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## About this book

John Vince describes a range of mathematical topics to provide a foundation for an undergraduate course in computer science, starting with a review of number systems and their relevance to digital computers, and finishing with differential and integral calculus. Readers will find that the author's visual approach will greatly improve their understanding as to why certain mathematical structures exist, together with how they are used in real-world applications.

Each chapter includes full-colour illustrations to clarify the mathematical descriptions, and in some cases, equations are also coloured to reveal vital algebraic patterns. The numerous worked examples will consolidate comprehension of abstract mathematical concepts.

Foundation Mathematics for Computer Science covers number systems, algebra, logic, trigonometry, coordinate systems, determinants, vectors, matrices, geometric matrix transforms, differential and integral calculus, and reveals the names of the mathematicians behind such inventions. During this journey, John Vince touches upon more esoteric topics such as quaternions, octonions, Grassmann algebra, Barycentric coordinates, transfinite sets and prime numbers. Whether you intend to pursue a career in programming, scientific visualisation, systems design, or real-time computing, you should find the author’s literary style refreshingly lucid and engaging, and prepare you for more advanced texts.

## Table of Contents

### Chapter 1. Visual Mathematics

Abstract
I consider myself a “visual” person, as pictures help me understand complex problems. I also don’t find it too difficult to visualise objects from different view points. I remember learning about electrons, neutrons and protons for the first time, where our planetary system provided a simple model to visualise the hidden structure of matter. My mental image of electrons was one of small orange spheres, spinning around a small, central nucleus containing blue protons and grey neutrons. I had still to discover the mechanism of colour, and although this visual model was seriously flawed, it provided a first step towards understanding the structure of matter.
John Vince

### Chapter 2. Numbers

Abstract
Our brain’s visual cortex possesses some incredible image processing features. For example, children know instinctively when they are given less sweets than another child, and adults know instinctively when they are short-changed by a Parisian taxi driver, or driven around the Arc de Triumph several times, on the way to the airport! Intuitively, we can assess how many donkeys are in a field without counting them, and generally, we seem to know within a second or two, whether there are just a few, dozens, or hundreds of something. But when accuracy is required, one can’t beat counting.
John Vince

### Chapter 3. Algebra

Abstract
Some people, including me, find learning a foreign language a real challenge; one of the reasons being the inconsistent rules associated with its syntax. For example, why is a table feminine in French, “la table”, and a bed masculine, “le lit”? They both have four legs! The rules governing natural language are continuously being changed by each generation, whereas mathematics appears to be logical and consistent. The reason for this consistency is due to the rules associated with numbers and the way they are combined together and manipulated at an abstract level. Such rules, or axioms, generally make our life easy, however, as we saw with the invention of negative numbers, extra rules have to be introduced, such as “two negatives make a positive”, which is easily remembered. However, as we explore mathematics, we discover all sorts of inconsistencies, such as there is no real value associated with the square-root of a negative number. It’s forbidden to divide a number by zero. Zero divided by zero gives inconsistent results. Nevertheless, such conditions are easy to recognise and avoided. At least in mathematics, we don’t have to worry about masculine and feminine numbers!.
John Vince

### Chapter 4. Logic

Abstract
The English mathematician George Boole (1815–1864) is regarded as the “father” of symbolic logic, which is why it bears his name. He did not associate logic with mathematics, but wanted to devise a logical framework for expressing and analysing logical statements. A logical statement contains one or more premises (or propositions), that form the basis of an argument. However, not all premises are true, and starting from an incorrect premise is not a good strategy for winning an argument, therefore one must anticipate the existence of valid and invalid premises. Complex arguments often combine individual premises using the logical connectives negation (NOT), conjunction (AND), inclusive disjunction (OR) and the exclusive disjunction (XOR). Today, logic is considered to play a central role in mathematics, and Russell believed that mathematics could be derived entirely from logic.
John Vince

