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Pure mathematics

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.

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OU qualifications are modular in structure; the credits from this undergraduate module could count towards a certificate of higher education, diploma of higher education, foundation degree or honours degree.

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Module

Module code
M208
Credits

Credits

  • Credits measure the student workload required for the successful completion of a module or qualification.
  • One credit represents about 10 hours of study over the duration of the course.
  • You are awarded credits after you have successfully completed a module.
  • For example, if you study a 60-credit module and successfully pass it, you will be awarded 60 credits.
60
Study level

Across the UK, there are two parallel frameworks for higher education qualifications, the Framework for Higher Education Qualifications in England, Northern Ireland and Wales (FHEQ) and the Scottish Credit and Qualifications Framework (SCQF). These define a hierarchy of levels and describe the achievement expected at each level. The information provided shows how OU module levels correspond to these frameworks.

OU SCQF FHEQ
2 9 5
Study method
Distance Learning
Module cost
See Module registration
Entry requirements
See Am I ready?

Student Reviews

This was a really exciting look into pure maths and a good introduction to group theory and vector spaces. The...
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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.

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

Vocational relevance

Successful study of this module will improve your skills in working with abstract concepts, constructing solutions to problems logically and communicating mathematical ideas clearly. In addition, you’ll learn how to critically analyse short pieces of mathematical text; identifying, explaining and correcting errors. These skills are highly valued by a wide range of employers.

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.

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 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.

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.

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.

    Course work includes:

    7 Tutor-marked assignments (TMAs)
    Examination
    No residential school

    Course satisfaction survey

    See the satisfaction survey results for this course.


    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).

    Register

    Start End England fee Register
    06 Oct 2018 Jun 2019 £2928.00

    Registration closes 13/09/18 (places subject to availability)

    Register
    This module is expected to start for the last time in October 2027.

    Additional Costs

    Study costs

    There may be extra costs on top of the tuition fee, such as a laptop, travel to tutorials, set books and internet access.

    If you're on a low income you might be eligible for help with some of these costs after your module has started.

    Ways to pay for this module

    Open University Student Budget Account

    The Open University Student Budget Accounts Ltd (OUSBA) offers a convenient 'pay as you go' option to pay your OU fees, which is a secure, quick and easy way to pay. Please note that The Open University works exclusively with OUSBA and is not able to offer you credit facilities from any other provider. All credit is subject to status and proof that you can afford the repayments.

    You pay the OU through OUSBA in one of the following ways:

    • Register now, pay later – OUSBA pays your module fee direct to the OU. You then repay OUSBA interest-free and in full just before your module starts. 0% APR representative. This option could give you the extra time you may need to secure the funding to repay OUSBA.
    • Pay by instalments – OUSBA calculates your monthly fee and number of instalments based on the cost of the module you are studying. APR 5.1% representative.

    Joint loan applications

    If you feel you would be unable to obtain an OUSBA loan on your own due to credit history or affordability issues, OUSBA offers the option to apply for a joint loan application with a third party. For example, your husband, wife, partner, parent, sibling or friend. In such cases, OUSBA will be required to carry out additional affordability checks separately and/or collectively for both joint applicants who will be jointly and severally liable for loan repayments.

    As additional affordability checks are required when processing joint loan applications, unfortunately, an instant decision cannot be given. On average the processing time for a joint loan application is five working days from receipt of the required documentation.

    Read more about Open University Student Budget Accounts (OUSBA).  

    Employer sponsorship

    Studying with The Open University can boost your employability. OU courses are recognised and respected by employers for their excellence and the commitment they take to complete. They also value the skills that students learn and can apply in the workplace.

    More than one in ten OU students are sponsored by their employer, and over 30,000 employers have used the OU to develop staff so far. If the module you’ve chosen is geared towards your job or developing your career, you could approach your employer to see if they will sponsor you by paying some or all of the fees. 

    • Your employer just needs to complete a simple form to confirm how much they will be paying and we will invoice them.
    • You won’t need to get your employer to complete the form until after you’ve chosen your module.  

    Credit/debit card

    You can pay part or all of your tuition fees upfront with a debit or credit card when you register for each module. 

    We accept American Express, Maestro (UK only), Mastercard, Visa/Delta and Visa Electron. 

    Mixed payments

    We know that sometimes you may want to combine payment options. For example, you may wish to pay part of your tuition fee with a debit card and pay the remainder in instalments through an Open University Student Budget Account (OUSBA).


    For more information about combining payment options, speak to an adviser or book a call back at a time convenient to you.


    Please note: your permanent address/domicile will affect your fee status and therefore the fees you are charged and any financial support available to you. The fees and funding information provided here is valid for modules starting before 31 July 2019. Fees normally increase annually in line with inflation and the University's strategic approach to fees. 

    This information was provided on 15/07/2018.

    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. 

    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.

    To find out more about what kind of support and adjustments might be available, contact us or visit our Overcoming barriers to study if you have a disability or health condition website.