^ BC YEAR EVENT c.2400 The people of Mesopotamia (now Iraq) developed a positional numbering (place value) system, in which the value of a digit depends on its position in a number. c.2000 Mesopotamian mathematicians solved quadratic equations (algebraic equations in which the highest power of a variable is 2). 876 A symbol of zero was used for the first time, in India. c.550 Greek mathematician Pythagoras formulated a theorem relating the lengths of the sides of a right-angled triangle. The theorem was already known by earlier mathematicians in China, Mesopotamia, and Egypt. c.450 Hipparcos of Metapontum discovered that some numbers are irrational (cannot be expressed as the ration of two integers). 300 Euclid laid out the laws of geometry in his book "Elements", which was to remain a standard text for 2,000 years. c.230 Eratosthenes developed a method for finding all prime numbers. c.190 Chinese mathematicians used powers of 10 to express magnitudes. c.100 Chinese mathematicians began using negative numbers. AD YEAR EVENT c. 210 Diophantus of Alexandria wrote the first book on algebra. c.600 A decimal number system was developed in India. 829 Persian mathematician Al-Khowarizmi published a work on algebra that made use of the decimal number system. 1202 Italian mathematician Leonardo Fibonacci studied the sequence of numbers in which each number is the sum of the two preceding ones. 1550 In Germany, Rheticus published trigonometrical tables that simplified calculations involving triangles. 1614 Scottish mathematician John Napier invented logarithms, which enable lengthy calculations involving multiplication and division to be carried out by addition and subtraction. 1623 Wilhelm Schickardt invented the mechanical calculating machine. 1637 French mathematician and philosopher René Descartes introduced co-ordinate geometry. 1654 In France, Blaise Pascal and Pierre de Fermat developed probability theory. 1666 Isaac Newton developed differential calculus, a method of calculating rates of change. 1675 German mathematician Gottfried Wilhelm Leibniz introduced the modern notation for integral calculus, a method of calculating volumes. 1679 Leibniz introduced binary arithmetic, in which only two symbols are used to represent all numbers. 1684 Leibniz published the first account of differential calculus. 1718 Jacques Bernoulli in Switzerland published his work on the calculus of variations (the study of functions that are close to their minimum or maximum values). 1746 In France, Jean le Rond d'Alembert developed the theory of complex numbers. 1747 D'Alembert used partial differential equations in mathematical physics. 1798 Norwegian mathematician Caspar Wessel introduced the vector representation of complex numbers. 1799 Karl Friedrich Gauss of Germany proved the fundamental theorem of algebra: the number of solutions of an algebraic equations is the same as the exponent of the highest term. 1810 In France, Jean-Baptiste Joseph Fourier published his method of representing functions by a series of trigonometric functions. 1822 French mathematician Pierre Simon Laplace published the first complete account of probability theory. 1827 Gauss introduced differential geometry, in which small features of curves are described by analytical methods. 1829 In Russia, Nikolai Ivanonvich Lobachevski developed hyperbolic geometry, in which a plane is regarded as part of a hyperbolic surface, shaped like a saddle. In France, Evariste Galois introduced the theory of groups (collections whose members obey certain simple rules of addition and multiplication). 1843 French mathematician Joseph Liouville found the first transcendental number, which cannot be expressed as an algebraic equation with rational coefficients. In Germany, Hermann Grassman studied vectors with more than three dimensions. 1854 George Boole in the UK published his system of symbolic logic, now called Boolean algebra. 1858 English mathematician Arthur Cayley developed calculations using ordered tables called matrices. 1865 August Ferdinand Möbius in Germany described how a strip of paper can have only one side and one edge. 1892 German mathematician Georg Cantor showed that there are different kinds of infinity and studied transfinite numbers. 1895 Jules-Henri Poincaré published the first paper on topology, often called 'the geometry of rubber sheets'. 1931 In the USA, Austrian-born mathematician Kurt Gödel proved that any formal system strong enough to include the laws of arithmetic is either incomplete or inconsistent. 1937 English Mathematician Alan Turing published the mathematical theory of computing. 1944 Jon Von Neumann and Oscar Morgenstern developed game theory in the USA. 1945 The first general purpose, fully electronic digital computer, ENIAC (Electronic Numerator, Integrator, Analyser, and Computer), was built at the University of Pennsylvania, USA. 1961 Meteorologist Edward Lorenz a the Massachusetts Institute of Technology, USA, discovered a mathematical system with chaotic behaviour, leading to a new branch of mathematics chaos theory. 1962 Benoit Mandelbrot in the USA invented fractal images, using a computer that repeats the same mathematical pattern over and over again. 1975 US mathematician Mitchell Feigenbaum discovered a new fundamental constant (approximately 4.669201609103), which plays an important role in chaos theory. 1980 Mathematicians worldwide completed the classification of all finite and simple groups, a task that took over a hundred mathematicians more than 35 years to complete and whose results took up more than 14,000 pages in mathematical journals. 1989 A team of US computer mathematicians at Amdahl Corporation, California, discovered the highest known prime number (it contains 65,087 digits). 1993 US mathematician Andrew Wiles published a 1,000-page proof of Fermat's last theorem, one of the most baffling challenges in pure mathematics.