It’s perhaps strange that for a subject that relies so strongly on mathematical proof, there is no right or wrong answer when it comes to answering the question, “what is mathematics?” Aristotle defined mathematics as “the science of quantity”, while Isidore Auguste Comte preferred calling it “the science of indirect measurement” and Benjamin Peirce “the science that draws necessary conclusions”.
The answer changes depending on the philosophical stance of the definer, and on the branch of mathematics s/he wishes to focus on. And, as new branches of mathematics are discovered and developed, the definition also continues to develop, adapt and change accordingly.
This is all, of course, part of the appeal of mathematics degrees. Follow our guide to find out more about the world’s top universities for mathematics, high-level mathematics topics and potential careers with a mathematics degree.
Common skills gained from a mathematics degree include:
Relying upon math experts and enthusiasts to define the subject is likely to lead to a variety of conflicting and wide-ranging answers. Let’s be safe, then, and call upon dictionary solutions to this question. Most non-specialist dictionaries define mathematics by summarizing the main mathematics topics and methods.
The Oxford English Dictionary states that mathematics is an “abstract science which investigates deductively the conclusions implicit in the elementary concepts of spatial and numerical relations, and which includes as its main divisions geometry, arithmetic, and algebra”. The American Heritage Dictionary sums up the subject as the “study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols”.
If you’re studying mathematics at undergraduate level, you’ll probably undertake a Bachelor of Science (BSc) or Bachelor of Arts (BA) in Mathematics. A few institutions in Australia, Canada, India, Russia, the US and the Philippines also award the Bachelor of Mathematics (BMath) degree – but the difference is usually only in the name. Note that the undergraduate mathematics degree at the University of Cambridge is referred to as the ‘Mathematical Tripos’.
Most undergraduate mathematics degrees take three or four years to complete with full-time study, with both China and Australia offering the fourth year as an “honors” year. Some institutions offer a Masters in Mathematics (MMath) as a first degree, which allows students to enroll to study mathematics to a more advanced level straight after completing secondary education. Some institutions arrange placement years for students to work in industry, providing opportunities to apply mathematics skills and knowledge in a real-world setting.
Mathematics is typically taught through a combination of lectures and seminars, with students spending a lot of time working independently to solve problems sets. Assessments vary depending on the institution; you may be assessed based on examinations, practical coursework or a combination of both.
A typical mathematics degree program involves a combination of pure (theory and abstract) mathematics and applied (practical application to the world) mathematics. Some institutions also offer pure and applied mathematics as separate degrees, so you can choose to focus on just one. Mathematics is also often offered as a joint-honors degree, paired with subjects including business management, computer science, economics, finance, history, music, philosophy, physics, sports science and statistics.
Entry requirements for mathematics degrees usually only emphasize an academic background in mathematics. Applicants may be required to have studied some or all of the following: further mathematics, pure mathematics, mechanics and complex numbers. Experience of studying other scientific subjects may also be welcomed, and can help provide an additional dimension to your studies.
Some universities in the UK (such as Cambridge and Warwick) require students to take the Sixth Term Examination Papers (known as the STEP exam) or the Advanced Extension Award (AEA). You may also need to prove your proficiency in the language you will study in, by taking an approved language proficiency test, and some institutions provide pre-sessional language courses. Other preparatory courses are also available, including the option of taking a foundation mathematics program if your mathematics is below the level required for undergraduate study.
If you have exceptional pre-university grades, some institutions allow exemption from the first year’s study so you can enter directly into the second year, or enroll in an ‘advanced entry’ program – both options will allow you to complete your undergraduate mathematics degree in one year less than usual.
You can expect to study a range of introductory courses in your first year, covering key mathematics topics such as abstract algebra, calculus, complex numbers, differential equations, geometry, number theory, probability and statistics. You’ll then move on to more advanced study, and will need to choose from a range of elective courses. Popular mathematics topics include:
Complex analysis involves investigating the functions of complex numbers – numbers which can be expressed in a form which allows for the combination of real and imaginary numbers. Complex analysis is useful in many branches of mathematics, including algebraic geometry, number theory and applied mathematics, so it is an essential starting point for the further study of mathematics. You’ll learn about the analytic functions of complex variables, complex functions and differentiation of complex functions, how complex variables can be applied to the real world and cover the many theorems surrounding complex functions such as Cauchy’s theorem, Morera’s theorem, Rouché’s theorem, Cauchy-Riemann equations and the Riemann sphere to name a few.
