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Mathematics Today

Mathematics Today is the membership publication of the Institute of Mathematics and its Applications.

Issued six times a year, this general interest mathematics publication provides articles, reports, reviews and news for mathematicians.

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Mathematics Today reaches a worldwide readership of over 4,400 professional mathematicians six times a year. It carries advertising for mathematics books, software, job vacancies, financial services and a whole range of products of interest to our readers.

In addition to competitive rates, a 10% discount is available to Institute members, Corporate Affiliates and publishers. Discounts may also be provided for multiple purchases. All adverts are in full colour. Advertising is also available on the IMA and Mathscareers websites (for job adverts only). Those advertising through all three media qualify for a 20% discount. Those advertising in two places qualify for a 10% discount.

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Content from the current issue


If you are reading this editorial, the chances are that you take it as read that to think mathematically affords a powerful means to understand and control one’s social and physical reality. Yet despite some twelve or so years of compulsory mathematical education, most adults in the developed world leave school with only limited access to mathematical ideas, or without much affinity with the idea of taking a mathematical point of view. In many cultures – the UK is particularly guilty – it is quite acceptable to admit ignorance of things mathematical in ways which would be inexcusable in relation to art, literature or music.

IMA Presidential Address 2015

The underlying theme of my presidential address was to outline the evidence of the importance of mathematics and some of the challenges that need to be faced if we are to increase participation in the subject. The first, overarching and enduring challenge is the invisibility of the subject. Most people just do not understand what the subject is about and why they are studying it for so many hours over so many years. I will point to some dangers if we do not rise to the challenge. The second challenge is to acknowledge, and then to address the complexities of teaching and learning mathematics: it is not a matter of mere exposition of procedures.

Software Tools That Do More with Less

Over the past decade we have been engaged in creating and investigating innovative learning environments in differential equations (DEs) that make use of a wide range of technologies, from applets to mainstream software such as Maple and Mathematica (see Rasmussen and Kwon [1] for an overview of this work). A key distinguishing feature of the applets we use is that they do much less for users than many of the prepackaged, commercial software tools.

A Diagrammatic Construction of Indefinite Integrals

Multiple representations of mathematical concepts reveal different aspects of their properties. Constructing links between pictorial and algebraic representations allows for deeper understanding and is therefore a powerful tool to extend learning in the mathematics classroom. The interpretation of definite integrals as the area under the curve is common practice but indefinite integrals are usually approached using algebra only. This article examines a series of learning activities around the gradient function from my Year 12 Curve Sketching Summer School at UCL.

Full contents page of the December 2015 printed issue
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Content from the October issue


One hundred years ago, Albert Einstein presented his field equations of gravitation in a plenary talk to the Prussian Academy of Sciences in Berlin. These equations form the foundations for his general theory of relativity, which models gravity in terms of space–time curvature that depends on local energy and momentum rather than as a force acting upon a mass. It formalises the relationships between the presence of matter and geometry in space and time. His equations have major implications for astrophysics, space travel and satellite navigation, including the Global Positioning System (GPS) and accurate timekeeping.

The Origins of General Relativity

AlexOne hundred years ago, Einstein completed his general theory of relativity. Here we re­flect on his ten-year journey of discovery which followed the publication of special relativ­ity in 1905. This article is based on Chapter 12 of From Eudoxus to Einstein, where the reader will find further details and extensive references.

In the latter half of the 19th century, tables used to determine Mercury’s position were notoriously inaccurate, in contrast to those for all the other planets. Mercury’s orbit can be thought of as an ellipse which slowly rotates so that the point of closest approach, the perihelion, slowly precesses around the sun. Much of this perihelion advance is due to the perturbative effect of the other planets, notably Venus and Jupiter, and calculations showed that these effects caused an advance of roughly 530” per century (about 1 1/4” per orbit) leaving 43” per century of the observed value unexplained.

A Century of General Relativity

E ThorringtonIn 1900 physics was in a quandary. Well-tested Newtonian physics was incompatible with Maxwell’s laws for electromagnetism, which implied a fixed speed of light, c. In Newtonian kinematics measured speed was relative to the observer’s motion. It was thus natural to assume, based on knowledge of other waves, that light waves had to be supported by a physical medium, the ‘aether’, at rest in Newton’s fixed space, and that the derivation of c applied only in a frame at rest in the aether.

Stellar aberration, discovered by Bradley in the 1720s, showed that the earth’s surface had to be moving through any aether. The 1880s Michelson-Morley experiments gave the contradictory result that it did not, creating the quandary mentioned. This could be resolved, rather unsatisfactorily, by assuming that measuring devices (‘rods and clocks’) in motion altered their size or clock rate in the specific way given by the Lorentz transformations. Then light always appeared to have the speed c.