### Chapter 5. Trigonometry

Abstract
This chapter covers some basic features of trigonometry such as angular measure, trigonometric ratios, inverse ratios, trigonometric identities and various rules, with which the reader should be familiar. The word “trigonometry” divides into three parts: “tri”, “gon”, “metry”, which means the measurement of three-sided polygons, i.e. triangles. It is an ancient subject and is used across all branches of mathematics.
John Vince

### Chapter 6. Coordinate Systems

Abstract
In this chapter we revise Cartesian coordinates, axial systems, the distance between two points in space, and the area of simple 2D shapes. It also covers polar, spherical polar and cylindrical coordinate systems. René Descartes is often credited with the invention of the xy-plane, but the French lawyer and mathematician Pierre de Fermat (1601–1665) was probably the first inventor. In 1636 Fermat was working on a treatise titled Ad locus planos et solidos isagoge, which outlined what we now call “analytic geometry”. Unfortunately, Fermat never published his treatise, although he shared his ideas with other mathematicians such as Blaise Pascal (1623–1662). At the same time, Descartes devised his own system of analytic geometry and in 1637 published his results in the prestigious journal Géométrie. In the eyes of the scientific world, the publication date of a technical paper determines when a new idea or invention is released into the public domain. Consequently, ever since this publication Descartes has been associated with the xy-plane, which is why it is called the Cartesian plane.
John Vince

### Chapter 7. Determinants

Abstract
When patterns of numbers or symbols occur over and over again, mathematicians often devise a way to simplify their description and assign a name to them.
John Vince

### Chapter 8. Vectors

Abstract
Vectors are a relative new invention in the world of mathematics, dating only from the 19th century. They enable us to solve complex geometric problems, the dynamics of moving objects, and problems involving forces and fields. We often only require a single number to represent quantities used in our daily lives such as height, age, shoe size, waist and chest measurement. The magnitude of these numbers depends on our age and whether we use metric or imperial units. Such quantities are called scalars. On the other hand, there are some things that require more than one number to represent them: wind, force, weight, velocity and sound are just a few examples. For example, any sailor knows that wind has a magnitude and a direction. The force we use to lift an object also has a value and a direction. Similarly, the velocity of a moving object is measured in terms of its speed (e.g. miles per hour), and a direction such as north-west. Sound, too, has intensity and a direction. Such quantities are called vectors.
John Vince

### Chapter 9. Matrices

Abstract
Matrices, like determinants, have their background in algebra and offer another way to represent and manipulate equations. Matrices can be added, subtracted and multiplied together, and even inverted, however, they must give the same result obtained through traditional algebraic techniques. A useful way to introduce the subject is via geometric transforms, which we examine first.
John Vince

### Chapter 10. Geometric Matrix Transforms

Abstract
Geometric matrix transforms are an intuitive way of defining and building geometric operations such as scale, translate, reflect, shear and rotate. In 2D, such operations are generally associated with images and text, and widely used in internet browsers, image-processing software, smart phones and watches. In 3D, they are used in computer games, computer animation, film special effects, virtual reality and scientific visualisation. They have proved so useful that they are incorporated in hardware to provide the highest possible execution speeds and real-time performance.
John Vince

### Chapter 11. Calculus: Derivatives

Abstract
Some quantities, such as the area of a circle or an ellipse, cannot be written precisely, as they incorporate $$\pi$$, which is transcendental. However, an approximate value can be obtained by devising a definition that includes a parameter that is made infinitesimally small. The techniques of limits and infinitesimals have been used in mathematics for over two-thousand years, and paved the way towards today’s calculus.
John Vince

### Chapter 12. Calculus: Integration

Abstract
In this chapter we develop the idea that integration is the inverse of differentiation, and explore the standard algebraic strategies for integrating functions, where the derivative is unknown; these include simple algebraic manipulation, trigonometric identities, integration by parts, integration by substitution and integration using partial fractions.
John Vince
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