Foundation Mathematics for Computer Science A Visual Approach
In this second edition of Foundation Mathematics for Computer Science, John Vince has reviewed and edited the original book and written new chapters on combinatorics, probability, modular arithmetic and complex numbers. These subjects complement the existing chapters on number systems, algebra, logi...
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| Main Author | |
|---|---|
| Format | eBook Book |
| Language | English |
| Published |
Cham
Springer Nature
2020
Springer Springer International Publishing AG Springer International Publishing |
| Edition | 2 |
| Subjects | |
| Online Access | Get full text |
| ISBN | 9783030420789 3030420787 3030420779 9783030420772 |
| DOI | 10.1007/978-3-030-42078-9 |
Cover
Table of Contents:
- Intro -- Preface -- Contents -- 1 Visual Mathematics -- 1.1 Visual Brains Versus Analytic Brains -- 1.2 Learning Mathematics -- 1.3 What Makes Mathematics Difficult? -- 1.4 Does Mathematics Exist Outside Our Brains? -- 1.5 Symbols and Notation -- 2 Numbers -- 2.1 Introduction -- 2.2 Counting -- 2.3 Sets of Numbers -- 2.4 Zero -- 2.5 Negative Numbers -- 2.5.1 The Arithmetic of Positive and Negative Numbers -- 2.6 Observations and Axioms -- 2.6.1 Commutative Law -- 2.6.2 Associative Law -- 2.6.3 Distributive Law -- 2.7 The Base of a Number System -- 2.7.1 Background -- 2.7.2 Octal Numbers -- 2.7.3 Binary Numbers -- 2.7.4 Hexadecimal Numbers -- 2.7.5 Adding Binary Numbers -- 2.7.6 Subtracting Binary Numbers -- 2.8 Types of Numbers -- 2.8.1 Natural Numbers -- 2.8.2 Integers -- 2.8.3 Rational Numbers -- 2.8.4 Irrational Numbers -- 2.8.5 Real Numbers -- 2.8.6 Algebraic and Transcendental Numbers -- 2.8.7 Imaginary Numbers -- 2.8.8 Complex Numbers -- 2.8.9 Quaternions and Octonions -- 2.8.10 Transcendental and Algebraic Numbers -- 2.9 Prime Numbers -- 2.9.1 The Fundamental Theorem of Arithmetic -- 2.9.2 Is 1 a Prime? -- 2.9.3 Prime Number Distribution -- 2.9.4 Infinity of Primes -- 2.9.5 Perfect Numbers -- 2.9.6 Mersenne Numbers -- 2.10 Infinity -- 2.11 Worked Examples -- 2.11.1 Algebraic Expansion -- 2.11.2 Binary Subtraction -- 2.11.3 Complex Numbers -- 2.11.4 Complex Rotation -- 2.11.5 Quaternions -- References -- 3 Algebra -- 3.1 Introduction -- 3.2 Background -- 3.3 Notation -- 3.3.1 Solving the Roots of a Quadratic Equation -- 3.4 Indices -- 3.4.1 Laws of Indices -- 3.5 Logarithms -- 3.6 Further Notation -- 3.7 Functions -- 3.7.1 Explicit and Implicit Equations -- 3.7.2 Function Notation -- 3.7.3 Intervals -- 3.7.4 Function Domains and Ranges -- 3.7.5 Odd and Even Functions -- 3.7.6 Power Functions -- 3.8 Worked Examples
- 3.8.1 Algebraic Manipulation -- 3.8.2 Solving a Quadratic Equation -- 3.8.3 Factorising -- 4 Logic -- 4.1 Introduction -- 4.2 Background -- 4.3 Truth Tables -- 4.3.1 Logical Connectives -- 4.4 Logical Premises -- 4.4.1 Material Equivalence -- 4.4.2 Implication -- 4.4.3 Negation -- 4.4.4 Conjunction -- 4.4.5 Inclusive Disjunction -- 4.4.6 Exclusive Disjunction -- 4.4.7 Idempotence -- 4.4.8 Commutativity -- 4.4.9 Associativity -- 4.4.10 Distributivity -- 4.4.11 de Morgan's Laws -- 4.4.12 Simplification -- 4.4.13 Excluded Middle -- 4.4.14 Contradiction -- 4.4.15 Double Negation -- 4.4.16 Implication and Equivalence -- 4.4.17 Exportation -- 4.4.18 Contrapositive -- 4.4.19 Reductio Ad Absurdum -- 4.4.20 Modus