Foundation mathematics for computer science. A visual approach

416 Pages·2020·5.053 MB·English

by  Vince J

John Vince
Foundation
Mathematics
for Computer
Science
A Visual Approach
Second Edition
Foundation Mathematics for Computer Science
John Vince
Foundation Mathematics
for Computer Science
A Visual Approach
Second Edition
123
JohnVince
BournemouthUniversity
Breinton, Hereford,UK
ISBN978-3-030-42077-2 ISBN978-3-030-42078-9 (eBook)
https://doi.org/10.1007/978-3-030-42078-9
1stedition:©SpringerInternationalPublishingSwitzerland2015
2ndedition:©SpringerNatureSwitzerlandAG2020
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of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors and the editors are safe to assume that the advice and information in this
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authors or the editors give a warranty, expressed or implied, with respect to the material contained
hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard
tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
This book is dedicated to my wife and best
friend, Heidi.
Preface
Computer science is a very large subject, and graduates will pursue a wide variety
of careers, including programming, systems design, cryptography, website design,
real-time systems, computer animation, computer games, data visualisation, etc.
Consequently, itisvirtually impossible towrite amathematicsbookthatcatersfor
allofthesepotentialcareerpaths.Nevertheless,Ihaveattemptedtodescribearange
of mathematical topics that I believe are relevant, and have helped me during my
own career in computer science. The book’s subtitle “A Visual Approach” reflects
theimportanceIplaceoncolouredillustrationsandfunctiongraphs,ofwhichthere
are over 160. Each chapter contains a variety of worked examples.
This second edition remains an introductory text, and is aimed at students
studying for an undergraduate degree in computer science. There are four extra
chapters on combinatorics, probability, modular arithmetic and complex numbers,
whichtogetherwiththeoriginaltwelvechaptersshouldprovidereaderswithasolid
foundation, upon which more advanced topics of mathematics can be studied.
Throughout the book I have referenced the key people behind the various
mathematical discoveries covered, which I hope adds a human dimension to the
subject. I have found it very interesting and entertaining to discover how some
mathematicians ridiculed their fellow peers, when they could not comprehend the
significanceofanewinvention—Cantor’sSetTheory,beinganexcellentexample.
There is no way I could have written this book without the assistance of the
Internet and my books previously published by Springer Verlag. In particular,
I would like to acknowledge Wikipedia and Richard Elwes’ excellent book Maths
1001. I prepared this book on an Apple iMac, using LATEX2e, Pages and the
Grapher package, and would recommend this combination to anyone considering
writingabookonmathematics.Idohopeyouenjoyreadingthisbook,andthatyou
are tempted to study mathematics to a deeper level.
Breinton, Herefordshire, UK John Vince
March 2020
vii
Contents
1 Visual Mathematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Visual Brains Versus Analytic Brains. . . . . . . . . . . . . . . . . . . 1
1.2 Learning Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 What Makes Mathematics Difficult?. . . . . . . . . . . . . . . . . . . . 2
1.4 Does Mathematics Exist Outside Our Brains?. . . . . . . . . . . . . 3
1.5 Symbols and Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Sets of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 Negative Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5.1 The Arithmetic of Positive and Negative
Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.6 Observations and Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.6.1 Commutative Law. . . . . . . . . . . . . . . . . . . . . . . . . 10
2.6.2 Associative Law . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.6.3 Distributive Law. . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.7 The Base of a Number System . . . . . . . . . . . . . . . . . . . . . . . 11
2.7.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.7.2 Octal Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.7.3 Binary Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.7.4 Hexadecimal Numbers. . . . . . . . . . . . . . . . . . . . . . 13
2.7.5 Adding Binary Numbers . . . . . . . . . . . . . . . . . . . . 17
2.7.6 Subtracting Binary Numbers . . . . . . . . . . . . . . . . . 18
2.8 Types of Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.8.1 Natural Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.8.2 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.8.3 Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 20
ix
x Contents
2.8.4 Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . 20
2.8.5 Real Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.8.6 Algebraic and Transcendental Numbers . . . . . . . . . 20
2.8.7 Imaginary Numbers. . . . . . . . . . . . . . . . . . . . . . . . 21
2.8.8 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . 22
2.8.9 Quaternions and Octonions . . . . . . . . . . . . . . . . . . 23
2.8.10 Transcendental and Algebraic Numbers . . . . . . . . . 24
2.9 Prime Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.9.1 The Fundamental Theorem of Arithmetic . . . . . . . . 26
2.9.2 Is 1 a Prime? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.9.3 Prime Number Distribution . . . . . . . . . . . . . . . . . . 27
2.9.4 Infinity of Primes . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.9.5 Perfect Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.9.6 Mersenne Numbers . . . . . . . . . . . . . . . . . . . . . . . . 30
2.10 Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.11 Worked Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.11.1 Algebraic Expansion . . . . . . . . . . . . . . . . . . . . . . . 31
2.11.2 Binary Subtraction . . . . . . . . . . . . . . . . . . . . . . . . 31
2.11.3 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . 32
2.11.4 Complex Rotation. . . . . . . . . . . . . . . . . . . . . . . . . 32
2.11.5 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3.1 Solving the Roots of a Quadratic Equation. . . . . . . 38
