Course options
Key information
Duration: 4 years full time
UCAS code: G10F
Institution code: R72
Campus: Egham
The course
BSc Mathematics (with Integrated Foundation Year)
This course is available to Home (UK) students and students from the EU who meet the English Language requirements.
Our Integrated Foundation Year for science is a thorough, skills-building course that will give you everything you need to start your study of BSc Mathematics with confidence.
Engineering, Physical, Computational and Mathematical sciences underpin modern technological society and can help us provide answers to fundamental questions. Our Foundation Year sets you up so that you’re ready to take on those questions - providing you with opportunities to gain knowledge and understanding of how to get started in studying the sciences at university, including your chosen degree subject.
Learning from friendly, expert tutors, you’ll explore modules designed to provide familiarity with Mathematics and computation – the language of modern science and technology, and key for success in science, technology and engineering.
Once you have completed your Foundation year, you will normally progress onto the full degree course, BSc Mathematics. There may also be flexibility to move onto a degree in another department (see end of section, below).
Mathematics is in everything: we use it in every aspect of our lives – from managing household finances and investments to working at the cutting edge of digital communications, engineering, aviation, satellite navigation, medical science, weather forecasting, psychology, sociology, insurance, or the financial markets of the global economy. It is one of the oldest academic disciplines and yet it sits at the heart of our 21st century lives.
Our flagship BSc course brings the beauty and breadth of mathematics to life, inviting you to delve deep into the world of abstract structures and ideas, whilst also equipping you with the practical skills and experience that will set you apart in the world of work. Guided by experts in the field, you will receive a thorough grounding in the key methods and concepts that underpin our subject, with the flexibility to tailor your studies in years 2 and 3, thanks to the courses modular structure.
Our broad curriculum is influenced by the department’s world-class research activities. We are renowned for our work in pure mathematics, information security, statistics and theoretical physics, and our BSc course spans pure and applied mathematics, statistical analysis, financial mathematics, the mathematics of information, and more. We also offer an array of postgraduate opportunities, and provided you make good progress in year 1 you will have the option of transferring onto our four-year masters course to help take your mathematics to a higher level, particularly if you want to pursue a career in industry or research. It might also be possible to transfer sideways into the second year of one of our other undergraduate mathematics courses.
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 to guide you through your studies. We also offer CV writing workshops and a competitive work placement scheme. Our graduates are in great demand for their numeracy, analytical skills, data handling powers, logical thinking and creative problem-solving abilities.
On successful completion of your Foundation Year, you may be able to choose an alternative pathway which could include a degree from one of the other departments offering a Foundation Year within the School of Engineering, Physical and Mathematical Sciences. If you'd like to do this, you may take your Foundation Year Individual Project in one of these other departments. The degree course you choose to take after progression is likely to depend on the individual project you select during the foundation year. Please note however that you must take 'Foundation Skills (Mathematics)' and your individual project in the Department of Mathematics if you wish to join a full degree course in Mathematics.
Our flexible degree programmes enable you to apply to take a Placement Year, which can be spent studying abroad, working or carrying out voluntary work. You can even do all three if you want to (minimum of three months each)! To recognise the importance of this additional skills development and university experience, your Placement Year will be formally recognised on your degree certificate and will contribute to your overall result. Please note conditions may apply if your degree already includes an integrated year out, please contact the Careers & Employability Service for more information. Find out more
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.
Course structure
Core Modules
Foundation Year
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The aims of this module are to develop fundamental practical skills in computer programming and computational thinking using a hands-on approach. This is an entry-level module and is suitable for those with no prior programming experience. This module will contain foundational programming topics, including basic data structures, conditionals and loops, functions, and classes.
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This module aims to provide some key concepts in physical sciences that underpin all Physics and Engineering disciplines. The module will start with basic physics principles and will be gradually further developed in the Physical Sciences II module in the second term. Subjects explored on this module include Units and dimensions, Materials, Motion, Newton's laws, Connected particles, Static equilibrium, Fluids, Thermodynamics and Electricity.
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This module aims to provide the mathematics required for a foundation in engineering and physical science, enabling students to progress to subsequent calculus-based courses. It is designed to help students develop comfort and skill in using mathematical concepts and applying these concepts to problems in Engineering, Computer Science, Maths, and Physics. Additionally, the module highlights the application of mathematical techniques to topics in engineering and physics. The main mathematical topics and concepts covered in the module include algebra (simplification and rearrangement), sequences and series, number bases, logic, functions, graphing of functions, exponentials and logarithms, trigonometry, vectors/matrices, and complex numbers.
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This module aims to provide some key concepts in physical sciences that underpin all Physics and Engineering discipline. This module will cover the topics that are not covered in the Physical Sciences I in the first term. Topics covered include Electric Fields, Magnetic fields, Electromagnetic induction, Capacitors, Waves and optics, Circular motion, Simple harmonic motion, and Quantum physics.
A group project is a key element of assessment and provides an important focus on the development of learner autonomy. Students have to design and create a physical model that illustrates their chosen topic and present it to the examiners at the end of the term. The aim is to develop the ability to work as a team to plan and organise effectively to achieve the project goals.
