Foundation Mathematics for Computer Science: A Visual Approach

Preface
Contents
1 Visual Mathematics
	1.1 Introduction
	1.2 Visual Brains Versus Analytic Brains
	1.3 Learning Mathematics
	1.4 What Makes Mathematics Difficult?
	1.5 Does Mathematics Exist Outside Our Brains?
	1.6 Symbols and Notation
	Reference
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 Types of Numbers
		2.7.1 Natural Numbers
		2.7.2 Integers
		2.7.3 Rational Numbers
		2.7.4 Irrational Numbers
		2.7.5 Real Numbers
		2.7.6 Algebraic and Transcendental Numbers
		2.7.7 Imaginary Numbers
		2.7.8 Complex Numbers
		2.7.9 Quaternions and Octonions
	2.8 Prime Numbers
		2.8.1 The Fundamental Theorem of Arithmetic
		2.8.2 Is 1 a Prime?
		2.8.3 The Goldbach Conjecture
		2.8.4 Prime Number Distribution
		2.8.5 Infinity of Primes
		2.8.6 Mersenne Numbers
	2.9 Perfect Numbers
	2.10 Triangular Numbers
	2.11 Infinity
	2.12 Worked Examples
		2.12.1 Algebraic Expansion
		2.12.2 Complex Numbers
		2.12.3 Quaternions
	References
3 Systems of Counting
	3.1 Introduction
	3.2 Decimal Positional System
		3.2.1 Background
		3.2.2 Binary Numbers
		3.2.3 Octal Numbers
		3.2.4 Hexadecimal Numbers
	3.3 Converting Decimal to Binary, Octal and Hexadecimal
		3.3.1 Converting Decimal to Binary
		3.3.2 Converting Decimal to Octal
		3.3.3 Converting Decimal to Hexadecimal
	3.4 Converting Between Binary and Octal Numbers
	3.5 Converting Between Binary and Hexadecimal Numbers
	3.6 Adding and Subtracting Binary Numbers
		3.6.1 Adding Binary Numbers
		3.6.2 Subtracting Binary Numbers Using Two's Complement
	3.7 Adding and Subtracting Decimal Numbers
		3.7.1 Adding Decimal Numbers
		3.7.2 Subtracting Decimal Numbers Using Ten's Complement
	3.8 Adding and Subtracting Octal Numbers
		3.8.1 Adding Octal Numbers
		3.8.2 Subtracting Octal Numbers Using Eight's Complement
	3.9 Summary
	3.10 Worked Examples
		3.10.1 Convert a Decimal Number into Binary
		3.10.2 Convert a Decimal Number into Binary Using an Algorithm
		3.10.3 Convert a Binary Number into Decimal
		3.10.4 Convert a Binary Number into Octal
		3.10.5 Convert an Octal Number into Binary
		3.10.6 Convert an Octal Number into Hexadecimal
		3.10.7 Convert a Hexadecimal Number into Octal
		3.10.8 Convert a Decimal Number into Octal
		3.10.9 Convert a Decimal Number into Octal Using an Algorithm
		3.10.10 Convert a Decimal Number into Hexadecimal
		3.10.11 Add Binary Numbers
		3.10.12 Subtract Binary Numbers
		3.10.13 Add Octal Numbers
		3.10.14 Subtract Octal Numbers
		3.10.15 Add Hexadecimal Numbers
4 Algebra
	4.1 Introduction
	4.2 Background
	4.3 Notation
		4.3.1 Solving the Roots of a Quadratic Equation
	4.4 Indices
		4.4.1 Laws of Indices
	4.5 Logarithms
	4.6 Further Notation
	4.7 Functions
		4.7.1 Explicit and Implicit Equations
		4.7.2 Function Notation
		4.7.3 Intervals
		4.7.4 Function Domains and Ranges
		4.7.5 Odd and Even Functions
		4.7.6 Power Functions
	4.8 Series
	4.9 Binomial Theorem
	4.10 Summary
	4.11 Worked Examples
		4.11.1 Algebraic Manipulation
		4.11.2 Solving a Quadratic Equation
		4.11.3 Factorising