Discrete mathematics involves mathematical structures that are fundamentally discrete (with finite, distinct, separate values) rather than continuous. This includes topics such as integers, graphs, trees, sets, chromatic numbers, recurrence relations and mathematical logic. Discrete mathematics usually involves examining the interrelations between probability and combinatorics. You’ll also learn about the complexity of algorithms, how to use algorithmic thinking in problem solving, algorithmic applications of random processes, asymptotic analysis, finite calculus and partitions. You’ll learn how discrete mathematics is applied to other topics within mathematics, and you’ll also look into broader academic fields such as computer science.
Mechanics is concerned with the study of forces that act on bodies and any resultant motion that they experience. Advanced study of mechanics involves quantum mechanics and relativity, covering topics such as electromagnetism, the Schrödinger equation, the Dirac equation and its transformation properties, the Klein-Gordon equation, pair production, Gamma matrix algebra, equivalence transformations and negative energy states. You’ll also look at how relativistic quantum mechanics can be used to explain physical phenomena such as spin, the gyromagnetic ratios of the electron and the fine structures of the hydrogen atom. You could also study statistical mechanics, which covers topics such as inference, multivariate complex systems, state variables, fluctuations, equilibrium systems, transport models, dynamical ordering and phase transitions, and emergent behavior in non-equilibrium systems.
Measure theory originates from real analysis and is used in many areas of mathematics such as geometry, probability theory, dynamical systems and functional analysis. It is concerned with notions of length, area or volume, with a measure within a set being a systematic way to assign a number to a subset of that set. You’ll look at the definition of a measurable space, additive measures, construction of measures, measurable functions, integrals with respect to a measure, differentiability of monotone functions, k-dimensional measures in n-dimensional space, Lebesgue-Stieltjes measure and Lebesgue measure. Theorems you will cover include Lusin’s theorem, Egoroff’s theorem, Fatou’s lemma, monotone convergence theorem, dominated convergence theorem, Fubini’s theorem, Radon-Nikodym theorem, Riesz representation theorem and divergence theorem.
The mathematical concept of ‘fractals’ is difficult to formally define, even for mathematicians! Fractals are geometric forms that display self-similar patterns on all scales of magnification, making them look the same when seen from near as from far. Fractal geometry looks at the mathematical theory behind fractals, the definition and properties of Hausdorff dimensioning and iterated function systems. You’ll gain intimacy with forms such as the middle third Cantor set, the Mandelbrot set and the von Koch snowflake curve.
Useful for students interested in engineering and aerospace, fluid dynamics addresses fluid phenomena of various scales from a mathematical viewpoint. You’ll apply mathematics topics such as ordinary and partial differential equations, basic mechanics and multivariable calculus, and will learn about governing equations, how to deduce the equations of motion from conservation laws (mass, momentum, energy), vorticity, dimensional analysis, scale-invariant solutions, universal turbulence spectra, gravity and rotation in atmospheric and oceanic dynamics, equations of motion such as boundary layer equations, flow kinematics, classical and simple laminar flows and flow instabilities. You’ll cover Euler’s equation, Navier-Stokes equation, Bernoulli’s equation, Kelvin’s circulation theorem, Taylor-Proudman theorem, Reynold’s number, Rayleigh number, Ekman number and Prandtl’s boundary layer theory.
Other mathematics topics you can choose from include: algorithms, applied mathematics, calculus, commutative algebra, computational mathematics, computer game technology, cryptography, differential equations, financial mathematics, financial modelling, functional analysis, geometry, knot theory, linear algebra, linear equations, mathematical biology, mathematical modelling, matrix analysis, multivariable calculus, number theory, numerical analysis, probability, pure mathematics, qualitative theory, real analysis, set theory, statistics, theoretical physics, topology and vectors.
Mathematics graduates go on to pursue many different career paths, often shaped by the mathematics topics they’ve chosen to focus on and the level of academic study they reach – as well as other interests with which they choose to combine their mathematics skills. The long list of possible careers with a mathematics degree includes roles in scientific research, engineering, business and finance, teaching, defense, computing and various types of analysis.
