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Pure mathematics

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Pure mathematics is one of the oldest creative human activities; this module introduces its central 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 from the links between these topics. Analysis, the foundation of calculus, covers operations such as differentiation and integration arising from infinite limiting processes.

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 pure 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 whether and how certain types of equations can be solved in each 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 normal subgroups
revises the Group theory 1 units and looks at how a group can be split into ‘shifts’ of any one of its subgroups. Quotient groups and conjugacy looks at how we can sometimes ‘divide’ a group by one of its subgroups to obtain another group, and 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 can find the full content list on the Open mathematics and statistics website.

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

You must have passed the following module:

Or be able to provide evidence you have the required mathematical skills.

You can check you’re ready for M208 and see the topics it covers here.

Talk to an advisor if you’re not sure you’re ready.

Preparatory work

You should aim to be confident and fluent with the concepts covered in the Are you ready? quiz here, and follow the advice in the quiz.

The key topics to revise include:

  • algebraic manipulation
  • coordinate geometry
  • trigonometry
  • functions
  • differentiation and integration
  • mathematical language
  • proof.

Essential mathematics 2 (MST125) is ideal preparation.

What's included

You’ll have access to a module website, which includes:

  • a week-by-week study planner
  • course-specific module materials
  • audio and video content
  • assessment details, instructions and guidance
  • online tutorial access
  • access to student and tutor group forums.

You’ll be provided with six printed module books, each covering one block of study, with many worked examples and exercises. You’ll also receive a printed handbook summarising the whole module, which you can refer to throughout your study and use during the exam.

Computing requirements

You’ll need broadband internet access and a desktop or laptop computer with an up-to-date version of Windows (10 or 11) or macOS Ventura or higher.

Any additional software will be provided or is generally freely available.

To join in spoken conversations in tutorials, we recommend a wired headset (headphones/earphones with a built-in microphone).

Our module websites comply with web standards, and any modern browser is suitable for most activities.

Our OU Study mobile app will operate on all current, supported versions of Android and iOS. It’s not available on Kindle.

It’s also possible to access some module materials on a mobile phone, tablet device or Chromebook. However, as you may be asked to install additional software or use certain applications, you’ll also require a desktop or laptop, as described above.

Teaching and assessment

Support from your tutor

Throughout your module studies, you’ll get help and support from your assigned module tutor. They’ll help you by:

  • Marking your assignments (TMAs) and providing detailed feedback for you to improve.
  • Guiding you to additional learning resources.
  • Providing individual guidance, whether that’s for general study skills or specific module content.
  • Facilitating online discussions between you and your fellow students, in the dedicated module and tutor group forums.

Module tutors also run online tutorials throughout the module. Where possible, recordings of online tutorials will be made available to students. While these tutorials won’t be compulsory for you to complete the module, you’re strongly encouraged to take part.


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

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 (M208) starts once a year – in October.

This page describes the module that will start in October 2024.

We expect it to start for the last time in October 2027.

Course work includes:

7 Tutor-marked assignments (TMAs)

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