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Mathematical Studies

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  1. Royal Holloway's institution code: R72
  2. Make a note of the UCAS code for the course you want to apply for:

    • Mathematical Studies BSc - G150
  3. Click on the link below to apply via the UCAS website:
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Mathematical Studies


Key information

Duration: 3 years full time

UCAS code: G150

Institution code: R72

Campus: Egham

The course

Mathematical Studies (BSc)

“The greatest mathematicians, as Archimedes, Newton and Gauss, always united theory and applications in equal measure.” – Felix Klein

Mathematics is intrinsically beautiful and satisfying to study but perhaps the true skills lie in applying it to the challenges and intricacies of the world we all live in. Our Mathematical Studies course is firmly rooted in the key concepts and techniques of mathematics, from the abstract to the practical, but it also allows you to pursue studies in related fields and to build a skill set that is unique to you and your personal interests. In years 2 and 3 you may be able to choose up to a quarter of your courses from other departments, such as Computer Science, Philosophy, Sociology, Physics or Management.

So many aspects of our daily lives rely on the skills of mathematicians – from computing and digital communications to engineering, aviation and satellite navigation, the physical sciences, economics, management, and the social sciences. Your solid grounding in mathematics will help you in any academic discipline or career that you choose to pursue. Our varied and flexible curriculum is influenced by our world-class research activities. We are internationally renowned for our work in pure mathematics, information security, statistics and theoretical physics. Provided you make good progress in year 1 you will have the option of transferring onto our four-year Mathematics MSci course (G103) or to transfer onto the second or third year of one of our other undergraduate mathematics courses, such as the more focused Mathematics BSc (G100).

Join our friendly and inspiring department and you will benefit from a thoroughly supportive learning environment. We offer small group tutorials, problem-solving sessions, practical workshops and IT classes, as well as generous staff office hours and a dedicated personal adviser who will help you with any queries or difficulties and guide you with your choice of courses and career. We also offer CV writing workshops and a competitive work placement scheme. Graduates from our department are in great demand for their numeracy, analytical skills, data handling powers, creative and logical thinking and problem-solving abilities.

  • Enjoy the chance to pursue academic interests in related fields, whilst keeping mathematics at the heart of your degree.
  • Learn from inspirational teachers.
  • Feel at home in a lively, friendly department with a strong focus on small group teaching, where you will be known as an individual.

From time to time, we make changes to our courses to improve the student and learning experience. If we make a significant change to your chosen course, we’ll let you know as soon as possible.

Core Modules

Year 1
  • In this module you will develop an understanding of the notion of probability and the basic theory and methods of statistics. You will look at random variables and their distributions, calculate probabilities of events that arise from standard distributions, estimate means and variances, and carry out t tests for means and differences of means. You will also consider the notions of types of error, power and significance levels, gaining experience in sorting a variety of data sets in a scientific way.

  • This module aims to develop the students’ confidence and skill in dealing with mathematical expressions, to extend their understanding of calculus, and to introduce some topics which they may not have met at A-level. It also aims to ease the transition to university work and to encourage the student to develop good study skills.

  • In this module you will develop an understanding of the calculus functions of more than one variable and how it may be used in areas such as geometry and optimisation. You learn how to manipulate partial derivatives, construct and manipulate line integrals, represent curves and surfaces in higher dimensions, calculate areas under a curve and volumes between surfaces, and evaluate double integrals, including the use of change of order of integration and change of coordinates.

  • In this module you will develop an understanding of the fundamental algebraic structures, including familiar integers and polynomial rings. You will learn how to apply Euclid's algorithm to find the greatest common divisor of two integers, and use mathematical induction to prove simple results. You will examine the use of arithmetic operations on complex numbers, extract roots of complex numbers, prove De Morgan's laws, and determine whether a given mapping is bijective.

  • In this module you will develop an understanding of basic linear algebra, in particular the use of matrices and vectors. You will look at the basic theoretical and computational techniques of matrix theory, examining the power of vector methods and how they may be used to describe three-dimensional space. You will consider the notions of field, vector space and subspace, and learn how to calculate the determinant of an n x n matrix.

Year 2
  • In this module you will develop an understanding of the basic principles of the mathematical theory of probability. You will use the fundamental laws of probability to solve a range of problems, and prove simple theorems involving discrete and continuous random variables. You will learn how to forumulate an explain fundamental limit theorems, such as the weak law of large numbers and the central limit theorem.

