You will study core mathematics under the three headings of Analytical and Computational Foundations, Calculus, and Linear Mathematics; this includes an introduction to the computer package MATLAB. You also begin studying the three main subject areas within mathematics, namely pure mathematics, applied mathematics, and probability and statistics.
You will benefit from our Peer-Assisted Study Support (PASS) scheme, designed specifically to help you settle in. PASS Leaders, who are current maths students, will provide you with a friendly face at the start of your first year and then academic support during that year, through regular PASS sessions.
Analytical and Computational Foundations
The idea of proof is fundamental to all mathematics. We’ll look at mathematical reasoning using techniques from logic to deal with sets, functions, sequences and series.
This module links directly with your study in Calculus and Linear Mathematics. It provides you with the foundations for the broader area of Mathematical Analysis. This includes the rigorous study of the infinite and the infinitesimal.
You will also learn the basics of computer programming. This will give you the chance to use computational algorithms to explore many of the mathematical results you’ll encounter in your core modules.
Your study will include:
- propositional and predicate logic; set theory, countability
- proof: direct, indirect and induction
- sequences and infinite series (convergence and divergence)
- limits and continuity of functions
- programming in Python
Calculus
How do we define calculus? How is it used in the modern world?
The concept can be explained as the mathematics of continuous change. It allows us to analyse motion and change in time and space.
You will cover techniques for differentiating, integrating and solving differential equations. You’ll learn about the theorems which prove why calculus works. We will explore the theory and how it can be applied in the real world.
Your study will include:
- functions: limits, continuity and differentiability, rules of differentiation
- techniques for integration, fundamental theorem of calculus
- solution of linear and nonlinear differential equations
- multivariate calculus, Lagrange multipliers, stationary points
- multiple integrals, changes of variables, Jacobians
This module gives you the mathematical tools required for later modules which involve modelling with differential equations. These include:
- mathematical physics
- mathematical medicine and biology
- scientific computation
Linear Mathematics
Vectors, matrices and complex numbers are familiar topics from A level Mathematics and Further Mathematics. Their common feature is linearity. A linear mathematical operation is one which is compatible with addition and scaling.
As well as these topics you’ll study the concept of a vector space, which is fundamental to later study in abstract algebra. We will also investigate practical aspects, such as methods for solving linear systems of equations.
The module will give you the tools to analyse large systems of equations that arise in mathematical, statistical and computational models. For example, in areas such as:
- fluid and solid mechanics
- mathematical medicine and biology
- mathematical finance
Your study will include:
- complex numbers, vector algebra and geometry
- matrix algebra, inverses, determinants
- vector spaces, subspaces, bases
- linear systems of simultaneous equations, Gaussian elimination
- eigenvalues and eigenvectors, matrix diagonalisation
- linear transformations, inner product spaces
Applied Mathematics
How can the flight-path of a spacecraft to another planet be planned? How many fish can we catch without depleting the oceans? How long would it take a lake to recover after its pollution is stopped?
The real world is often too complicated to get exact information. Instead, mathematical models can help by providing estimates. In this module, you’ll learn how to construct and analyse differential equations which model real-life applications.
Your study will include:
- modelling with differential equations
- kinematics and dynamics of moving bodies
- Newton’s laws, balance of forces
- oscillating systems, springs, simple harmonic motion
- work, energy and motion
You'll be able to expand on these techniques later in your degree through topics such as:
- black holes, quantum theory
- fluid and solid mechanics
- mathematical medicine and biology
- mathematical finance
Probability
What is the importance of probability in the modern world?
It allows us to assess risk when calculating insurance premiums. It can help when making investment decisions. It can be used to estimate the impact that government policy will have on climate change or the spread of disease.
We will look at the theory and practice of discrete and continuous probability. Your study will include:
- sample spaces, events and counting problems
- conditional probability, independence, Bayes’ theorem
- random variables, expectation, variance
- discrete and continuous probability distributions
- multivariate random variables
- sums of random variables, central limit theorem
These topics will help you prepare for later modules in:
- probability methods
- stochastic models
- uncertainty quantification
- mathematical finance
Statistics
Statistics is concerned with methods for collecting, organising, summarising, presenting and analysing data. It enables us to draw valid conclusions and make reasonable decisions based on the analysis. It can be used to answer a diverse range of questions such as:
- Do the results of a clinical trial indicate that a new drug works?
- Is the HS2 rail project likely to be cost-effective?
- Should a company lend money to a customer with a given credit history?
In this module you’ll study statistical inference and learn how to analyse, interpret and report data. You’ll learn about the widely used statistical computer language R.
Your study will include:
- exploratory data analysis
- point estimators, confidence intervals
- hypothesis testing
- correlation, statistical inference
- linear regression, chi-squared tests
These first-year topics give you the foundations for later related modules in:
- statistical models and methods
- data analysis and modelling
- statistical machine learning
Foundations of Pure Mathematics
Pure mathematics at university is typically very different to the pure mathematics you've learnt at school or college. You'll use the language of sets, functions and relations to study some very abstract mathematical ideas.
In this module, we'll develop the skills of reading and writing the language of pure mathematics. You will learn techniques to build mathematical proofs in an abstract setting.
Your study will include:
- the language of set theory
- relations and functions
- rational and irrational numbers
- modular arithmetic
- prime factorisation
These topics will provide you with the basics you need for subsequent modules in algebra, number theory and group theory.