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Pure mathematics

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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 3-dimensional 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.

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 higher-level modules in pure mathematics, or if you teach mathematics (it includes a good deal of background to the A-level mathematics syllabuses, for example).

You will become familiar with new mathematical ideas mainly by using pencil and paper and by thinking.

Sets, functions and vectors
revises these important foundations of pure mathematics and the mathematical language used to describe them. 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’ arithmetic, and looks at when and how certain types of equations can be solved in the system. Mathematical language and proof covers the writing of pure mathematics and some of the methods used to construct proofs, and as a further introduction to abstract mathematical thinking equivalence relations are introduced. Real functions, graphs and conics is a reminder of the principles underlying the sketching of graphs of functions and other curves.

Group theory 1
Symmetry and groups
studies the symmetry of plane figures and solids, and shows how this topic leads to the definition of a group, which is a set of elements that can be combined with each other in a way that has four basic properties called group axioms. Subgroups and isomorphisms looks at subgroups, which are groups that lie inside other groups, and also at cyclic groups, which are groups whose elements can all be obtained by repeatedly combining a single element with itself. It also investigates groups that appear different but have identical structures. Permutations studies functions that rearrange the elements of a set: it shows how these functions form groups and looks at some of their properties. Lagrange’s Theorem and small groups introduces a fundamental theorem about groups, and uses it to investigate the structures of groups that have only a few elements, before focusing on improving skills in understanding theorems and proofs in the context of group theory.

Linear Algebra
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 three-dimensional 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 1
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 2
Cosets and quotient groups
looks at how we can sometimes ‘divide’ a group by one of its subgroups to obtain another group. Conjugacy looks at how in any group some elements and some subgroups are similar to each other in a particular sense. Homomorphisms looks at functions that map groups to other groups in a way that respects at least some of the structure of the groups. Group actions studies how group elements can sometimes be applied to elements of other sets in natural ways.  This leads to a method of counting how many different objects there are of certain types, such as how many different coloured cubes can be produced if their faces can be painted any of three different colours.

Analysis 2
introduces the epsilon-delta approach to limits and continuity, and relates these to the sequential approach to limits of functions. Differentiation studies differentiable functions and gives L’Hôpital’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.

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.

Entry requirements

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 A-level 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).

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 online-discussion 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.


The assessment details for this module can be found in the facts box above.

You can choose whether to submit your tutor-marked 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.

If you have a disability

The OU strives to make all aspects of study accessible to everyone and this Accessibility Statement outlines what studying M208 involves. You should use this information to inform your study preparations and any discussions with us about how we can meet your needs.

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.

Course work includes:

7 Tutor-marked assignments (TMAs)
No residential school

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