Undergraduate

Pure mathematics

On this page

• What you will study
• Entry
• If you have a disability or additional requirement
• Study materials
• Teaching and assessment
• Professional recognition
• Students also studied
• Future availability

Pure mathematics is one of the oldest creative human activities and this course 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. You’ll need a sound knowledge of mathematics as developed in Using mathematics (MST121) to study this course.

Course facts

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This course is available for study in the countries shown. Fees and financial support may vary by country.

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 course is suitable whether you want a basic understanding of mathematics without taking the subject further, or to prepare for higher-level courses 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. You will need a scientific calculator but will not need it in the examination. You do not need a computer, though there are many opportunities to use one to reinforce your understanding of new topics if you so wish. 

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 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 (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 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 course should improve your skills in working with abstract concepts, constructing solutions to problems logically and communicating mathematical ideas clearly.

Entry

This is a Level 2 course and you need a good knowledge of the subject area, obtained either from Level 1 study with the OU or from equivalent work at another university. 

You should have a good basic knowledge of elementary algebra, coordinate geometry, Euclidean geometry, trigonometry, functions, differentiation and integration. It would also be helpful to have met vectors, matrices and groups, though these are not essential. The ideal preparation would be good passes in Using mathematics (MST121) and Exploring mathematics (MS221). Students are more likely to complete this course successfully if they have acquired their prerequisite knowledge through passing these courses. Our diagnostic quiz Am I ready to start M208? will help you to determine whether you are adequately prepared for this course.

If you have any doubt about the suitability of the course, please contact our Student Registration & Enquiry Service. 

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).

If you have a disability or additional requirement

Please be aware that the course 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 audio-visual material. The books are available in a comb-bound format. Our Services for disabled students website has the latest information about availability.

If you are a new student, or new to courses using a computer or the internet, you will need to inform us of your particular needs as soon as possible, as some of our support services may take several weeks to arrange. Details of how to do this and our range of support services are described in our publication Meeting Your Needs.

You can also find information about accessible study materials, financial support and the Disabled Students' Allowance, equipment and other services, on our Services for disabled students website. It also includes our contact details for advice and support both before you register and while you are studying.

Study materials

What's included

Course books, DVDs, CDs, website.

You will need

DVD and CD players.

Computing requirements

This course includes online computer activities – you can access these using a web browser that can play Flash and Shockwave.

You will need internet access and a computer. If you have purchased a new computer since 2005 it should meet your course computing requirements. Check our Technical Requirements section if your computer is older than this or is otherwise unusual.

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 course.  Contact our Student Registration & Enquiry Service if you want to know more about study with The Open University before you register.

Assessment

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

Please note that TMAs for all undergraduate mathematics and statistics courses must be submitted on paper as – due to technical reasons – we are unable to accept TMAs via our eTMA system.

Professional recognition

This course may help you to gain recognition from a professional body. You can view or download our Recognition leaflets 3.3 Professional Engineering Institutions, Recognition leaflet 3.6 Institute of Mathematics and its Applications and Recognition leaflet 3.11 Other Professional Bodies for further information.

Students also studied

Students who studied this course also studied at some time:

• Exploring mathematics (MS221)
• Using mathematics (MST121)
• Mathematical methods and models (MST209)
• Graphs, networks and design (MT365)
• Analysing data (M248)

Future availability

The details given here are for the course that starts in October 2012. We expect it to be available once a year.

Fees 2012

See fees and funding options for study from September 2012.

Study explained

Financial support

- find out if you qualify for support with your fees with our eligibility checker.

Study explained

- all you need to know about distance learning with the OU.

A taste of OU study

Student Reviews

A well-designed course covering many aspects of Pure Maths. The module texts are well-written with many examples. The accompanying audio-visual ...
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A nice module covering Analysis, Group Theory and an introduction to Linear Algebra. The course notes are very well written ...
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