  • This course introduces you to the basics of Python programming for building solutions to mathematics-based tasks. This will encourage a deeper understanding of the mathematics that you will learn across your degree, developing general mathematical skills and group working. Additionally, the module will provide guidance through the process of applying for a summer internship or job, as well as reviewing the range of career options available to Mathematicians upon graduation.

  • In this module you will develop an understanding of vectors and matrices within the context of vector spaces, with a focus on deriving and using various decompositions of matrices, including eigenvalue decompositions and the so-called normal forms. You will learn how these abstract notions can be used to solve problems encountered in other fields of science and mathematics, such as optimisation theory.

Year 3
  • The aim of this module is to enable you to plan and carry out longer assignments, which require significant preparation and/or advanced data analysis. This module will lay the groundwork for the comprehensive project work undertaken in your final year project whilst introducing you to some of the research activities within the Department. You will establish skills including report writing and oral presentations in the context of employment.

Optional Modules

There are a number of optional course modules available during your degree studies. The following is a selection of optional course modules that are likely to be available. Please note that although the College will keep changes to a minimum, new modules may be offered or existing modules may be withdrawn, for example, in response to a change in staff. Applicants will be informed if any significant changes need to be made.

Year 1
  • All modules are core
Year 2
  • In this module the vector calculus methods are applied to a variety of problems in the physical sciences, with a focus on electromagnetism and optics. On completion of the module, the student should be able to calculate relevant physical quantities such as field strengths, forces, and energy distributions in static as well as dynamical electromagnetic systems and be able to treat mathematically the interactions between moving electrical charges, magnets and optical fields.

  • In this module you will develop an understanding of statistical modelling, becoming familiar with the theory and the application of linear models. You will learn how to use the classic simple linear regression model and its generalisations for modelling dependence between variables. You will examine how to apply non-parametric methods, such as the Wilxocon and Kolmogorov-Smirnov goodness-of-fit tests, and learn to use the R open source software package.

  • In this module you will develop an understanding of the basic principles of the mathematical theory of probability. You will use the fundamental laws of probability to solve a range of problems, and prove simple theorems involving discrete and continuous random variables. You will learn how to forumulate an explain fundamental limit theorems, such as the weak law of large numbers and the central limit theorem.

  • In this module you will develop an understanding of the concepts arising when the boundary conditions of a differential equation involves two points. You will look at eigenvalues and eigenfunctions in trigonometric differential equations, and determine the Fourier series for a periodic function. You will learn how to manipulate the Dirac delta-function and apply the Fourier transform. You will also examine how to solve differential equations where the coefficients are variable.

  • In this module you will develop an understanding of ring theory and how this area of algebra can be used to address the problem of factorising integers into primes. You will look at how these ideas can be extended to develop notions of 'prime factorisation' for other mathematical objects, such as polynomials. You will investigate the structure of explicit rings and learn how to recognise and construct ring homomorphisms and quotients. You will examine the Gaussian integers as an example of a Euclidean ring, Kronecker's theorem on field extensions, and the Chinese Remainder Theorem.

  • In this module you will develop an understanding of the convergence of series. You will look at the Weierstrass definition of a limit and use standard tests to investigate the convergence of commonly occuring series. You will consider the power series of standard functions, and analyse the Intermediate Value and Mean Value Theorems. You will also examine the properties of the Riemann integral.

Year 3
  • You will carry out a detailed investigation on a topic of your choosing, guided by an academic supervisor. You will prepare a written report around 7,000 words in length, and give a ten-minute presentation outlining your findings.

  • In this module you will develop an understanding of a range of methods for teaching children up to A-level standard. You will act act as a role model for pupils, devising appropriate ways to convey the principles and concepts of mathematics. You will spend one session a week in a local school, taking responsibility for preparing lesson plans, putting together relevant learning aids, and delivering some of the classes. You will work with a specific teacher, who will act as a trainer and mentor, gaining valuable transferable skills.

  • In this module you will develop an understanding of how prime numbers are the building blocks of the integers 0, ±1, ±2, … You will look at how simple equations using integers can be solved, and examine whether a number like 2017 should be written as a sum of two integer squares. You will also see how Number Theory can be used in other areas such as Cryptography, Computer Science and Field Theory.

  • In this module you will develop an understanding of quantum theory, and the development of the field to explain the behaviour of particles at the atomic level. You will look at the mathematical foundations of the theory, including the Schrodinger equation. You will examine how the theory is applied to one and three dimensional systems, including the hydrogen atom, and see how a probabilistic theory is required to interpret what is measured.