Ponens -- 4.4.21 Proof by Cases -- 4.5 Set Theory -- 4.5.1 Empty Set -- 4.5.2 Membership and Cardinality of a Set -- 4.5.3 Subsets, Supersets and the Universal Set -- 4.5.4 Set Building -- 4.5.5 Union -- 4.5.6 Intersection -- 4.5.7 Relative Complement -- 4.5.8 Absolute Complement -- 4.5.9 Power Set -- 4.6 Worked Examples -- 4.6.1 Truth Tables -- 4.6.2 Set Building -- 4.6.3 Sets -- 4.6.4 Power Set -- 5 Combinatorics -- 5.1 Introduction -- 5.2 Permutations -- 5.3 Permutations of Multisets -- 5.4 Combinations -- 5.5 Worked Examples -- 5.5.1 Eight-Permutations of a Multiset -- 5.5.2 Eight-Permutations of a Multiset -- 5.5.3 Number of Permutations -- 5.5.4 Number of Five-Card Hands -- 5.5.5 Hand Shakes with 100 People -- 5.5.6 Permutations of MISSISSIPPI -- 6 Probability -- 6.1 Introduction -- 6.2 Definition and Notation -- 6.2.1 Independent Events -- 6.2.2 Dependent Events -- 6.2.3 Mutually Exclusive Events -- 6.2.4 Inclusive Events -- 6.2.5 Probability Using Combinations -- 6.3 Worked Examples -- 6.3.1 Product of Probabilities -- 6.3.2 Book Arrangements -- 6.3.3 Winning a Lottery -- 6.3.4 Rolling Two Dice -- 6.3.5 Two Dice Sum to 7
- 9.13.4 Spherical Polar Coordinates -- 9.13.5 Cylindrical Coordinates -- 9.13.6 Barycentric Coordinates -- Reference -- 10 Determinants -- 10.1 Introduction -- 10.2 Background -- 10.3 Linear Equations with Two Variables -- 10.4 Linear Equations with Three Variables -- 10.4.1 Sarrus's Rule -- 10.5 Mathematical Notation -- 10.5.1 Matrix -- 10.5.2 Order of a Determinant -- 10.5.3 Value of a Determinant -- 10.5.4 Properties of Determinants -- 10.6 Worked Examples -- 10.6.1 Determinant Expansion -- 10.6.2 Complex Determinant -- 10.6.3 Simple Expansion -- 10.6.4 Simultaneous Equations -- 11 Vectors -- 11.1 Introduction -- 11.2 Background -- 11.3 2D Vectors -- 11.3.1 Vector Notation -- 11.3.2 Graphical Representation of Vectors -- 11.3.3 Magnitude of a Vector -- 11.4 3D Vectors -- 11.4.1 Vector Manipulation -- 11.4.2 Scaling a Vector -- 11.4.3 Vector Addition and Subtraction -- 11.4.4 Position Vectors -- 11.4.5 Unit Vectors -- 11.4.6 Cartesian Vectors -- 11.4.7 Products -- 11.4.8 Scalar Product -- 11.4.9 The Vector Product -- 11.4.10 The Right-Hand Rule -- 11.5 Deriving a Unit Normal Vector for a Triangle -- 11.6 Surface Areas -- 11.6.1 Calculating 2D Areas -- 11.7 Worked Examples -- 11.7.1 Position Vector -- 11.7.2 Unit Vector -- 11.7.3 Vector Magnitude -- 11.7.4 Angle Between Two Vectors -- 11.7.5 Vector Product -- Reference -- 12 Complex Numbers -- 12.1 Introduction -- 12.2 Representing Complex Numbers -- 12.2.1 Complex Numbers -- 12.2.2 Real and Imaginary Parts -- 12.2.3 The Complex Plane -- 12.3 Complex Algebra -- 12.3.1 Algebraic Laws -- 12.3.2 Complex Conjugate -- 12.3.3 Complex Division -- 12.3.4 Powers of i -- 12.3.5 Rotational Qualities of i -- 12.3.6 Modulus and Argument -- 12.3.7 Complex Norm -- 12.3.8 Complex Inverse -- 12.3.9 Complex Exponentials -- 12.3.10 de Moivre's Theorem -- 12.3.11 nth Root of Unity
- 14.1 Introduction