Mathematics graduates are highly sought-after by employers in many sectors as they are perceived as having proven intellectual rigor, strong analytical and problem-solving skills and an ability to tackle complex tasks. So, even if your decision to study mathematics at university is motivated solely by your love of the subject, it seems likely that your degree will nonetheless provide a strong foundation for future career options. Some popular careers with a mathematics degree include:
Accountancy careers involve providing professional advice on financial matters to clients. This might involve financial reporting, taxation, auditing, forensic accounting, corporate finance, businesses recovery, accounting systems and accounting processes. You’ll be relied on to manage financial systems and budgets, prepare accounts, budget plans and tax returns, administer payrolls, provide professional advice based on financial audits, and review your client’s systems and analyzing risks.
You’ll need to carry out tests to check your client’s financial information and systems and advise your clients on tax planning according to legislation, on business transactions, and on preventing fraud. You’ll also need to maintain accounting records and prepare reports and budget plans to present to your client. You may need to manage junior colleagues.
A mathematics degree could also be the starting point for many different roles within engineering careers. Most engineers work as part of a multi-disciplinary project team, with a range of specialists. As such, you’re likely to need excellent team-working and communication skills – as well as the ability to apply your mathematics skills in a very practical environment.
Potential engineering careers with a mathematics degree include roles in mechanical and electrical engineering, within sectors including manufacturing, energy, construction, transport, healthcare, computing and technology. You may be involved in all stages of product development or focus on just one aspect – such as research, design, testing, manufacture, installation and maintenance.
There are a range of banking careers that may be suitable for mathematics graduates due to their strong focus on numbers and analytics. Two of the major pathways are investment banking and retail banking. Investment banking careers involve gathering, analyzing and interpreting complex numerical and financial information, then assessing and predicting financial risks and returns in order to provide investment advice and recommendations to clients. Retail banking careers involve providing financial services to customers, including assessing and reviewing the financial circumstances of individual customers, implementing new products, processes and services, maintaining statistical and financial records, meeting sales targets and managing budgets.
Actuarial careers involve using mathematical and statistical modelling to predict future events that will have a financial impact on the organization you are employed by. This involved high levels of mathematics skills, combined with an understanding of business and economics. You’ll use probability theory, investment theory, statistical concepts and mathematical modelling techniques to analyze statistical data in order to assess risks. You’ll prepare reports on your findings, give advice, ensure compliance with the requirements of relevant regulatory bodies and communicate with clients and external stakeholders.
Careers in mathematics research are available within both the private and public sectors, with employers including private or government research laboratories, commercial manufacturing companies and universities. A research mathematician is able to study, create and apply new mathematical methods to achieve solutions to problems, including deep and abstract theorems.
The role also involves keeping up to date with new mathematical developments, producing original mathematics research, using specialist mathematical software and sharing your research through regular reports and papers. Your job will vary depending on the sector you work in, but some tasks may involve developing mathematical descriptions and models to explain or predict real life phenomena, applying mathematical principles to identify trends in data sets or applying your research to develop a commercial product or predict business trends and market developments.
A statistician collects, analyzes, interprets and presents quantitative information, obtained through the use of experiments and surveys on behalf of a client. You’ll probably work alongside professionals from other disciplines, so interpersonal and communication skills are important, as well as the ability to explain statistical information to non-statisticians.
Typical tasks include consulting with your client to agree on what data to collect and how, designing data acquisition trials such as surveys and experiments while taking into account ethical and legislative concerns, assessing results and analyzing predictable trends and advising your client on future strategy. You might also advise policymakers on key issues, collecting and analyzing data to monitor relevant issues and predicting demand for products and services. Statistician careers are available in a range of sectors including health, education, government, finance, transportation and market research, and you may also teach statistics in an academic setting.
In many countries, governments are calling for more mathematics graduates to go into teaching. Often this requires completing a postgraduate qualification in teaching, though this depends on the level and type of institution you teach at. Duties will involve instructing students, creating lesson plans, assigning and correcting homework, managing students in the classroom, communicating with students and parents and helping student prepare for standardized testing.
Other popular careers for mathematics graduates include: