This module is based around six books, each developing a particular topic in pure mathematics. A feature of the module is that we identify a core route through each book. This is designed to help time-poor students identify the material necessary to understand later parts of the module. It is possible to follow the core route for some weeks and the standard route for others. Following the core route through the whole module should enable you to pass, but may not allow you to gain the higher grades.
Number Theory
In the first book, we study integers and prime numbers. In particular, we look at classical problems that require integer solutions, such as finding all integer solutions to the equation 2x + 11y = 5. We also develop methods for solving linear congruences such as ax2 ≡ b (mod n), and in the final chapter, we study the classical theorems of Fermat and Wilson.
Groups
In the second book we consolidate and build on the group theory presented at level 2 of our curriculum in Pure mathematics (M208). The book leads up to the classification of all finite abelian groups and ends with an introduction to the problem of classifying groups that are not abelian. On completion, you will be able to use the Sylow Theorems to analyse the structure of appropriate finite groups.
Numbers and Rings
In the first half of this book, we consider the solution of quadratic congruences, ax2 ≡ b (mod n). In the second half, we use our knowledge of integers to define and study the abstract algebraic structures known as rings.
Metric Spaces I
In this book, motivated by our understanding of how distance works for points in the plane, we define metrics, which can be used to give us an idea of distance between arbitrary objects (such as words or fractals). This allows us to generalise what it means for a function to be continuous.
Rings and Fields
In this book, we return to our study of algebra. We start by looking at polynomial rings and then continue our investigation of abstract algebraic structures. This unexpectedly resolves some famous problems of antiquity, such as ‘squaring the circle’ or ‘trisecting the angle’. The final chapter shows how algebraic ideas underlie the modern theory of cryptography.
Metric Spaces II
In this book, we return to metric spaces. We examine the implications of our new definition of distance for understanding what it means for something to be connected. The book culminates in an introduction to the theory of fractals.
The module website includes a non-assessed reader that provides an overview of the historical development of topological and metric spaces and modern algebra. Where appropriate, the reader includes information and/or links about modern applications and unsolved/recently solved problems.
The full content list is on the Open mathematics and statistics website.
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:
Each module book has an associated online quiz. You can attempt these quizzes as many times as you wish; they don't count towards your final grade.
Each TMA is associated with a particular module book and consists of a mixture of questions: some contribute to your final grade, and some are developmental. The feedback you receive will help you improve your knowledge and understanding of the study material and develop essential skills.
You’ll have access to a module website, which includes:
Additionally, the website includes:
We also provide physical:
You can study this module on its own or use the credits you gain towards an Open University qualification.
M303 is an option module in our:
Further pure mathematics (M303) starts once a year – in October.
It will next start in October 2026.
We expect it to start for the last time in October 2029.
As a student of The Open University, you should be aware of the content of the academic regulations, which are available on our Student Policies and Regulations website.
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There may be extra costs on top of the tuition fee, such as set books, a computer and internet access.
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