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.
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.
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.
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.
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.
This module may help you to gain membership of the Institute of Mathematics and its Applications (IMA). For further information, see the IMA website.
Module books, DVDs, CDs, website.
You will need
DVD and CD players.
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.