Foundation Mathematics for Computer Science: A Visual Approach

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
		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
		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
		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
	14.1 Introduction
	14.2 Matrix Transforms
		14.2.1 2D Translation
		14.2.2 2D Scaling
		14.2.3 2D Reflections
		14.2.4 2D Shearing
		14.2.5 2D Rotation
		14.2.6 2D Scaling
		14.2.7 2D Reflection
		14.2.8 2D Rotation About an Arbitrary Point
	14.3 3D Transforms
		14.3.1 3D Translation
		14.3.2 3D Scaling
		14.3.3 3D Rotation
		14.3.4 Rotating About an Axis
		14.3.5 3D Reflections
	14.4 Rotating a Point About an Arbitrary Axis
		14.4.1 Matrices
	14.5 Determinant of a Transform
	14.6 Perspective Projection
	14.7 Worked Examples
		14.7.1 2D Scale and Translate
		14.7.2 2D Rotation
		14.7.3 Determinant of the Rotate Transform
		14.7.4 Determinant of the Shear Transform
		14.7.5 Yaw, Pitch and Roll Transforms
		14.7.6 Rotation About an Arbitrary Axis
		14.7.7 3D Rotation Transform Matrix
		14.7.8 Perspective Projection
15 Calculus: Derivatives
	15.1 Introduction
	15.2 Background
	15.3 Small Numerical Quantities
	15.4 Equations and Limits
		15.4.1 Quadratic Function
		15.4.2 Cubic Equation
		15.4.3 Functions and Limits
		15.4.4 Graphical Interpretation of the Derivative
		15.4.5 Derivatives and Differentials
		15.4.6 Integration and Antiderivatives
	15.5 Function Types
	15.6 Differentiating Groups of Functions
		15.6.1 Sums of Functions
		15.6.2 Function of a Function
		15.6.3 Function Products
		15.6.4 Function Quotients
	15.7 Differentiating Implicit Functions
	15.8 Differentiating Exponential and Logarithmic Functions
		15.8.1 Exponential Functions
		15.8.2 Logarithmic Functions
	15.9 Differentiating Trigonometric Functions
		15.9.1 Differentiating tan
		15.9.2 Differentiating csc
		15.9.3 Differentiating sec
		15.9.4 Differentiating cot
		15.9.5 Differentiating arcsin, arccos and arctan
		15.9.6 Differentiating arccsc, arcsec and arccot
	15.10 Differentiating Hyperbolic Functions
		15.10.1 Differentiating sinh, cosh and tanh
	15.11 Higher Derivatives
	15.12 Higher Derivatives of a Polynomial
	15.13 Identifying a Local Maximum or Minimum
	15.14 Partial Derivatives
		15.14.1 Visualising Partial Derivatives
		15.14.2 Mixed Partial Derivatives
	15.15 Chain Rule
	15.16 Total Derivative
	15.17 Power Series
	15.18 Worked Examples
		15.18.1 Antiderivative 1
		15.18.2 Antiderivative 2
		15.18.3 Differentiating Sums of Functions
		15.18.4 Differentiating a Function Product
		15.18.5 Differentiating an Implicit Function
		15.18.6 Differentiating a General Implicit Function
		15.18.7 Local Maximum or Minimum
		15.18.8 Partial Derivatives
		15.18.9 Mixed Partial Derivative 1
		15.18.10 Mixed Partial Derivative 2
		15.18.11 Total Derivative
16 Calculus: Integration
	16.1 Introduction
	16.2 Indefinite Integral
	16.3 Integration Techniques
		16.3.1 Continuous Functions
		16.3.2 Difficult Functions
	16.4 Trigonometric Identities
		16.4.1 Exponent Notation
		16.4.2 Completing the Square
		16.4.3 The Integrand Contains a Derivative
		16.4.4 Converting the Integrand into a Series of Fractions
		16.4.5 Integration by Parts
		16.4.6 Integration by Substitution
		16.4.7 Partial Fractions
	16.5 Area Under a Graph
	16.6 Calculating Areas
	16.7 Positive and Negative Areas
	16.8 Area Between Two Functions
	16.9 Areas with the y-Axis
	16.10 Area with Parametric Functions
	16.11 The Riemann Sum
	16.12 Worked Examples
		16.12.1 Integrating a Function Containing Its Own Derivative
		16.12.2 Dividing an Integral into Several Integrals
		16.12.3 Integrating by Parts 1
		16.12.4 Integrating by Parts 2
		16.12.5 Integrating by Substitution 1
		16.12.6 Integrating by Substitution 2
		16.12.7 Integrating by Substitution 3
		16.12.8 Integrating with Partial Fractions
Appendix A Limit of (sinθ)/θ
Appendix B Integrating cosnθ
Index