from The Open University
Alternatively you can skip the navigation by pressing 'Enter'.
Wartime Farm Episode 2Tuesday, 5th May 2015 11:00 - YesterdayAs our wartime farmers continue to get to grips with agriculture under fire, they pitch in for silage and meet the... Read more: Wartime Farm - Episode 2
Wastemen: The Home FrontAvailable until Thursday, 4th June 2015 00:00In the first of the series, Christmas is coming. And the most magical time of the year brings with it enormous... Read more: Wastemen: The Home Front
OU on the BBC: Frozen PlanetA stunning portrait of life at the poles, presented by David Attenborough Read more: OU on the BBC: Frozen Planet
Take the photographic memory testCan you capture scenes just by looking at them? Find out with our photographic memory test. Launch now: Take the photographic memory test
Ratio, proportion and percentagesFrom politics to cookery, ratios, proportions and percentages are part of everyday life. This... Try: Ratio, proportion and percentages now
Succeed with maths – Part 1[BETA] If you feel that maths is a mystery that you want to unravel then this short 8-week course... Try: Succeed with maths – Part 1 now
Number systems and the rules for combining numbers can be daunting. This unit will help...
Number systems and the rules for combining numbers can be daunting. This unit will help you to understand the detail of rational and real numbers, complex numbers and integers. You will also be introduced to modular arithmetic and the concept of a relation between elements of a set.
After studying this unit you should be able to:
- understand the arithmetical properties of the rational and real numbers;
- understand the definition of a complex number;
- perform arithmetical operations with complex numbers;
- represent complex numbers as points in the complex plane;
- determine the polar form of a complex number;
- use de Moivre's Theorem to find the nth roots of a complex number and to find some trigonometric identities;
- understand the definition of ez, where z is a complex variable;
- explain the terms modular addition and modular multiplication;
- use Euclid's Algorithm to find multiplicative inverses in modular arithmetic, where these exist;
- explain the meanings of a relation defined on a set, an equivalence relation and a partition of a set;
- determine whether a given relation defined on a given set is an equivalence relation by checking the reflexive, symmetric and transitive properties;
- understand that an equivalence relation partitions a set into equivalence classes;
- determine the equivalence classes for a given equivalence relation.
- Current section: Introduction
- Learning outcomes
- 1 Real numbers
- 2 Complex numbers
- 3 Modular arithmetic
- 4 Equivalence relations
In this unit we look at some different systems of numbers, and the rules for combining numbers in these systems. For each system we consider the question of which elements have additive and/or multiplicative inverses in the system. We look at solving certain equations in the system, such as linear, quadratic and other polynomial equations.
In Section 1 we start by revising the notation used for the rational numbers and the real numbers, and we list their arithmetical properties. You will meet other properties of these numbers in the analysis units, as the study of real functions depends on properties of the real numbers. We note that some quadratic equations with rational coefficients, such as x2 = 2, have solutions which are real but not rational.
In Section 2 we introduce the set of complex numbers. This system of numbers enables us to solve all polynomial equations, including those with no real solutions, such as x2 + 1 = 0. Just as real numbers correspond to points on the real line, so complex numbers correspond to points in a plane, known as the complex plane.
In Section 3 we look further at some properties of the integers, and introduce modular arithmetic. This will be useful in the group theory units, as some sets of numbers with the operation of modular addition or modular multiplication form groups.
In Section 4 we introduce the concept of a relation between elements of a set. This is a more general idea than that of a function, and leads us to a mathematical structure known as an equivalence relation. An equivalence relation on a set classifies elements of the set, separating them into disjoint subsets called equivalence classes.
This unit is an adapted extract from the Open University course