamathematicsbookthatcaterstoallofthesecareer
paths.Nevertheless,Ihaveattemptedtodescribearangeofmathematicaltopicsthat
Ibelievearerelevant,andhavehelpedmeduringmyowncareerincomputerscience.
Thebook’ssubtitle‘AVisualApproach’reflectstheimportanceIplaceoncoloured
illustrationsandfunctiongraphs,ofwhichthereareover210and90tables.Many
chapterscontainavarietyofworkedexamples.
Thisthirdeditionremainsanintroductorytext,andisaimedatstudentsstudying
for an undergraduate degree in computer science. There are now nineteen chap-
tersonnumbers,counting,algebra,logic,combinatorics,probability,modulararith-
metic, trigonometry, coordinate systems, determinants, vectors, complex numbers,
matrices,geometricmatrixtransforms,differentiation,integration,areaandvolume,
whichshouldprovidereaderswithasolidfoundation,uponwhichmoreadvanced
topicsofmathematicscanbestudied.
I have referenced the key people behind the various mathematical discoveries
covered,whichIhopeaddsahumandimensiontothesubject.Ihavefounditvery
interesting and entertaining to discover how some mathematicians ridiculed their
fellowpeers,whentheycouldnotcomprehendthesignificanceofanewinvention.
ThereisnowayIcouldhavewrittenthisbookwithouttheassistanceoftheInternet
andmybookspreviouslypublishedbySpringerVerlag.Inparticular,Iwouldliketo
acknowledgeWikipediaandRichardElwes’excellentbookMaths1001.Iprepared
this book on an Apple iMac, using LaTeX2e, Pages and the Grapher package,
and would recommend this combination to anyone considering writing a book on
mathematics. I do hope you enjoy reading this book, and that you are tempted to
studymathematicstoadeeperlevel.
Breinton,Herefordshire,UK JohnVince
October2022
vii
Contents
1 VisualMathematics ........................................... 1
1.1 Introduction ............................................ 1
1.2 VisualBrainsVersusAnalyticBrains ....................... 1
1.3 LearningMathematics ................................... 2
1.4 WhatMakesMathematicsDifficult? ....................... 2
1.5 DoesMathematicsExistOutsideOurBrains? ............... 3
1.6 SymbolsandNotation ................................... 3
Reference ..................................................... 5
2 Numbers ..................................................... 7
2.1 Introduction ............................................ 7
2.2 Counting ............................................... 7
2.3 SetsofNumbers ........................................ 8
2.4 Zero ................................................... 9
2.5 NegativeNumbers ....................................... 10
2.5.1 The Arithmetic of Positive and Negative
Numbers ..................................... 11
2.6 ObservationsandAxioms ................................ 12
2.6.1 CommutativeLaw ............................. 12
2.6.2 AssociativeLaw ............................... 13
2.6.3 DistributiveLaw .............................. 13
2.7 TypesofNumbers ....................................... 14
2.7.1 NaturalNumbers .............................. 14
2.7.2 Integers ...................................... 14
2.7.3 RationalNumbers ............................. 14
2.7.4 IrrationalNumbers ............................. 15
2.7.5 RealNumbers ................................. 15
2.7.6 AlgebraicandTranscendentalNumbers ........... 15
2.7.7 ImaginaryNumbers ............................ 17
2.7.8 ComplexNumbers ............................. 17
2.7.9 QuaternionsandOctonions ..................... 19
ix
x Contents
2.8 PrimeNumbers ......................................... 21
2.8.1 TheFundamentalTheoremofArithmetic ......... 21
2.8.2 Is1aPrime? .................................. 22
2.8.3 TheGoldbachConjecture ....................... 23
2.8.4 PrimeNumberDistribution ..................... 23
2.8.5 InfinityofPrimes .............................. 24
2.8.6 MersenneNumbers ............................ 25
2.9 PerfectNumbers ........................................ 25
2.10 TriangularNumbers ..................................... 26
2.11 Infinity ................................................ 26
2.12 WorkedExamples ....................................... 28
2.12.1 AlgebraicExpansion ........................... 28
2.12.2 ComplexNumbers ............................. 28
2.12.3 Quaternions .................................. 28
References .................................................... 29
3 SystemsofCounting ........................................... 31
3.1 Introduction ............................................ 31
3.2 DecimalPositionalSystem ............................... 31
3.2.1 Background .................................. 31
3.2.2 BinaryNumbers ............................... 32
3.2.3 OctalNumbers ................................ 33
3.2.4 HexadecimalNumbers ......................... 33
3.3 ConvertingDecimaltoBinary,OctalandHexadecimal ....... 34
3.3.1 ConvertingDecimaltoBinary ................... 34
3.3.2 ConvertingDecimaltoOctal .................... 35
3.3.3 ConvertingDecimaltoHexadecimal ............. 36
3.4 ConvertingBetweenBinaryandOctalNumbers ............. 38
3.5 ConvertingBetweenBinaryandHexadecimalNumbers ...... 39
3.6 AddingandSubtractingBinaryNumbers ................... 41
3.6.1 AddingBinaryNumbers ........................ 41
3.6.2 SubtractingBinaryNumbersUsingTwo’s
Complement .................................. 42
3.7 AddingandSubtractingDecimalNumbers .................. 42
