When considering the contributions of mathematicians of the past to the modern world, it is useful to distinguish between the contributions which help us to understand mathematics, and those which help us to use it to solve practical problems. In some cases, of course, individuals contributed to both aspects of the development of the subject.

The very basic idea of a real number is not as simple as one might think, and mathematics itself would not be possible without a rigorous account of what a real number is. This was given by Dedekind and his work ensures that we can use the concept of a real number with confidence, for example, knowing that there are no 'holes' in the real number line, and understanding exactly the nature of the square root of two. One soon comes across the idea of 'infinity', and the idea that there may be more than one type of infinity.

Amongst other work, Cantor put such ideas on firm foundations on which future progress could be made.

With the real numbers understood, one remaining question concerns the idea of the square root of -1, or indeed any negative number. This problem rears its head again in a different guise through the Fundamental Theorem of Algebra, which says that every polynomial equation of degree n has n roots. But what about z^3-1=0? There is a root at z=1, but to find the other two (and to resolve the question on square roots) we need the concept of complex numbers. Complex numbers are much more than a mathematical curiosity; they find direct use particularly in engineering and physics, but are liable to crop up anywhere! Complex numbers were investigated particularly be De Moivre and Cauchy (who developed the ideas of the theory of a function of a complex variable), and Euler who discovered the connection with trigonometric functions.

Numbers continue to fascinate, and the study of numbers has become a major part of mathematics. Contributors to the field include Fermat, Chebyshev, Hilbert, Lagrange and Ramanujan. It is interesting to note that one distinguished mathematician said in 1940 that there were no uses for number theory. In fact, it forms the basis for modern coding and cryptography, and thus for secure data transfer over the internet etc., and is very useful indeed. (One of the companies sponsoring the Mathematics @ Work initiative is one of the leaders in this field, and so number theory is impinging on you too!)

Moving on from numbers to ideas of space and objects in space and their properties, in other words, we come to geometry.

Many significant advances in geometry were made in very early times, when the framework of mathematics was also first considered, leading to the concept of proof. Geometry, or at least geometrical awareness, is in everyday use by just about everyone, with some more complex ideas exploited by designers of cars, gardens, theatrical scenery, clothing, computer games and graphics, domestic equipment and so on. A major development came when Descartes developed analytical, or co-ordinate geometry, meaning that algebraic techniques could be applied to solve geometric problems. So, for example, a circle and a line are each described by an equation: if they intersect then the points of intersection are found as the solution of a set of simultaneous equations. Thus, in more recent times, analytical geometry has also provided the means by which computers could be used to solve geometrical problems, and so aid (amongst others) designers in various fields.

Newton and Liebnitz independently discovered differential calculus, and thus gave us the framework for the description of motion. Newton also showed how forces cause motion, and his ideas remain the foundation of mechanics. Using ideas developed by him, we can predict the motion of the planets, the behaviour of a car as it is driven, or the motion of a wobbly bridge. More recently Einstein realised that `Newtonian' mechanics was not giving accurate predictions for particles which moved at speeds near to that of light. To account for this he developed his theory of relativity and this is at the heart of modern physics. His famous equation E=mc^2, relating energy to mass and the speed of light gave us the first step on the road to the development of nuclear power.

Calculus is a very basic tool in many forms of applied mathematics, not just those involving the motion of bodies. One example is the important field of optimisation, where the optimum (cheapest, least energy-consuming etc) solution or strategy is sought. Many people have contributed to its development and application, including all three Bernoulli brothers, Euler, Gauss, Green, Hilbert, L'Hopital, Lagrange, Laplace, Poincare and Taylor.

Building on an understanding of the motion of bodies or particles, and with a knowledge of calculus, we might move on to the motion of fluids. Being able to predict such motion is necessary for the design of car engines, aircraft and rockets...and golf-balls. It is also necessary to be able to calculate the flow of blood in an artery or of the air in a lung.

Boole, who gave his name to Boolean algebra, provided the necessary structure for building logic circuits, which are at the heart of modern electronic equipment. Whilst Fourier devised the Fourier Series technique for solving problems in heat conduction, the ideas provide the essential mathematical tools to support modern electronic communication. For example, using ideas from Fourier analysis, it is possible to understand how a radio is tuned, and how many telephone conversations can be simultaneously transmitted through a single link.

The world is full of uncertainty, and one way of coping with this depends on the idea of probability. Fermat and Pascal, followed by Jacques Bernoulli and De Moivre, all contributed to our understanding of this important subject. Probability theory is now a highly developed subject with many sophisticated applications in fields from medicine to finance, engineering, marketing insurance and so on. In addition, we all use the basic ideas to support our own decisions in the face of uncertainty, and bookmakers earn a handsome living from them!

Nigel C Steele, Professor of Mathematics, School of Mathematical and Information Sciences, Coventry University