		4.11.4 Binomial Theorem
	References
5 Logic
	5.1 Introduction
	5.2 Background
	5.3 Truth Tables
		5.3.1 Logical Connectives
	5.4 Logical Premises
		5.4.1 Material Equivalence
		5.4.2 Implication
		5.4.3 Negation
		5.4.4 Conjunction
		5.4.5 Inclusive Disjunction
		5.4.6 Exclusive Disjunction
		5.4.7 Idempotence
		5.4.8 Commutativity
		5.4.9 Associativity
		5.4.10 Distributivity
		5.4.11 de Morgan's Laws
		5.4.12 Simplification
		5.4.13 Excluded Middle
		5.4.14 Contradiction
		5.4.15 Double Negation
		5.4.16 Implication and Equivalence
		5.4.17 Exportation
		5.4.18 Contrapositive
		5.4.19 Reductio Ad Absurdum
		5.4.20 Modus Ponens
		5.4.21 Proof by Cases
	5.5 Set Theory
		5.5.1 Empty Set
		5.5.2 Membership and Cardinality of a Set
		5.5.3 Subsets, Supersets and the Universal Set
		5.5.4 Set Building
		5.5.5 Union
		5.5.6 Intersection
		5.5.7 Relative Complement
		5.5.8 Absolute Complement
		5.5.9 Power Set
	5.6 Worked Examples
		5.6.1 Truth Tables
		5.6.2 Set Building
		5.6.3 Sets
		5.6.4 Power Set
6 Combinatorics
	6.1 Introduction
	6.2 Permutations
	6.3 Permutations of Multisets
	6.4 Combinations
	6.5 Worked Examples
		6.5.1 Eight-Permutations of a Multiset
		6.5.2 Eight-Permutations of a Multiset
		6.5.3 Number of Permutations
		6.5.4 Number of Five-Card Hands
		6.5.5 Hand Shakes with 100 People
		6.5.6 Permutations of MISSISSIPPI
7 Probability
	7.1 Introduction
	7.2 Definition and Notation
		7.2.1 Independent Events
		7.2.2 Dependent Events
		7.2.3 Mutually Exclusive Events
		7.2.4 Inclusive Events
		7.2.5 Probability Using Combinations
	7.3 Worked Examples
		7.3.1 Product of Probabilities
		7.3.2 Book Arrangements
		7.3.3 Winning a Lottery
		7.3.4 Rolling Two Dice
		7.3.5 Two Dice Sum to 7
		7.3.6 Two Dice Sum to 4
		7.3.7 Dealing a Red Ace
		7.3.8 Selecting Four Aces in Succession
		7.3.9 Selecting Cards
		7.3.10 Selecting Four Balls from a Bag
		7.3.11 Forming Teams
		7.3.12 Dealing Five Cards
8 Modular Arithmetic
	8.1 Introduction
	8.2 Informal Definition
	8.3 Notation
	8.4 Congruence
	8.5 Negative Numbers
	8.6 Arithmetic Operations
		8.6.1 Sums of Numbers
		8.6.2 Products
		8.6.3 Multiplying by a Constant
		8.6.4 Congruent Pairs
		8.6.5 Multiplicative Inverse
		8.6.6 Modulo a Prime
		8.6.7 Fermat's Little Theorem
	8.7 Applications of Modular Arithmetic
		8.7.1 ISBN Parity Check
		8.7.2 IBAN Check Digits
	8.8 Worked Examples
		8.8.1 Negative Numbers
		8.8.2 Sums of Numbers
		8.8.3 Remainders of Products
		8.8.4 Multiplicative Inverse
		8.8.5 Product Table for Modulo 13
		8.8.6 ISBN Check Digit
	Reference
9 Trigonometry
	9.1 Introduction
	9.2 Background
	9.3 Units of Angular Measurement
	9.4 The Trigonometric Ratios
		9.4.1 Domains and Ranges
	9.5 Inverse Trigonometric Ratios
	9.6 Trigonometric Identities
	9.7 The Sine Rule
	9.8 The Cosine Rule
	9.9 Compound-Angle Identities
		9.9.1 Double-Angle Identities
		9.9.2 Multiple-Angle Identities
		9.9.3 Half-Angle Identities
	9.10 Perimeter Relationships
	9.11 Worked Examples
		9.11.1 Degrees to Radians
		9.11.2 Sine Rule
		9.11.3 Cosine Rule
		9.11.4 Compound Angle