  • In this module you will develop an understanding of non-linear dynamical systems. You will investigate whether the behaviour of a non-linear system can be predicted from the corresponding linear system and see how dynamical systems can be used to analyse mechanisms such as the spread of disease, the stability of the universe, and the evolution of economic systems. You will gain an insight into the 'secrets' of the non-linear world and the appearance of chaos.

  • In this module you will develop an understanding of some of the techniques and concepts of combinatorics, including methods of counting, generating functions, probabilistic methods, permutations, and Ramsey theory. You will see how algebra and probability can be used to count abstract mathematical objects, and how to calculate sets by inclusion and exclusion. You will examine the applications of number theory and consider the use of simple probabilistic tools for solving combinatorial problems.

  • In this module you will develop an understanding of the algebraic structures known as groups. You will look at how groups represent symmetries in the world around us, examining examples that arise from the theory of matrices and permutations. You will see how groups are ubiquitous and used in many different fields of human study, including mathematics, physics, the study of crystals and atoms, public key cryptography, and music theory. You also will also consider how various counting problems concerning discrete patterns can be solved by means of group actions.

  • In this module you will develop an understanding of geometric objects and their properties. You will look at objects that are preserved under continuous deformation, such as through stretching or twisting, and will examine knots and surfaces. You will see how colouring a knot can be used to determine whether or not it can be transformed into the unknot without any threading. You will also consider why topologists do not distinguish between a cup and a donut.

  • In this module you will develop an understanding of the algebraic structures known as groups. You will look at how groups represent symmetries in the world around us, examining examples that arise from the theory of matrices and permutations. Groups are ubiquitous and used in many different fields of human study, including mathematics, physics, the study of crystals and atoms, public key cryptography, and music theory.

  • In this module you will develop an understanding of the probabilistic methods used to model systems with uncertain behaviour. You will look at the structure and concepts of discrete and continuous time Markov chains with countable stable space, and consider the methods of conditional expectation. You will learn how to generate functions, and construct a probability model for a variety of problems.

  • Optimisation problems usually seek to maximise or minimise some function subject to a number of constraints; for example, how products should be shipped from factories to shops so as to minimise cost, or finding the shortest route a delivery driver should take. Linear programming is concerned with solving optimisation problems whose requirements are represented by linear relationships. This module introduces different types of optimisation problems and the appropriate approaches for solving each type of problem.

  • In this module you will develop an understanding of the main priciples and methods of statstics, in particular the theory of parametric estimation and hypotheses testing.You will learn how to formulate statistical problems in mathematical terms, looking at concepts such as Bayes estimators, the Neyman-Pearson framework, likelihood ratio tests, and decision theory.

  • This module shows how mathematics is used by those who work in securities markets. The topics in the module are chosen to demonstrate important applications, and may include the basic asset pricing models, the general behaviour of markets, the theory of derivative securities and elements of portfolio theory.

  • This module continues the study of financial mathematics begun in Financial Mathematics I. In this module, students will develop a further understanding of the role of mathematics in securities markets. The topics in the module may include models of interest rates, elements of portfolio theory and models for financial returns.

  • This module will analyse ways in which mathematics can be applied to situations that involve making strategic decisions, and will apply game theory to a wide variety of situations including to examples from business and economics.

  • In this module you will develop an understanding of how the behaviour of quantum systems can be harnessed to perform information processing tasks that are otherwise difficult, or impossible, to carry out. You will look at basic phenomena such as quantum entanglement and the no-cloning principle, seeing how these can be used to perform, for example, quantum key distribution. You will also examine a number of basic quantum computing algorithms, observing how they outperform their classical counterparts when run on a quantum computer.

The course has a flexible, modular structure and you will take a total of 12 course units at a rate of four, 30-credit modules per year. In addition to our core modules you will be free to choose from a number of optional module. Some contribute 15 credits to your overall award while others contribute 30.

We use a variety of teaching methods and there is a strong focus on small group teaching in the department. You will attend 12 to 15 hours of formal teaching in a typical week, including lectures, tutorials, problem solving workshops and practical sessions in statistics and computational mathematics. You will also be expected to work on worksheets, revision and project work outside of these times. In year 2, teaching will mainly be delivered through lectures and workshops and in year 3, mostly through relatively small group lectures.

Our modules are mostly examined by written exams taken in the summer term, but some of our statistics and computational modules also have project components, and between 10% and 30% of your final mark for each core module in year 1 will come from coursework. Some of the first year modules also include tests that contribute 10% of the final mark. In the first term of year 2 you will work in small groups to prepare a report and an oral presentation on a mathematical topic of your choice, which will contribute to one of your core module grades. CV writing skills are also embedded into that module. In year 3 there are two optional modules which are examined solely by a project and presentation.

A Levels: ABB-ABC

Required subjects:

  • A-level Mathematics at grade A
  • At least five GCSEs at grade A*-C or 9-4 including English and Mathematics

Where an applicant is taking the EPQ alongside A-levels, the EPQ will be taken into consideration and result in lower A-level grades being required. For students who are from backgrounds or personal circumstances that mean they are generally less likely to go to university, you may be eligible for an alternative lower offer. Follow the link to learn more about our contextual offers.


We accept T-levels for admission to our undergraduate courses, with the following grades regarded as equivalent to our standard A-level requirements:

  • AAA* – Distinction (A* on the core and distinction in the occupational specialism)
  • AAA – Distinction
  • BBB – Merit
  • CCC – Pass (C or above on the core)
  • DDD – Pass (D or E on the core)

Where a course specifies subject-specific requirements at A-level, T-level applicants are likely to be asked to offer this A-level alongside their T-level studies.

English language requirements

All teaching at Royal Holloway (apart from some language courses) is in English. You will therefore need to have good enough written and spoken English to cope with your studies right from the start.

The scores we require
  • IELTS: 6.0 overall. No subscore lower than 5.5.
  • Pearson Test of English: 61 overall. No subscore lower than 51.
  • Trinity College London Integrated Skills in English (ISE): ISE III.
  • Cambridge English: Advanced (CAE) grade C.

Country-specific requirements

For more information about country-specific entry requirements for your country please visit here.

Undergraduate preparation programme

For international students who do not meet the direct entry requirements, for this undergraduate degree, the Royal Holloway International Study Centre offers an International Foundation Year programme designed to develop your academic and English language skills.

Upon successful completion, you can progress to this degree at Royal Holloway, University of London.

Graduates from this course are in high demand for their broad and deep understanding of mathematical methods and concepts, their ability to handle complex data sets, approach problems creatively and logically and undertake specialist research. We have a strong track record of producing high achievers who go on to enjoy rewarding and high profile careers. Our department is part of the School of Engineering, Physical and Mathematical  Sciences and we enjoy strong ties with both the information security sector and industry at large. Graduates from our department have successfully secured positions in business management, IT consultancy, computer analysis and programming, accountancy, the civil service, teaching, actuarial science, finance, risk analysis, research and engineering. We have graduates working for organisations such as KPMG, Ernst & Young, the Ministry of Defence, Barclays Bank, Lloyds Banking Group, the Department of Health, Logica, McLaren and TowersWatson, and in research teams tackling problems as diverse as aircraft design, operational research and cryptography. 

We offer a competitive work experience scheme at the end of year 2, with short-term placements available during the summer holidays. You will also attend a CV writing workshop as part of your core modules in year 2, and your personal adviser and the campus Careers team will be on hand to offer advice and guidance on your chosen career. The University of London Careers Advisory Service also offers tailored sessions for Mathematics students, on finding summer internships or holiday jobs and securing employment after graduation.

  • Our strong ties with industry mean we understand the needs of employers. 
  • Take advantage of our summer work placement scheme and learn to fine-tune your CV. 
  • Benefit from a personal adviser who will guide you through your studies

Home (UK) students tuition fee per year*: £9,250

EU and international students tuition fee per year**: £27,500

Other essential costs***: There are no single associated costs greater than £50 per item on this course

How do I pay for it? Find out more about funding options, including loans, scholarships and bursaries. UK students who have already taken out a tuition fee loan for undergraduate study should check their eligibility for additional funding directly with the relevant awards body.

*The tuition fee for UK undergraduates is controlled by Government regulations. The fee for the academic year 2024/25 is £9,250 and is provided here as a guide. The fee for UK undergraduates starting in 2025/26 has not yet been set, but will be advertised here once confirmed. 

**This figure is the fee for EU and international students starting a degree in the academic year 2025/26.  

Royal Holloway reserves the right to increase tuition fees annually for overseas fee-paying students. The increase for continuing students who start their degree in 2025/26 will be 5%.  For further information see fees and funding and the terms and conditions.

*** These estimated costs relate to studying this particular degree at Royal Holloway during the 2025/26 academic year and are included as a guide. Costs, such as accommodation, food, books and other learning materials and printing, have not been included. 


Institute of Mathematics and its Applications

This course is accredited by the Institute of Mathematics and its Applications (IMA). On successful completion of the programme, you will meet, in part, the educational requirements for Chartered Mathematician status.

Top 30

UK Mathematics Department

Source: The Guardian University Guide, 2024

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