Pure mathematics is one of the oldest creative human activities and this module introduces its main topics. Group Theory explores sets of mathematical objects that can be combined – such as numbers, which can be added or multiplied, or rotations and reflections of a shape, which can be performed in succession. Linear Algebra explores 2 and 3dimensional space and systems of linear equations, and develops themes arising from the links between these topics. Analysis, the foundation of calculus, covers operations such as differentiation and integration, arising from infinite limiting processes. To study this module you should have a sound knowledge of relevant mathematics as provided by the appropriate OU level 1 study.
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This module is expected to start for the last time in October 2027. 
What you will study
Pure mathematics can be studied for its own sake, because of its intrinsic elegance and powerful ideas, but it also provides many of the principles that underlie applications of mathematics.
This module is suitable whether you want a basic understanding of mathematics without taking the subject further, or to prepare for higherlevel modules in pure mathematics, or if you teach mathematics (it includes a good deal of background to the Alevel mathematics syllabuses, for example).
You will become familiar with new mathematical ideas mainly by using pencil and paper and by thinking.
Introduction
Real Functions and Graphs is a reminder of the principles underlying the sketching of graphs of functions and other curves. Mathematical Language covers the writing of pure mathematics and some of the methods used to construct proofs. Number Systems looks at the systems of numbers most widely used in mathematics: the integers, rational numbers, real numbers, complex numbers and modular or ‘clock’ arithmetics.
Group Theory (A)
Symmetry studies the symmetries of plane figures and solids, including the five ‘Platonic solids’, and leads to the definition of a group. Groups and Subgroups introduces the idea of a cyclic group, using a geometric viewpoint, as well as isomorphisms between groups. Permutations introduces permutations, the cycle decomposition of permutations, odd and even permutations, and the notion of conjugacy. Cosets and Lagrange’s Theorem is about ‘blocking’ a group table, and leads to the notions of normal subgroup and quotient group.
Linear Algebra
Vectors and Conics is an introduction to vectors and to the properties of conic sections. Linear Equations and Matrices explains why simultaneous equations may have different numbers of solutions, and also explains the use of matrices. Vector Spaces generalises the plane and threedimensional space, providing a common structure for studying seemingly different problems. Linear Transformations is about mappings between vector spaces that preserve many geometric and algebraic properties. Eigenvectors leads to the diagonal representation of a linear transformation, and applications to conics and quadric surfaces.
Analysis (A)
Numbers deals with real numbers as decimals, rational and irrational numbers, and goes on to show how to manipulate inequalities between real numbers. Sequences explains the ‘null sequence’ approach, used to make rigorous the idea of convergence of sequences, leading to the definitions of pi and e. Series covers the convergence of series of real numbers and the use of series to define the exponential function. Continuity describes the sequential definition of continuity, some key properties of continuous functions, and their applications.
Group Theory (B)
Conjugacy looks at conjugate elements and conjugate subgroups, and returns to the idea of normal subgroups in this context. Homomorphisms is a generalisation of isomorphisms, which leads to a greater understanding of normal subgroups. Group Actions is a way of relating groups to geometry, which can be used to count the number of different ways a symmetric object can be coloured.
Analysis (B)
Limits introduces the epsilondelta approach to limits and continuity, and relates these to the sequential approach to limits of functions. Differentiation studies differentiable functions and gives l’HÃ´pital’s rule for evaluating limits. Integration explains the fundamental theorem of calculus, the Maclaurin integral test and Stirling’s formula. Power Series is about finding power series representations of functions, their properties and applications.
You will learn
Successful study of this module should improve your skills in working with abstract concepts, constructing solutions to problems logically and communicating mathematical ideas clearly.
Entry
Normally, to study this module you should have completed the OU level 1 module Essential mathematics 2 (MST125) or the discontinued module MS221. This level 1 module is ideal preparation. It provides you with a good basic knowledge of elementary algebra, coordinate geometry, Euclidean geometry, trigonometry, functions, differentiation and integration.
You can try our diagnostic quiz to help you determine whether you are adequately prepared for this module.
There may be circumstances in which you can study M208 without having first studied MST125 (or MS221), but you should speak to an adviser to discuss this before registering on this module.
Preparatory work
If you need to revise the subjects described in Entry, or you want to do some preparatory work, try reading some current Alevel textbooks, such as the MEI Structured Mathematics texts on Pure Mathematics and Further Pure Mathematics published by Hodder. They contain plenty of exercises to get you used to regular study.
For an exciting and accessible introduction to pure mathematics, try From Here to Infinity by Ian Stewart (Oxford Paperbacks).
Qualifications
M208 is a compulsory module in our:
M208 is an optional module in our:
It can also count towards most of our other degrees at bachelors level, where it is equally appropriate to a BA or BSc. We advise you to refer to the relevant qualification descriptions for information on the circumstances in which this module can count towards these qualifications because from time to time the structure and requirements may change.
Excluded combinations
Sometimes you will not be able to count a module towards a qualification if you have already taken another module with similar content. To check any excluded combinations relating to this module, visit our excluded combination finder or check with an adviser before registering.
If you have a disability
Please be aware that the module contains a large number of diagrams. The study materials are available in Adobe Portable Document Format (PDF). Some Adobe PDF components may not be available or fully accessible using a screen reader and mathematical, scientific, and foreign language materials may be particularly difficult to read in this way. Written transcripts are available for the audiovisual material. The books are available in a combbound format.
If you have particular study requirements please tell us as soon as possible, as some of our support services may take several weeks to arrange. Find out more about our services for disabled students..
Study materials
What's included
Module books, DVDs, CDs, website.
You will need
DVD and CD players.
Computing requirements
A computing device with a browser and broadband internet access is required for this module. Any modern browser will be suitable for most computer activities. Functionality may be limited on mobile devices.
Any additional software will be provided, or is generally freely available. However, some activities may have more specific requirements. For this reason, you will need to be able to install and run additional software on a device that meets the requirements below.
A desktop or laptop computer with either:
 Windows 7 or higher
 macOS 10.7 or higher
The screen of the device must have a resolution of at least 1024 pixels horizontally and 768 pixels vertically.
To participate in our onlinediscussion area you will need both a microphone and speakers/headphones.
Our Skills for OU study website has further information including computing skills for study, computer security, acquiring a computer and Microsoft software offers for students.
Teaching and assessment
Support from your tutor
You will have a tutor who will help you with the study material and mark and comment on your written work, and whom you can ask for advice and guidance. We may also be able to offer group tutorials or day schools that you are encouraged, but not obliged, to attend. Where your tutorials are held will depend on the distribution of students taking the module.
Contact us if you want to know more about study with The Open University before you register.
Assessment
The assessment details for this module can be found in the facts box above.
You can choose whether to submit your tutormarked assignments (TMAs) on paper or online through the eTMA system. You may want to use the eTMA system for some of your assignments but submit on paper for others. This is entirely your choice.
Professional recognition
This module may help you to gain membership of the Institute of Mathematics and its Applications (IMA). For further information, see the IMA website.
Students also studied
Students who studied this module also studied at some time:
Future availability
Pure mathematics starts once a year – in October. This page describes the module that will start in October 2018. We expect it to start for the last time in October 2027.
How to register
We regret that we are currently unable to accept registrations for this module. Where the module is to be presented again in the future, relevant registration information will be displayed on this page as soon as it becomes available.
Regulations
As a student of The Open University, you should be aware of the content of the academic regulations which are available on our
Essential Documents website.