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Introduction to Pure Mathematics - MAT00013C

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  • Department: Mathematics
  • Module co-ordinator: Dr. Emilie Dufresne
  • Credit value: 20 credits
  • Credit level: C
  • Academic year of delivery: 2024-25
    • See module specification for other years: 2023-24

Module summary

This module introduces students to a selection of topics in analysis, algebra and number theory by developing a range of concepts, tools and techniques in these areas.

Related modules

Co-requisite modules:
Foundations & Calculus

Post-requisite modules:
All Pure Mathematics modules



Module will run

Occurrence Teaching period
A Semester 1 2024-25

Module aims

In this module students explore topics from three core areas of Pure Mathematics: Number Theory, Algebra and Analysis.

In the Number Theory part, students will develop a coherent theory of the integers, including their axiomatic construction, congruences and prime numbers.

In the Algebra part, students will explore the basics of group theory - a fundamental area of modern mathematics.

In the Analysis part, students will learn the foundations of Real Analysis including a discussion of cardinality, the construction and properties of real numbers and a variety of fundamental tools and properties including the formal treatment of the limits of sequences and functions and the continuity of functions.

The module also performs a cultural transition to the rigorous development of mathematics which is vital at University level, and especially characteristic of Pure Mathematics

Module learning outcomes

By the end of this module, students will be able to:

  1. work with foundational concepts and tools in Pure mathematics and use these when solving problems at an appropriate level;

  2. use various proof techniques;

  3. write clear mathematical statements and rigorous proofs as well as distinguish correct from incorrect or sloppy mathematical reasoning.

Module content

  • Equivalence and order relations;

  • Integer, rational and real numbers;

  • Countable and uncountable sets;

  • Divisibility, Euclid’s algorithm and linear equations in integers;

  • Prime numbers and the Fundamental Theorem of Arithmetic;

  • Congruence equations, Chinese Remainder Theorem;

  • Modular arithmetic and residue classes;

  • Groups, subgroups and cosets, cyclic subgroups;

  • Fields, examples of finite and infinite fields;

  • Limits, continuity and uniform continuity of functions;

  • Intermediate Value Theorem, Inverse Function Theorem for Continuous Functions, Extreme Value Theorem;

  • Convergence of sequences and series of numbers, Cauchy’s criterion, Bolzano–Weierstrass theorem.


Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Closed exam : Introduction to Pure Mathematics
2 hours 80
Computation Exercises : Introduction to Pure Mathematics Computational Exercises
N/A 10
Coursework : Introduction to Pure Mathematics Coursework
N/A 10

Special assessment rules


Additional assessment information

Due to the pedagogical desire to provide speedy feedback in seminars, extensions to the written coursework are not possible.

To mitigate for exceptional circumstances, the written coursework grade will be the best 4 out of the 5 assignments. If more than one assignment is affected by exceptional circumstances, an ECA claim must be submitted (with evidence)


Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Closed exam : Introduction to Pure Mathematics Exam
2 hours 100

Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

Indicative reading


The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.