Studying at York
Introduction to Pure Mathematics - MAT00013C
- Department: Mathematics
- 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:
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work with foundational concepts and tools in Pure mathematics and use these when solving problems at an appropriate level;
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use various proof techniques;
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write clear mathematical statements and rigorous proofs as well as distinguish correct from incorrect or sloppy mathematical reasoning.
Module content
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Equivalence and order relations;
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Integer, rational and real numbers;
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Countable and uncountable sets;
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Divisibility, Euclid’s algorithm and linear equations in integers;
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Prime numbers and the Fundamental Theorem of Arithmetic;
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Congruence equations, Chinese Remainder Theorem;
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Modular arithmetic and residue classes;
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Groups, subgroups and cosets, cyclic subgroups;
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Fields, examples of finite and infinite fields;
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Limits, continuity and uniform continuity of functions;
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Intermediate Value Theorem, Inverse Function Theorem for Continuous Functions, Extreme Value Theorem;
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Convergence of sequences and series of numbers, Cauchy’s criterion, Bolzano–Weierstrass theorem.
Indicative assessment
| Task | % of module mark |
|---|---|
| Closed/in-person Exam (Centrally scheduled) | 80 |
| Essay/coursework | 20 |
Special assessment rules
None
Additional assessment information
The assessed coursework component mark will be calculated from a written coureswork and computer exercises, weighted 1:1 respectively.
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)
Indicative reassessment
| Task | % of module mark |
|---|---|
| Closed/in-person Exam (Centrally scheduled) | 80 |
| Essay/coursework | 20 |
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
TBA