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This module is built around global issues. Each topic will embody issues of global importance and be based on authentic and topical world events. The module is intended to deliver an interdisciplinary, academically authentic introduction to global issues, which satisfies a wide range of interests appealing to students wanting to progress onto a range of subjects such as maths, physics, computer science and electronic engineering. Therefore, readings and lectures will approach the topics from an engineering science perspective. Topics covered are likely to include Neural networks, Micro controllers\robotics, Computer architecture, Algorithms, Astrotechnology, The science of climate change and Industry 4.0 & Society 5.0.
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The aim of the module is to provide the calculus and statistics required for a foundation in engineering and physical science. The module will provide a foundation so that a student can apply calculus to real world problems. The module also aims to aid students in developing familiarity and skills in differentiation and integration. The main mathematical topics and concepts in the module are differentiation, integration, first-order ordinary differential equations, Probability, and Statistics.
Year 1
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This module aims to show how geometry has developed over the last 2500 years and use to use this material to motivate the study of various aspects of the subject.
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In this module, you will develop an understanding of how the techniques for solving differential equations can be applied to describe the real world. You will look at situations from balls flying through the air to planets orbiting the stars, including why the moon continues to orbit the Earth and not the Sun. You will consider the chatotic motion of a pendulum, and examine Einstein's theory of special relativity to describe the propagation of matter and light at high speeds.
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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.
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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.
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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.
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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.
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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.
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Real analysis studies the behaviour of the real numbers and real-valued functions. It provides the foundations of calculus, and is essential to both pure and applied mathematics. This module provides a user-friendly introduction to key ideas of real analysis, illustrated with copious examples. It examines topics such as the properties and axioms of the real numbers, and convergence of infinite sequences and infinite series.
Year 2
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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.
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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.
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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
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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
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All modules are core
Year 2
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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.
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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.
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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.
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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.
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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.
Year 3
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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.
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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.
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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.
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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.
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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.
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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.
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In this module you will develop an understanding of how error correcting codes are used to store and transmit information in technologies such as DVDs, telecommunication networks and digital television. You will look at the methods of elementary enumeration, linear algebra and finite fields, and consider the main coding theory problem. You will see how error correcting codes can be used to reconstruct the original information even if it has been altered or degraded.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
Teaching & assessment
In your Foundation Year, teaching methods include a mixture of lectures, practical classes and workshops, laboratory classes, individual tutorials, and supervisory sessions. Outside of the classroom you’ll undertake guided and independent practice. You will be assigned a Personal Tutor in the Department of Mathematics and will have regular scheduled sessions. In the Foundation Year, you’ll also be assigned a Personal Tutor in the Centre for the Development of Academic Skills (CeDAS). Assessments are varied; practical exercises, weekly problem sheets, set exercises, written examinations, laboratory reports, scientific poster preparation and presentation. In addition the Foundation Year offers a full range of skills-based training and also the opportunity to take a micro-placement to enhance your employability.
Once you progress onto your full degree course, it 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 compulsory core modules, you will be free to choose between a number of optional courses. Some contribute 15 credits to your overall award while others contribute the full 30.
We continue to 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 courses are mostly examined by written exams taken in the summer term, but some of our statistics and computational courses 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 course. In year 3 there are two optional courses which are examined solely by a project and presentation.
Entry requirements
A Levels: CCC
Required subjects:
- Mathematics
- At least five GCSEs at grade A*-C or 9-4 including English and Mathematics.
T-levels
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.
Other UK and Ireland Qualifications
EU requirements
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: 6.5 overall. Writing 7.0. No other subscore lower than 5.5.
- Pearson Test of English: 61 overall. Writing 69. No other subscore lower than 51.
- Trinity College London Integrated Skills in English (ISE): ISE III.
- Cambridge English: Advanced (CAE) grade C.
Your future career
With our internationally recognised Mathematics degree you will be in demand for your numerical and analytical skills, data handling powers, creative and logical thinking and problem solving abilities. We are 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. Our recent graduates have gone on to enjoy successful careers in: business management, IT consultancy, computer analysis and programming, accountancy, the civil service, teaching, actuarial science, finance, risk analysis, research and engineering. They work for organisations such as: KPMG, Ernst & Young, the Ministry of Defence, Barclays Bank, Lloyds Banking Group, the Department of Health, Logica, McLaren and Willis Towers Watson, and in research teams tackling problems as diverse as aircraft design, operational research and cryptography. Depending on your choice of courses, you could also be eligible for certain membership exemptions from the Institute of Actuaries and other professional bodies.
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.
- Our mathematicians are in great demand from employers for their valuable skills.
- Our strong ties with industry mean we understand the needs of employers.
- Take advantage of our summer work placement scheme and fine-tune your CV before you enter your final year.
- Benefit from a personal adviser who will guide you through your many options.
Fees, funding & scholarships
Home (UK) students tuition fee per year*: £9,250
Eligible EU students tuition fee per year**: £27,500
Foundation year 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 Integrated Foundation Year courses starting in September 2025 in the academic year 2025/26 will be £9,250 for that year. The fee for UK students in 2026/27 and beyond has not yet been set.
**This figure is the fee for EU 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.