3.7.1 AddingDecimalNumbers ...................... 42
3.7.2 SubtractingDecimalNumbersUsingTen’s
Complement .................................. 43
3.8 AddingandSubtractingOctalNumbers .................... 44
3.8.1 AddingOctalNumbers ......................... 44
3.8.2 SubtractingOctalNumbersUsingEight’s
Complement .................................. 44
3.9 Summary .............................................. 45
Contents xi
3.10 WorkedExamples ....................................... 45
3.10.1 ConvertaDecimalNumberintoBinary ........... 45
3.10.2 ConvertaDecimalNumberintoBinaryUsing
anAlgorithm ................................. 45
3.10.3 ConvertaBinaryNumberintoDecimal ........... 46
3.10.4 ConvertaBinaryNumberintoOctal ............. 46
3.10.5 ConvertanOctalNumberintoBinary ............ 46
3.10.6 ConvertanOctalNumberintoHexadecimal ....... 46
3.10.7 ConvertaHexadecimalNumberintoOctal ........ 47
3.10.8 ConvertaDecimalNumberintoOctal ............ 47
3.10.9 ConvertaDecimalNumberintoOctalUsing
anAlgorithm ................................. 47
3.10.10 ConvertaDecimalNumberintoHexadecimal ..... 48
3.10.11 AddBinaryNumbers .......................... 48
3.10.12 SubtractBinaryNumbers ....................... 49
3.10.13 AddOctalNumbers ............................ 49
3.10.14 SubtractOctalNumbers ........................ 49
3.10.15 AddHexadecimalNumbers ..................... 49
4 Algebra ...................................................... 51
4.1 Introduction ............................................ 51
4.2 Background ............................................ 51
4.3 Notation ............................................... 52
4.3.1 SolvingtheRootsofaQuadraticEquation ........ 54
4.4 Indices ................................................ 58
4.4.1 LawsofIndices ............................... 58
4.5 Logarithms ............................................. 58
4.6 FurtherNotation ........................................ 60
4.7 Functions .............................................. 61
4.7.1 ExplicitandImplicitEquations .................. 61
4.7.2 FunctionNotation ............................. 62
4.7.3 Intervals ...................................... 62
4.7.4 FunctionDomainsandRanges .................. 63
4.7.5 OddandEvenFunctions ........................ 65
4.7.6 PowerFunctions .............................. 66
4.8 Series ................................................. 66
4.9 BinomialTheorem ...................................... 67
4.10 Summary .............................................. 68
4.11 WorkedExamples ....................................... 69
4.11.1 AlgebraicManipulation ........................ 69
4.11.2 SolvingaQuadraticEquation ................... 70
4.11.3 Factorising ................................... 72
4.11.4 BinomialTheorem ............................. 72
References .................................................... 72
xii Contents
5 Logic ......................................................... 73
5.1 Introduction ............................................ 73
5.2 Background ............................................ 73
5.3 TruthTables ............................................ 74
5.3.1 LogicalConnectives ........................... 74
5.4 LogicalPremises ........................................ 75
5.4.1 MaterialEquivalence ........................... 75
5.4.2 Implication ................................... 76
5.4.3 Negation ..................................... 77
5.4.4 Conjunction .................................. 77
5.4.5 InclusiveDisjunction ........................... 78
5.4.6 ExclusiveDisjunction .......................... 79
5.4.7 Idempotence .................................. 79
5.4.8 Commutativity ................................ 79
5.4.9 Associativity .................................. 80
5.4.10 Distributivity ................................. 82
5.4.11 deMorgan’sLaws ............................. 83
5.4.12 Simplification ................................. 83
5.4.13 ExcludedMiddle .............................. 84
5.4.14 Contradiction ................................. 85
5.4.15 DoubleNegation .............................. 85
5.4.16 ImplicationandEquivalence .................... 85
5.4.17 Exportation ................................... 86
5.4.18 Contrapositive ................................ 86
5.4.19 ReductioAdAbsurdum ........................ 87
5.4.20 ModusPonens ................................ 88
5.4.21 ProofbyCases ................................ 89
5.5 SetTheory ............................................. 91
5.5.1 EmptySet .................................... 91
5.5.2 MembershipandCardinalityofaSet ............. 91
5.5.3 Subsets,SupersetsandtheUniversalSet .......... 92
5.5.4 SetBuilding .................................. 92
5.5.5 Union ........................................ 93
5.5.6 Intersection ................................... 94
5.5.7 RelativeComplement .......................... 94
5.5.8 AbsoluteComplement ......................... 96
5.5.9 PowerSet .................................... 96
5.6 WorkedExamples ....................................... 97
5.6.1 TruthTables .................................. 97
5.6.2 SetBuilding .................................. 97
5.6.3 Sets ......................................... 99
5.6.4 PowerSet .................................... 99
