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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: 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

• None

### Prohibited combinations

• None

Co-requisite modules:
Foundations & Calculus

Post-requisite modules:
All Pure Mathematics modules

## Module will run

Occurrence Teaching period
A Semester 1 2023-24

## Module aims

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

In the Analysis part, students will learn the foundations of Real Analysis including 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.

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

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

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;

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

## Assessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Introduction to Pure Mathematics
N/A 90
Essay/coursework
Introduction to Pure Mathematics
N/A 10

### Special assessment rules

None

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)

### Reassessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Introduction to Pure Mathematics
N/A 10

## 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.