3.4 Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4.1 Laws of Indices . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.5 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.6 Further Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.7 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.7.1 Explicit and Implicit Equations . . . . . . . . . . . . . . . 45
3.7.2 Function Notation. . . . . . . . . . . . . . . . . . . . . . . . . 45
3.7.3 Intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.7.4 Function Domains and Ranges. . . . . . . . . . . . . . . . 47
3.7.5 Odd and Even Functions. . . . . . . . . . . . . . . . . . . . 48
3.7.6 Power Functions. . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.8 Worked Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.8.1 Algebraic Manipulation. . . . . . . . . . . . . . . . . . . . . 50
3.8.2 Solving a Quadratic Equation . . . . . . . . . . . . . . . . 51
3.8.3 Factorising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Contents xi
4 Logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 Truth Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.1 Logical Connectives . . . . . . . . . . . . . . . . . . . . . . . 56
4.4 Logical Premises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4.1 Material Equivalence. . . . . . . . . . . . . . . . . . . . . . . 57
4.4.2 Implication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4.3 Negation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4.4 Conjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4.5 Inclusive Disjunction. . . . . . . . . . . . . . . . . . . . . . . 59
4.4.6 Exclusive Disjunction . . . . . . . . . . . . . . . . . . . . . . 59
4.4.7 Idempotence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.4.8 Commutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4.9 Associativity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4.10 Distributivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4.11 de Morgan’s Laws . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4.12 Simplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4.13 Excluded Middle. . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4.14 Contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4.15 Double Negation. . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4.16 Implication and Equivalence . . . . . . . . . . . . . . . . . 66
4.4.17 Exportation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4.18 Contrapositive. . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4.19 Reductio Ad Absurdum. . . . . . . . . . . . . . . . . . . . . 67
4.4.20 Modus Ponens . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4.21 Proof by Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5 Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.5.1 Empty Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.5.2 Membership and Cardinality of a Set . . . . . . . . . . . 71
4.5.3 Subsets, Supersets and the Universal Set . . . . . . . . 72
4.5.4 Set Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.5.5 Union. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.5.6 Intersection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.5.7 Relative Complement . . . . . . . . . . . . . . . . . . . . . . 74
4.5.8 Absolute Complement. . . . . . . . . . . . . . . . . . . . . . 75
4.5.9 Power Set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.6 Worked Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.6.1 Truth Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.6.2 Set Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.6.3 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.6.4 Power Set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
xii Contents
5 Combinatorics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2 Permutations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3 Permutations of Multisets . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.4 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.5 Worked Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.5.1 Eight-Permutations of a Multiset . . . . . . . . . . . . . . 85
5.5.2 Eight-Permutations of a Multiset . . . . . . . . . . . . . . 86
5.5.3 Number of Permutations . . . . . . . . . . . . . . . . . . . . 87
5.5.4 Number of Five-Card Hands . . . . . . . . . . . . . . . . . 87
5.5.5 Hand Shakes with 100 People . . . . . . . . . . . . . . . . 87
5.5.6 Permutations of MISSISSIPPI . . . . . . . . . . . . . . . . 88
6 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2 Definition and Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2.1 Independent Events. . . . . . . . . . . . . . . . . . . . . . . . 91
6.2.2 Dependent Events. . . . . . . . . . . . . . . . . . . . . . . . . 91
6.2.3 Mutually Exclusive Events . . . . . . . . . . . . . . . . . . 92
6.2.4 Inclusive Events . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.2.5 Probability Using Combinations. . . . . . . . . . . . . . . 93
6.3 Worked Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.3.1 Product of Probabilities. . . . . . . . . . . . . . . . . . . . . 95
6.3.2 Book Arrangements . . . . . . . . . . . . . . . . . . . . . . . 96
6.3.3 Winning a Lottery. . . . . . . . . . . . . . . . . . . . . . . . . 96
6.3.4 Rolling Two Dice. . . . . . . . . . . . . . . . . . . . . . . . . 96
6.3.5 Two Dice Sum to 7 . . . . . . . . . . . . . . . . . . . . . . . 96
6.3.6 Two Dice Sum to 4 . . . . . . . . . . . . . . . . . . . . . . . 97
6.3.7 Dealing a Red Ace . . . . . . . . . . . . . . . . . . . . . . . . 97
6.3.8 Selecting Four Aces in Succession. . . . . . . . . . . . . 97
6.3.9 Selecting Cards. . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.3.10 Selecting Four Balls from a Bag . . . . . . . . . . . . . . 98
6.3.11 Forming Teams. . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.3.12 Dealing Five Cards . . . . . . . . . . . . . . . . . . . . . . . . 99
7 Modular Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.2 Informal Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.4 Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.5 Negative Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.6 Arithmetic Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.6.1 Sums of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.6.2 Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105