Introductory description

This module provides students with a strong background in pure mathematics, particularly the theory of sets and functions, real and complex number systems, logic and proof, analysis of real-valued functions, and differential equations. This allows the students to develop a fluency with abstract mathematical reasoning, and gives a deeper understanding of techniques used in mathematical economics and econometrics.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

The module will typically cover the following topics:Set theory (notation, basic concepts), Real numbers (basic properties, interval notation), Complex numbers (basic definitions, Cartesian form, polar form, roots of unity, the Fundamental Theorem of Algebra), Functions (injectivity, surjectivity, composition), Counting (cardinality of finite and infinite sets, countability of the rational numbers, uncountability of the real numbers), Limits (basic definitions, the Sandwich Rule, boundedness), Continuity (basic definitions, the Intermediate Value Theorem, numerical methods for solving equations), Differentiation (basic definitions and properties, Rolle’s Theorem, the Mean Value Theorem), L’Hopital’s Rule (techniques and applications), Taylor’s Theorem (generalisation of the Mean Value Theorem, polynomial approximations to functions, convergence criteria), Integration (basic properties, the Newton-Leibniz definition, the Riemann definition, the Fundamental Theorem of Calculus, integration by parts, calculation of improper integrals), Differential equations (first-order separable equations, first- and second-order linear equations)

Learning outcomes

By the end of the module, students should be able to:

• Subject knowledge and understanding: … demonstrate an understanding of basic properties of real and complex numbers, functions, and finite and infinite sets. The teaching and learning methods that enable students to achieve this learning outcome are: Lectures, tutorials, problem sheets and independent study. The assessment methods that measure the achievement of this learning outcome are: Problem sheets and unseen examination.
• Subject knowledge and understanding: … demonstrate an understanding of basic topics in the analysis of real-valued functions, including limits, continuity, differentiation, Taylor-MacLaurin series, and integration. The teaching and learning methods that enable students to achieve this learning outcome are: Lectures, tutorials, problem sheets and independent study. The assessment methods that measure the achievement of this learning outcome are: Problem sheets and unseen examination.
• Key skills: …understand formal mathematical definitions and theorems, and apply them to prove statements about real-valued functions. The teaching and learning methods that enable students to achieve this learning outcome are: Lectures, tutorials, problem sheets and independent study. The assessment methods that measure the achievement of this learning outcome are: Problem sheets and unseen examination.

Indicative reading list

Please see Talis Aspire link for most up to date list.

View reading list on Talis Aspire

Subject specific skills

Students will have the opportunity to develop skills in:
Analytical thinking and communication
Analytical reasoning
Critical thinking
Problem-solving
Abstraction

Transferable skills

Students will have the opportunity to develop skills in:
Numeracy and quantitative skills
Written communication
Oral communication
Mathematical, statistical and data-based research skills