from The Open University
Alternatively you can skip the navigation by pressing 'Enter'.

TV, Radio & Events menu item
Coming up
Secret History Of... Deptford High St
Tuesday, 21st April 2015 20:00  BBC FourHow did the "Oxford Street of South London" come to be one of the poorest shopping streets in modern London? Read more: The Secret History Of Our Streets  Deptford High StreetSecret History Of... Deptford High St
Wednesday, 22nd April 2015 02:00  BBC FourThinking Allowed: The Ethnography Award winner 2015
Wednesday, 22nd April 2015 16:00  BBC Radio 4Thinking Allowed: The Ethnography Award winner 2015
Monday, 27th April 2015 00:15  BBC FourCatch up
Thinking Allowed: The Ethnography Award 'Shortlist' 2015
Available until Friday, 15th April 2016 10:30This week's Thinking Allowed hosts a special programme dedicated to academic research in ethnography. Read more: Thinking Allowed: The Ethnography Award 'Shortlist' 2015A History of Ideas  Descartes Cogito Ergo Sum
Available until Thursday, 14th April 2016 08:30A History of Ideas  Erving Goffman's Performed Self
Available until Thursday, 14th April 2016 08:15Thinking Allowed: The Ethnography Award 'Shortlist' 2015
Available until Friday, 15th April 2016 09:45 
Get Started menu item
Welcome to OpenLearn
Get the most out of OpenLearn  Sign in
OpenLearn is the home of free learning from the UK's largest university, The Open University.
We have over 600 free courses available for you to study right now. More about this siteInspire me
What does it mean to be me?
Watch these short and snappy animations on the subject of me: the individual, memory, 'self' and... Watch now: What does it mean to be me?Take the photographic memory test
Can you capture scenes just by looking at them? Find out with our photographic memory test. Launch now: Take the photographic memory testFree courses
Introduction to bookkeeping and accounting
This free course Introduction to bookkeeping and accounting provides an introduction to the... Try: Introduction to bookkeeping and accounting nowSucceed with maths – Part 1
[BETA] If you feel that maths is a mystery that you want to unravel then this short 8week course... Try: Succeed with maths – Part 1 nowBecome an OU student
Learning to learn
Try our free course and get the most out of all our free courses Try: Learning to learn now
Number systems
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.
 Duration 20 hours
 Updated Thursday 7th April 2011
 Intermediate level
 Posted under Mathematics
Contents
 Current section: Introduction
 Learning outcomes
 1 Real numbers
 2 Complex numbers
 3 Modular arithmetic
 4 Equivalence relations
 Acknowledgements
Number systems
Introduction
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 x^{2} = 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 x^{2} + 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 Pure mathematics (M208) [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)]
About this site
OpenLearn: free learning from The Open University