		9.11.5 Double-Angle Identity
		9.11.6 Perimeter Relationship
10 Coordinate Systems
	10.1 Introduction
	10.2 Background
	10.3 The Cartesian Plane
	10.4 Function Graphs
	10.5 Shape Representation
		10.5.1 2D Polygons
		10.5.2 Areas of Shapes
	10.6 Theorem of Pythagoras in 2D
		10.6.1 Pythagorean Triples
	10.7 3D Cartesian Coordinates
		10.7.1 Theorem of Pythagoras in 3D
	10.8 Polar Coordinates
	10.9 Spherical Polar Coordinates
	10.10 Cylindrical Coordinates
	10.11 Barycentric Coordinates
	10.12 Homogeneous Coordinates
	10.13 Worked Examples
		10.13.1 Area of a Shape
		10.13.2 Distance Between Two Points
		10.13.3 Polar Coordinates
		10.13.4 Spherical Polar Coordinates
		10.13.5 Cylindrical Coordinates
		10.13.6 Barycentric Coordinates
	Reference
11 Determinants
	11.1 Introduction
	11.2 Background
	11.3 Linear Equations with Two Variables
	11.4 Linear Equations with Three Variables
		11.4.1 Sarrus's Rule
	11.5 Mathematical Notation
		11.5.1 Matrix
		11.5.2 Order of a Determinant
		11.5.3 Value of a Determinant
		11.5.4 Properties of Determinants
	11.6 Worked Examples
		11.6.1 Determinant Expansion
		11.6.2 Complex Determinant
		11.6.3 Simple Expansion
		11.6.4 Simultaneous Equations
12 Vectors
	12.1 Introduction
	12.2 Background
	12.3 2D Vectors
		12.3.1 Vector Notation
		12.3.2 Graphical Representation of Vectors
		12.3.3 Magnitude of a Vector
	12.4 3D Vectors
		12.4.1 Vector Manipulation
		12.4.2 Scaling a Vector
		12.4.3 Vector Addition and Subtraction
		12.4.4 Position Vectors
		12.4.5 Unit Vectors
		12.4.6 Cartesian Vectors
		12.4.7 Products
		12.4.8 Scalar Product
		12.4.9 The Dot Product in Lighting Calculations
		12.4.10 The Scalar Product in Back-Face Detection
		12.4.11 The Vector Product
		12.4.12 The Right-Hand Rule
	12.5 Deriving a Unit Normal Vector for a Triangle
	12.6 Surface Areas
		12.6.1 Calculating 2D Areas
	12.7 Summary
	12.8 Worked Examples
		12.8.1 Position Vector
		12.8.2 Unit Vector
		12.8.3 Vector Magnitude
		12.8.4 Angle Between Two Vectors
		12.8.5 Vector Product
	References
13 Complex Numbers
	13.1 Introduction
	13.2 Representing Complex Numbers
		13.2.1 Complex Numbers
		13.2.2 Real and Imaginary Parts
		13.2.3 The Complex Plane
	13.3 Complex Algebra
		13.3.1 Algebraic Laws
		13.3.2 Complex Conjugate
		13.3.3 Complex Division
		13.3.4 Powers of i
		13.3.5 Rotational Qualities of i
		13.3.6 Modulus and Argument
		13.3.7 Complex Norm
		13.3.8 Complex Inverse
		13.3.9 Complex Exponentials
		13.3.10 de Moivre's Theorem
		13.3.11 nth Root of Unity
		13.3.12 nth Roots of a Complex Number
		13.3.13 Logarithm of a Complex Number
		13.3.14 Raising a Complex Number to a Complex Power
		13.3.15 Visualising Simple Complex Functions
		13.3.16 The Hyperbolic Functions
	13.4 Summary
	13.5 Worked Examples
		13.5.1 Complex Addition
		13.5.2 Complex Products
		13.5.3 Complex Division
		13.5.4 Complex Rotation
		13.5.5 Polar Notation
		13.5.6 Real and Imaginary Parts
		13.5.7 Magnitude of a Complex Number
		13.5.8 Complex Norm
		13.5.9 Complex Inverse
		13.5.10 de Moivre's Theorem
		13.5.11 nth Root of Unity