- 12.3.12 nth Roots of a Complex Number -- 12.3.13 Logarithm of a Complex Number -- 12.3.14 Raising a Complex Number to a Complex Power -- 12.3.15 Visualising Simple Complex Functions -- 12.3.16 The Hyperbolic Functions -- 12.4 Summary -- 12.5 Worked Examples -- 12.5.1 Complex Addition -- 12.5.2 Complex Products -- 12.5.3 Complex Division -- 12.5.4 Complex Rotation -- 12.5.5 Polar Notation -- 12.5.6 Real and Imaginary Parts -- 12.5.7 Magnitude of a Complex Number -- 12.5.8 Complex Norm -- 12.5.9 Complex Inverse -- 12.5.10 de Moivre's Theorem -- 12.5.11 nth Root of Unity -- 12.5.12 Roots of a Complex Number -- 12.5.13 Logarithm of a Complex Number -- 12.5.14 Raising a Number to a Complex Power -- References -- 13 Matrices -- 13.1 Introduction -- 13.2 Geometric Transforms -- 13.3 Transforms and Matrices -- 13.4 Matrix Notation -- 13.4.1 Matrix Dimension or Order -- 13.4.2 Square Matrix -- 13.4.3 Column Vector -- 13.4.4 Row Vector -- 13.4.5 Null Matrix -- 13.4.6 Unit Matrix -- 13.4.7 Trace -- 13.4.8 Determinant of a Matrix -- 13.4.9 Transpose -- 13.4.10 Symmetric Matrix -- 13.4.11 Antisymmetric Matrix -- 13.5 Matrix Addition and Subtraction -- 13.5.1 Scalar Multiplication -- 13.6 Matrix Products -- 13.6.1 Row and Column Vectors -- 13.6.2 Row Vector and a Matrix -- 13.6.3 Matrix and a Column Vector -- 13.6.4 Square Matrices -- 13.6.5 Rectangular Matrices -- 13.7 Inverse Matrix -- 13.7.1 Inverting a Pair of Matrices -- 13.8 Orthogonal Matrix -- 13.9 Diagonal Matrix -- 13.10 Worked Examples -- 13.10.1 Matrix Inversion -- 13.10.2 Identity Matrix -- 13.10.3 Solving Two Equations Using Matrices -- 13.10.4 Solving Three Equations Using Matrices -- 13.10.5 Solving Two Complex Equations -- 13.10.6 Solving Three Complex Equations -- 13.10.7 Solving Two Complex Equations -- 13.10.8 Solving Three Complex Equations -- 14 Geometric Matrix Transforms
- 6.3.6 Two Dice Sum to 4 -- 6.3.7 Dealing a Red Ace -- 6.3.8 Selecting Four Aces in Succession -- 6.3.9 Selecting Cards -- 6.3.10 Selecting Four Balls from a Bag -- 6.3.11 Forming Teams -- 6.3.12 Dealing Five Cards -- 7 Modular Arithmetic -- 7.1 Introduction -- 7.2 Informal Definition -- 7.3 Notation -- 7.4 Congruence -- 7.5 Negative Numbers -- 7.6 Arithmetic Operations -- 7.6.1 Sums of Numbers -- 7.6.2 Products -- 7.6.3 Multiplying by a Constant -- 7.6.4 Congruent Pairs -- 7.6.5 Multiplicative Inverse -- 7.6.6 Modulo a Prime -- 7.6.7 Fermat's Little Theorem -- 7.7 Applications of Modular Arithmetic -- 7.7.1 ISBN Parity Check -- 7.7.2 IBAN Check Digits -- 7.8 Worked Examples -- 7.8.1 Negative Numbers -- 7.8.2 Sums of Numbers -- 7.8.3 Remainders of Products -- 7.8.4 Multiplicative Inverse -- 7.8.5 Product Table for Modulo 13 -- 7.8.6 ISBN Check Digit -- References -- 8 Trigonometry -- 8.1 Introduction -- 8.2 Background -- 8.3 Units of Angular Measurement -- 8.4 The Trigonometric Ratios -- 8.4.1 Domains and Ranges -- 8.5 Inverse Trigonometric Ratios -- 8.6 Trigonometric Identities -- 8.7 The Sine Rule -- 8.8 The Cosine Rule -- 8.9 Compound-Angle Identities -- 8.9.1 Double-Angle Identities -- 8.9.2 Multiple-Angle Identities -- 8.9.3 Half-Angle Identities -- 8.10 Perimeter Relationships -- 9 Coordinate Systems -- 9.1 Introduction -- 9.2 Background -- 9.3 The Cartesian Plane -- 9.4 Function Graphs -- 9.5 Shape Representation -- 9.5.1 2D Polygons -- 9.5.2 Areas of Shapes -- 9.6 Theorem of Pythagoras in 2D -- 9.7 3D Cartesian Coordinates -- 9.7.1 Theorem of Pythagoras in 3D -- 9.8 Polar Coordinates -- 9.9 Spherical Polar Coordinates -- 9.10 Cylindrical Coordinates -- 9.11 Barycentric Coordinates -- 9.12 Homogeneous Coordinates -- 9.13 Worked Examples -- 9.13.1 Area of a Shape -- 9.13.2 Distance Between Two Points -- 9.13.3 Polar Coordinates