Principles of Pure Mathematics - 2025 entry

This module develops core mathematical skills essential for progression into a degree in mathematics or other quantitative disciplines. It lays the foundation of Algebra, Trigonometry, Calculus and complex numbers for more advanced mathematical studies by bringing you to a level of knowledge and competence equivalent to the pre-requisite for a first year mathematics at any quantitative degree programme. In this module you will get a grasp of Algebra, which is the study of symbolic representations and the rules for manipulating symbols such as the skills required in ‘backwards thinking’. You will develop competency to confidently manipulate algebraic expressions, to solve equations and inequalities, as well as to explore functional relationships. Calculus is another part of mathematics to cover in this module, which is concerned with the study of continuous changes, and has two branches, differential calculus (the study of measuring rates of change) and integral calculus (the study of accumulation of quantities), which are precisely linked by the Fundamental Theorem of Calculus. You will also learn about Trigonometry and complex numbers. Those skills are fundamental tools for the study of mathematics across the physical, engineering, life and environmental sciences. In this module you will also learn how to: use theories, definitions and properties; analyse mathematical statements; use logic and critical thinking to perform mathematics; present findings and communicate results in a coherent way.

 

On successful completion of this module you will be equipped with the skills to apply those mathematical concepts in different contexts, and you will have a sound understanding of fundamental mathematical techniques necessary to handle a diverse range of problems in mathematics, engineering and sciences.

Module Specific Skills and  Knowledge:

1.  Manipulate algebraic and numerical expressions accurately and with confidence

2.  Recognise and solve equations involving logarithmic, exponential, trigonometric and hyperbolic functions

3.  Sketch the graphs of a variety of functions of one variable

4.  Perform accurate calculus manipulations using a variety of standard techniques

5.  Understand, manipulate and analyse expressions involving complex numbers

 

Discipline Specific Skills and Knowledge:

6.  Manipulate basic mathematical objects necessary in order to progress to successful studies in mathematics, engineering and sciences

7.  Communicate mathematics effectively and clearly

8.  Demonstrate an ability to model a given problem mathematically, i.e. to find the mathematical formula which represents the problem

 

Personal and Key Transferable/ Employment Skills and Knowledge:

9.  Formulate and solve problems and communicate reasoning and solutions effectively in writing

10. Use learning resources appropriately

11. Communicate ideas and plans in a clear and concise way 

12. Exhibit self-management and time management skills

Algebra: exponent laws; algebraic expressions and operations, including polynomial long division remainder and factor theorems; binomial expansions; partial fraction decomposition.

 

Equations and inequalities: solving linear equations; solving quadratic equations by factorisation, computing the discriminant, or completing the square; solving linear and quadratic inequalities.

 

Functions: dependent and independent variables; domain, codomain and range; piecewise-defined functions; continuity; transformations; combining functions; inverse functions.

 

Elementary functions and graphs, including polynomial functions, exponential and logarithmic functions, and trigonometric and hyperbolic functions and identities including solving equations.

 

Complex numbers: cartesian form and the complex plane; operations on complex numbers; polar exponential forms; powers and roots of complex numbers; loci in the complex plane.

 

Differential calculus: definition of the derivative; standard derivatives; differentiation using the product, quotient, and chain rules; implicit differentiation; logarithmic differentiation.

 

Applications of differentiation: tangents and normals; increasing and decreasing functions; finding and classifying stationary points; Mean Value Theorem; Taylor series.

 

Integral calculus: the indefinite integral; Riemann sums and the definite integral; Fundamental Theorem of Calculus.

 

Integration methods: integration by substitution; integration using partial fractions; integration by parts; integration by Taylor series expansion

 

Applications of integration: areas under curves or between curves; volume of a solid of revolution; arc length; area of a surface of revolution