		13.5.12 Roots of a Complex Number
		13.5.13 Logarithm of a Complex Number
		13.5.14 Raising a Number to a Complex Power
	References
14 Matrices
	14.1 Introduction
	14.2 Geometric Transforms
	14.3 Transforms and Matrices
	14.4 Matrix Notation
		14.4.1 Matrix Dimension or Order
		14.4.2 Square Matrix
		14.4.3 Column Vector
		14.4.4 Row Vector
		14.4.5 Null Matrix
		14.4.6 Unit Matrix
		14.4.7 Trace
		14.4.8 Determinant of a Matrix
		14.4.9 Transpose
		14.4.10 Symmetric Matrix
		14.4.11 Antisymmetric Matrix
	14.5 Matrix Addition and Subtraction
		14.5.1 Scalar Multiplication
	14.6 Matrix Products
		14.6.1 Row and Column Vectors
		14.6.2 Row Vector and a Matrix
		14.6.3 Matrix and a Column Vector
		14.6.4 Square Matrices
		14.6.5 Rectangular Matrices
	14.7 Inverse Matrix
		14.7.1 Inverting a Pair of Matrices
	14.8 Orthogonal Matrix
	14.9 Diagonal Matrix
	14.10 Worked Examples
		14.10.1 Matrix Inversion
		14.10.2 Identity Matrix
		14.10.3 Solving Two Equations Using Matrices
		14.10.4 Solving Three Equations Using Matrices
		14.10.5 Solving Two Complex Equations
		14.10.6 Solving Three Complex Equations
		14.10.7 Solving Two Complex Equations
		14.10.8 Solving Three Complex Equations
15 Geometric Matrix Transforms
	15.1 Introduction
	15.2 Matrix Transforms
		15.2.1 2D Translation
		15.2.2 2D Scaling
		15.2.3 2D Reflections
		15.2.4 2D Shearing
		15.2.5 2D Rotation
		15.2.6 2D Scaling
		15.2.7 2D Reflection
		15.2.8 2D Rotation About an Arbitrary Point
	15.3 3D Transforms
		15.3.1 3D Translation
		15.3.2 3D Scaling
		15.3.3 3D Rotation
		15.3.4 Rotating About an Axis
		15.3.5 3D Reflections
	15.4 Rotating a Point About an Arbitrary Axis
		15.4.1 Matrices
	15.5 Determinant of a Transform
	15.6 Perspective Projection
	15.7 Worked Examples
		15.7.1 2D Scale and Translate
		15.7.2 2D Rotation
		15.7.3 Determinant of the Rotate Transform
		15.7.4 Determinant of the Shear Transform
		15.7.5 Yaw, Pitch and Roll Transforms
		15.7.6 Rotation About an Arbitrary Axis
		15.7.7 3D Rotation Transform Matrix
		15.7.8 Perspective Projection
16 Calculus: Derivatives
	16.1 Introduction
	16.2 Background
	16.3 Small Numerical Quantities
	16.4 Equations and Limits
		16.4.1 Quadratic Function
		16.4.2 Cubic Equation
		16.4.3 Functions and Limits
		16.4.4 Graphical Interpretation of the Derivative
		16.4.5 Derivatives and Differentials
		16.4.6 Integration and Antiderivatives
	16.5 Function Types
	16.6 Differentiating Groups of Functions
		16.6.1 Sums of Functions
		16.6.2 Function of a Function
		16.6.3 Function Products
		16.6.4 Function Quotients
	16.7 Differentiating Implicit Functions
	16.8 Differentiating Exponential and Logarithmic Functions
		16.8.1 Exponential Functions
		16.8.2 Logarithmic Functions
	16.9 Differentiating Trigonometric Functions
		16.9.1 Differentiating tan
		16.9.2 Differentiating csc
		16.9.3 Differentiating sec
		16.9.4 Differentiating cot
		16.9.5 Differentiating arcsin, arccos and arctan
		16.9.6 Differentiating arccsc, arcsec and arccot
	16.10 Differentiating Hyperbolic Functions
		16.10.1 Differentiating sinh, cosh and tanh
