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.
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.
Introduction
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
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 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
Limits 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 must have passed the following module:
Or be able to provide evidence that you have the required mathematical skills.
You can check you’re ready for M208 and see the topics it covers here.
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:
Essential mathematics 2 (MST125) is ideal preparation.
You’ll get help and support from an assigned tutor throughout your module.
They’ll help by:
Online tutorials run throughout the module. While they’re not compulsory, we strongly encourage you to participate. Where possible, we’ll make recordings available.
Course work includes:
We’re using a new examination verification process for this module. We may ask you to attend a 15-minute post-exam video discussion, where you’ll present a photo ID and discuss your answers to a small number of questions with a tutor or member of the module team. The discussion isn’t graded; it’s only to verify that you completed the exam yourself.
You’ll have access to a module website, which includes:
We also provide printed:
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.
To find out more about what kind of support and adjustments might be available, contact us or visit our Disability support website.
Pure mathematics (M208) starts once a year – in October.
It will next start in October 2026.
We expect it to start for the last time in October 2027.
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