In this opening programme Marcus du Sautoy looks at how fundamental mathematics is to our lives before exploring the mathematics of ancient Egypt, Mesopotamia, and Greece. In Egypt he uncovers use of a decimal system based on ten fingers of the hand, the Egyptians’ unusual method of multiplication and division, and their understanding of binary numbers, fractions, and solids such as the pyramid. He discovers that the way we tell the time today is based on the Babylonian Base 60 number system - so it is thanks to the Babylonians that we have 60 seconds in a minute, and 60 minutes in an hour - and shows how the Babylonians used quadratic equations to measure their land. In Greece, he looks at the contributions of some of the giants of mathematics including Plato, Euclid, Archimedes, and Pythagoras, who is credited with beginning the transformation of mathematics from a tool for counting into the analytical subject we know today. A controversial figure, Pythagoras’ teachings were considered suspect and his followers seen as a bizarre sect. Legend has it that one of his followers, Hippasus, was drowned when he announced his discovery of irrational numbers - a discovery that upset those who had held faith with the Pythagorean world view. As well as his ground-breaking work on the properties of right-angled triangles, Pythagoras developed another important theory after observing the properties of musical instruments: he discovered that the intervals between harmonious musical notes are always in whole number ratios to each other.

When ancient Greece fell into decline, mathematical progress stagnated as Europe fell under the shadow of the Dark Ages. But in the East, mathematics would reach new heights. In the second leg of his journey, Marcus du Sautoy visits China and explores how mathematics helped build imperial China and was at the heart of such amazing feats of engineering as the Great Wall. Here, he discovers the first use of a decimal place number system; the ancient Chinese fascination with patterns in numbers and the development of an early version of Sudoku; and their belief in the mystical powers of numbers, which still exists today. Marcus also learns how mathematics played a role in managing how the Emperor slept his way through the imperial harem to ensure the most favourable succession - and how internet cryptography encodes numbers using a branch of mathematics that has its origins in ancient Chinese work on equations. In India he discovers how the symbol for the number zero was invented - one of the great landmarks in the development of mathematics. He also examines Indian mathematicians’ understanding of the new concepts of infinity and negative numbers, and their invention of trigonometry. Next, he examines mathematical developments in the Middle East, looking at the invention of the new language of algebra, and the evolution of a solution to cubic equations. This leg of his journey ends in Italy, where he examines the spread of Eastern knowledge to the West through mathematicians such as Leonardo Fibonacci, creator of the Fibonacci Sequence.

By the seventeenth century Europe had taken over from the Middle East as the world’s power house of mathematical ideas. Great strides had been made in understanding the geometry of objects fixed in time and space. The race was now on to discover the mathematics that describes objects in motion. In this programme, Marcus du Sautoy visits France to look at the work of René Descartes, an outstanding mathematician and theoretical physicist as well as one of the great philosophers, who realised that it was possible to link algebra and geometry. His vital insight - that it was possible for curved lines to be described as equations - would change the course of the discipline forever. Marcus also examines the amazing properties of prime numbers discovered by Pierre Fermat, whose famous Last Theorem would puzzle mathematicians for more than 350 years. He shows how one of Fermat’s theorems is now the basis for the codes that protect credit card transactions on the internet. In England he looks at Isaac Newton’s development of calculus, a great breakthrough which is crucial to understanding the behaviour of moving objects and is used today by every engineer. He also goes in search of mathematical greats such as Leonard Euler, the father of topology or ‘bendy geometry’ and Carl Friedrich Gauss, who at the age of 24 was responsible for inventing modular arithmetic (a new way of handling equations). Gauss made major breakthroughs in our understanding of how prime numbers are distributed. This made a crucial contribution to the work of Bernhard Riemann, who developed important theories on prime numbers and had important insights into the properties of objects, which he saw as manifolds that could exist in multi-dimensional space.

In the last programme in the series, Marcus du Sautoy looks at some of the great unsolved problems that confronts mathematics in the twentieth century and tells the stories of the mathematicians who would try to crack them. Mathematicians like Georg Cantor, who investigated a subject that many of the finest mathematical minds had avoided – infinity. Cantor discovered that there were different kinds of infinity - and that some were bigger than others. Henri Poincaré was trying to solve one mathematical problem when he accidentally stumbled on chaos theory, which has led to a range of ‘smart’ technologies, including machines which control the regularity of heart beats. But in the middle of the twentieth century, mathematics was itself thrown into chaos. Kurt Gödel, an active member of the famous 'Vienna Circle’ of philosophers, detonated a 'logic bomb’ under 3,000 years of mathematics when he showed that it was impossible for mathematics to prove its own consistency - and that the unknowable is itself an integral part of mathematics. In this programme, Marcus looks at the startling discoveries of the American mathematician Paul Cohen, who established that there were several different sorts of mathematics in which conflicting answers to the same question were possible. He also examines the work of André Weil and his colleagues, who developed algebraic geometry, a field of study which helped to solve many of mathematics' toughest equations, including Fermat’s Last Theorem. He also reflects on the contributions of Alexander Grothendieck, whose ideas have had a major influence on current mathematical thinking about the hidden structures behind all mathematics. Marcus concludes his journey by considering the great unsolved problems of mathematics today, including the Riemann Hypothesis - a conjecture about the distribution of prime numbers – which are the atoms of the mathematical universe. There is now $1 million prize and a place in the history books for anyone who can prove Riemann’s theorem.