	16.11 Higher Derivatives
	16.12 Higher Derivatives of a Polynomial
	16.13 Identifying a Local Maximum or Minimum
	16.14 Partial Derivatives
		16.14.1 Visualising Partial Derivatives
		16.14.2 Mixed Partial Derivatives
	16.15 Chain Rule
	16.16 Total Derivative
	16.17 Power Series
	16.18 Worked Examples
		16.18.1 Antiderivative 1
		16.18.2 Antiderivative 2
		16.18.3 Differentiating Sums of Functions
		16.18.4 Differentiating a Function Product
		16.18.5 Differentiating an Implicit Function
		16.18.6 Differentiating a General Implicit Function
		16.18.7 Local Maximum or Minimum
		16.18.8 Partial Derivatives
		16.18.9 Mixed Partial Derivative 1
		16.18.10 Mixed Partial Derivative 2
		16.18.11 Total Derivative
17 Calculus: Integration
	17.1 Introduction
	17.2 Indefinite Integral
	17.3 Integration Techniques
		17.3.1 Continuous Functions
		17.3.2 Difficult Functions
	17.4 Trigonometric Identities
		17.4.1 Exponent Notation
		17.4.2 Completing the Square
		17.4.3 The Integrand Contains a Derivative
		17.4.4 Converting the Integrand into a Series of Fractions
		17.4.5 Integration by Parts
		17.4.6 Integration by Substitution
		17.4.7 Partial Fractions
	17.5 Summary
	17.6 Worked Examples
		17.6.1 Integrating a Function Containing its Own Derivative
		17.6.2 Dividing an Integral into Several Integrals
		17.6.3 Integrating by Parts 1
		17.6.4 Integrating by Parts 2
		17.6.5 Integrating by Substitution 1
		17.6.6 Integrating by Substitution 2
		17.6.7 Integrating by Substitution 3
		17.6.8 Integrating with Partial Fractions
18 Area
	18.1 Introduction
	18.2 Area Under a Graph
	18.3 Calculating Areas
	18.4 Positive and Negative Areas
	18.5 Area Between Two Functions
	18.6 Areas with the y-Axis
	18.7 Area with Parametric Functions
	18.8 The Riemann Sum
	18.9 Surface of Revolution
		18.9.1 Surface Area of a Cylinder
		18.9.2 Surface Area of a Right Cone
		18.9.3 Surface Area of a Sphere
		18.9.4 Surface Area of a Paraboloid
	18.10 Surface Area Using Parametric Functions
	18.11 Double Integrals
	18.12 Jacobians
		18.12.1 1D Jacobian
		18.12.2 2D Jacobian
		18.12.3 3D Jacobian
	18.13 Double Integrals for Calculating Area
	18.14 Summary
		18.14.1 Summary of Formulae
19 Volume
	19.1 Introduction
	19.2 Solid of Revolution: Disks
		19.2.1 Volume of a Cylinder
		19.2.2 Volume of a Right Cone
		19.2.3 Volume of a Right Conical Frustum
		19.2.4 Volume of a Sphere
		19.2.5 Volume of an Ellipsoid
		19.2.6 Volume of a Paraboloid
	19.3 Solid of Revolution: Shells
		19.3.1 Volume of a Cylinder
		19.3.2 Volume of a Right Cone
		19.3.3 Volume of a Sphere
		19.3.4 Volume of a Paraboloid
	19.4 Volumes with Double Integrals
		19.4.1 Objects with a Rectangular Base
		19.4.2 Rectangular Box
		19.4.3 Rectangular Prism
		19.4.4 Curved Top
		19.4.5 Objects with a Circular Base
		19.4.6 Cylinder
		19.4.7 Truncated Cylinder
	19.5 Volumes with Triple Integrals
		19.5.1 Rectangular Box
		19.5.2 Volume of a Cylinder
		19.5.3 Volume of a Sphere
		19.5.4 Volume of a Cone
	19.6 Summary
		19.6.1 Summary of Formulae
Appendix A Limit of (sinθ)/θ
Appendix B Integrating cosnθ
Index