Cambridge Mathematics for the IB Diploma Higher Level © Cambridge University Press, 2012 Option 10: Introduction 1

Option10: Discrete mathematics

Discrete mathematics includes several branches of mathematics concerned with the study of structures which are discrete, rather than continuous. Th is means that they are made up of discrete (separate) objects, such as whole numbers or vertices of a cube, rather than continuous quantities such as distance or time (represented by real numbers). Some aspects of discrete mathematics, mainly the theory of whole numbers, date back to antiquity. Many of the areas started developing in the 17th century and only found substantial applications in the 20th. One of the main drivers behind recent developments in the subject is its use in computer science, as computers are fundamentally discrete machines.

Five branches of Discrete Mathematics are studied within the IB syllabus:

Number Th eory is the fi rst part of this option. It is concerned with properties of whole numbers, particularly divisibility and prime numbers. Number theory was studied by Greek mathematicians,

Introductory problem

Suppose you have a large supply of 20 cent and 30 cent stamps. In how many diff erent ways can you make up $5 postage?

Introduction

In this Option you will learn:

about different methods of proof used in mathematics•

about divisibility of integers, and why prime numbers are so important•

how to fi nd integer solutions of linear equations in two variables (Diophantine • equations)

how to fi nd a remainder when one integer is divided by another, and how to perform • calculations and solve equations with remainders (modular arithmetic)

about various properties of remainders, including the Chinese Remainder Theorem and • Fermat’s Little Theorem

about a new mathematical structure called a graph, which consists of points joined by • lines, and can be used to model networks

how to solve optimisation problems on graphs, for example fi nding the shortest route • between two points

how to fi nd a formula for a sequence given by a recurrence relation.•

2 Cambridge Mathematics for the IB Diploma Higher Level © Cambridge University Press, 2012 Option 10: Introduction

especially around 3rd century AD, and a little bit later by Indian and Islamic mathematicians. One of the main topics of interest was Diophantine equations – equations where we are only interested in integer solutions. We will study these in chapter 30 of this option. Number theory has held the interest of mathematicians all over the world for many centuries. Th is is because it is full of problems that have remained unsolved for a long time. Some examples of these problems are: the Goldbach conjecture (that every even integer greater than 2 is a sum of two primes); the twin prime conjecture (that there are infi nitely many pairs of consecutive odd numbers which are both prime); and Fermat’s Last Th eorem (that there are no integers satisfying x y zn n n=yn when n is greater than 2), which was fi rst stated in 1637 but only proved in the 1990s. Prime numbers are a topic of active research because they have recently found applications in cryptography, but there are still many things we don’t know about their distribution and how to fi nd them.

Recurrence relations are equations that describe how to get from one term in a sequence to the next. Given a recurrence relation and a suffi cient number of starting terms we can use these rules to build up the sequence. However, for analysing the behaviour of the sequence it is more useful to know how the value of each term depends on its position in the sequence. Finding an equation for the nth term of a sequence can be remarkably diffi cult and in many cases even impossible. In the fi nal chapter of this option we will learn how to solve some types of recurrence relations. We will also use recurrence relations to solve counting problems and model fi nancial situations involving compound interest and debt repayment.

Logic is the study of rules and principles of reasoning and structure of arguments. It is a part of both mathematics and philosophy. As well as being of philosophical interest, it is important in electronic circuit design, computer science and the study of artifi cial intelligence. Within the IB syllabus, logic is covered in Mathematical Studies and in Th eory of Knowledge.

Graph Th eory is concerned with problems that can be modelled on a network. Th is is when a set of points are joined by lines, possibly with a length assigned to each line. Th ese graphs could represent road networks, molecular structures, electronic circuits or planning fl owcharts, for example. One class of problems involves studying the properties of graphs, such as various ways of getting from one point to another. Another important class is optimisation problems, where graph theory is required to develop an algorithm to fi nd, for example, the shortest possible route between two points, or the optimal sequence of operations to complete a task in the shortest possible time. Th is is very diff erent from continuous optimisation problems, which are solved using calculus. Graph theory was fi rst studied in the 18th century, but many of the developments are more recent and have applications in such diverse fi elds as computer science, physics, chemistry, sociology and linguistics. We will study graph theory in the second part of this option.

Group Th eory is an abstract study of the underlying structure of a collection of objects, rather than objects themselves. For example, counting hours on a 12-hour clock, rotations by 30°, and remainders when numbers are divided by 12 all have the same structure, in that they repeat aft er 12 steps. Group theory is an example of how a single abstract mathematical concept can be applied in many concrete situations. It is important within mathematics in the theory of polynomial equations and topology (the study of certain properties of shapes), but it also has applications in physics and chemistry because it can be used to describe symmetries. Groups are covered in the Sets, relations and groups option.

In the fi rst chapter of this option we will introduce various methods of proof which will be needed in subsequent chapters. Chapters 28–31 cover Number theory, chapters 32 and 33 Graph theory, and chapter 34 Recurrence relations. Chapter 35 contains a selection of examination-style questions, some of which combine material from diff erent chapters.