Full contents page of the October 2015 printed issue
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Content from the August issue

AlexRemarkable! There I was, reading about shoelaces in Professor Ian Stewart’s brilliant book The Magical Maze [1], when a simple diagram that looks something like Figure 1 below jumped out at me. Citing a 1995 paper published by Professor John Halton (University of North Carolina), he refers to these graphs respectively as the American, European and shoe-store patterns for tying shoelaces. These patterns connect eight pairs of eyelets from top to toe of a shoe, alternating between left and right. The aim is to use the least amount of shoelace possible and, by cleverly using reflective symmetry for these graphs, Halton proved that the American pattern is the most efficient of all such arrangements, regardless of the number of pairs of eyelets, the distance between successive eyelets and the gap between left and right eyelets. He also proved that the European pattern is necessarily less efficient, albeit more decorative, and that the shoe-store pattern always uses even more shoelace.


Domineering: Comments on a Game of No Chance

AlexThe game of crosscram was created by Goran Andersson and discussed in an article [1] by Martin Gardner; the game is now more usually referred to as domineering and has attracted the attention of a number of mathematicians since its original presentation in 1974. An analysis of the various configurations is given in the book Winning Ways for Your Mathematical Plays [2] and the result obtained for best play on boards of different dimensions has been given in an article [3]. However, the mathematical formalism is probably rather difficult for individuals who are not familiar with the notation used in game theory. Consequently, we adopt, here, a simpler approach, which should enable readers to play interesting games and possibly to explore some of the mathematical ideas behind the sequential placements of the pieces without becoming too enmeshed in the mathematical formalities. We have used the game successfully in a number of maths masterclass sessions and found that the competitive interactions and strategic aspects are very positive for pupil groups of all ages above ten.

Don’t Switch! Why Mathematicians’ Answer to the Monty Hall Problem is Wrong

E ThorringtonThe Monty Hall problem is one of those rare curiosities – a mathematical problem that has made the front pages of national news. Everyone now knows, or thinks they know, the answer but a realistic look at the problem demonstrates that the standard mathematician’s answer is wrong. The mathematics is fine, of course, but the assumptions are unrealistic in the context in which they are set. In fact, it is not clear that this problem can be appropriately addressed using the standard tools of probability theory and this raises questions about what we think probabilities are and the way we teach them.

Full contents page of the August 2015 printed issue
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Content from the June issue


Yes, dear Reader, this is an editorial, wrote David Broomhead in August 2002, and so began the tradition of editorials for Mathematics Today. Dave’s last editorial was in April 2011, and between these dates Dave helped recre­ate the magazine as a regularly enjoyable and informative read for members of the IMA. Dave brought the same enthusiasm and commitment to Mathematics Today as he brought to his collaborations and friendships, his students and his academic thoughts. He influenced science policy in the UK and beyond through his work for the Research Councils and his internationally renowned scientific contributions. A chemist by training, he became a Professor of Mathematics and his death last year at the age of 63 was tragically early.

IMA Public Lecture: Hawking and Green

Every year the UK’s pure mathematicians and applied mathematicians hold major conferences, the British Mathematical Colloquium (BMC) and the British Applied Mathematics Colloquium (BAMC) respectively, but once every five years these conferences come together in one super joint meeting. This year, on 30 March to 2 April 2015, it was the turn of Cambridge University to hold what was the fourth such joint meeting, and I had the good fortune (if that is the right phrase) to chair the Local Organising Committee. We were delighted that the meeting generated enormous interest, and involved more than 600 participants from the UK and overseas, giving in excess of 300 invited and contributed talks on a vast range of topics from all across the mathematical sciences.

How to Model Honeybee Colonies
E Thorrington

Mathematics is a way of travelling logically from A to B, where A is a set of precisely stated rules or axioms and B is a set of predictions logically arising from these rules. Mathematical modelling is the art of mapping the real world onto these sets A and B. These are the two basic components of applied mathematics. One part is an abstract, rigorous toolkit for investigating logical truth, the other part is an attempt to align that truth with a more messy reality.

Using Tropical Maths to Model Ribosome Dynamics

If two activities must be performed consecutively then the time required to complete both is the sum of the individual times, but if they may be performed concurrently then the time required is the maximum of the individual times. For instance, imagine you are designing a train timetable. What is the earliest time at which a train can depart a given station? Certainly, it cannot leave until after it has arrived (despite what departure boards are prone to suggest!), but what other factors should be considered? Suppose that there are passengers arriving at the station on connecting trains from three other towns. We would therefore like to schedule our train to depart after these three trains have arrived. We don’t need to worry about the order in which the three connecting trains arrive; we only care that the departure time of our train occurs after the maximum of the three arrival times. To give the passengers a chance to change platforms and board, we should also add on a fixed amount of time to this maximum. Of course, in reality, rail networks involve huge numbers of trains with more complicated systems of dependencies between them. This sort of timing-analysis will therefore yield large systems of non-linear equations involving sums and maxima.

Full contents page of the June 2015 printed issue
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Content from the April 2015 issue

There was a smart chap called Einstein,
Whose theories curved way out of line,
He claimed light is bent,
No one knew what he meant,
But now we all think he’s divine.

Yes, I know, don’t give up the day job! Anyway, pick a word from the first line of the limerick. Now count the number of letters it contains and progress that number of words. For example, ‘Einstein’ contains eight letters and maps onto ‘He’. Repeat this process to the end of the limerick and I guarantee ... [drum roll] … that you will end up on the word ‘divine’. This exercise seems to demonstrate the convergence of a Markov chain, though we can do better than that. Wherever you started, you selected the word ‘light’ en route and this is my theme for this issue of Mathematics Today. The United Nations has declared 2015 to be the International Year of Light and Light-based Technologies (IYL), and with very good reason. This year celebrates several anniversaries of events that changed our perceptions of light significantly.

An Interview with Alexandra Randolph

Alexandra Randolph née Neville CMath MIMA is a teacher at St Paul’s Girls’ School in London. Her PhD was on the reaction and diffusion of chemicals within tumours. When Alex became a chartered mathematician, Rick Crawford CMath MIMA caught up with her for this interview.

I’d be interested to know about your voluntary work for the UK Mathematics Trust.

I help mark the Maclaurin Olympiad paper, set questions for the Junior Mathematical Olympiad and Senior Team Challenge and help at teacher meetings. I’ve also helped out at their summer schools and given a lecture to 200 teachers about stretch and challenge in the classroom. It’s really good to work closely with such great mathematical material for the classroom, but also to meet mathematicians from a range of backgrounds.

Improving Access for State-School Students

E ThorringtonIn a Guardian article [1] in November 2014, Paul Mason described how students from private schools have an advantage over state-educated students when applying for courses at leading universities. We support his conclusion that there should be a system that offers clear and transparent information about the requirements for courses, so that all students, whatever their background or type of school or college, understand what is required to access prestigious undergraduate degree courses.

Urban Maths: Mind Your S’s and D’s

I still spend some time writing computer code, but not half as much as I used to do when I first began my career. Back in those days, which are longer ago than I would care to mention, I typed many more lines of code than I did lines of text. And the nature of the code I was writing, and the tools that I was using, meant that I typed string quite a lot.  Now, that wouldn’t necessarily be a problem, but that sequence of characters has become permanently lodged in the muscle memory of my fingers, so much so that when I try to write strong I often end up with string instead.

Full contents page of the April 2015 printed issue
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Content from the February 2015 issue


The opportunity of editing Mathematics Today was one that I seized gratefully. For the past thirty years, I have enjoyed receiving copies of this general interest mathematics publication (formerly IMA Bulletin), so I am delighted to have this chance of contributing to its ongoing success. It had already been published for twenty years before I subscribed and early issues are still well worth reading.

My predecessor, Professor Linton, maintained and improved the quality of Mathematics Today and I echo the vote of thanks that appeared in December’s issue. All members of the Editorial Board generously and voluntarily donate considerable time and effort, as do the many authors of feature articles, reviews and correspondence. Of course, much credit is due to the Editorial Officer, Rebecca Waters, and other professional IMA staff.

University Liaison: Industrial Placements

Happy New Year! I hope everyone had a good holiday and success if you had start of year exams. Good luck for those of you having interviews this term.

We are really fortunate to have for this issue, two pieces by Waleed Backler from University of Greenwich about his one year internship with the Department of Health and NHS England and his short PLACE (Patient-Led Assessments of the Care Environment) assignment at Leeds Teaching Hospitals.

Are You Paying Too Much for Your Car Insurance?

E ThorringtonInsurers are risk-takers, accepting premiums to cover unknown, but potentially very large, future insured events; insurers have to balance complex theoretical mathematics with commercial considerations; they must be profitable but yet offer commercially acceptable premiums. In particular, in the car insurance market insurers have to take into account the possibility of selling the customer other products, real time pricing (as a result of price comparison websites) and new EU-wide Solvency II rules for capital management.

Full contents page of the February 2015 printed issue
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Mathematics Today
Supplementary Downloads

  • Domineering (August 2015)
  • Symbolic Solvers (October 2012)
  • Quadrature Programme (December 2010)