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Clay tablets dealing with fractions, algebra and equations Rhind Papyrus (instruction manual in arithmetic, geometry, unit fractions, etc) �Sulba Sutra� lists several Pythagorean triples and simplified Pythagorean theorem for the sides of a square and a rectangle, quite accurate approximation to √2 Lo Shu order three (3 x 3) �magic square� in which each row, column and diagonal sums to 15 Expansion of geometry, rigorous approach building from first principles, square and triangular numbers, Pythagoras� theorem First systematic compilation of geometrical knowledge, Lune of Hippocrates Developments in geometry and fractions, volume of a cone Platonic solids, statement of the Three Classical Problems, influential teacher and popularizer of mathematics, insistence on rigorous proof and logical methods Definitive statement of classical (Euclidean) geometry, use of axioms and postulates, many formulas, proofs and theorems including Euclid�s Theorem on infinitude of primes Formulas for areas of regular shapes, �method of exhaustion� for approximating areas and value of π, comparison of infinities Work on geometry, especially on cones and conic sections (ellipse, parabola, hyperbola) Heron�s Formula for finding the area of a triangle from its side lengths, Heron�s Method for iteratively computing a square root Diophantine Analysis of complex algebraic problems, to find rational solutions to equations with several unknowns Solved linear equations using a matrices (similar to Gaussian elimination), leaving roots unevaluated, calculated value of π correct to five decimal places, early forms of integral and differential calculus �Surya Siddhanta� contains roots of modern trigonometry, including first real use of sines, cosines, inverse sines, tangents and secants Definitions of trigonometric functions, complete and accurate sine and versine tables, solutions to simultaneous quadratic equations, accurate approximation for π (and recognition that π is an irrational number) Basic mathematical rules for dealing with zero (+, - and x), negative numbers, negative roots of quadratic equations, solution of quadratic equations with two unknowns First to write numbers in Hindu-Arabic decimal system with a circle for zero, remarkably accurate approximation of the sine function Advocacy of the Hindu numerals 1 - 9 and 0 in Islamic world, foundations of modern algebra, including algebraic methods of �reduction� and �balancing�, solution of polynomial equations up to second degree First use of proof by mathematical induction, including to prove the binomial theorem Derived a formula for the sum of fourth powers using a readily generalizable method, �Alhazen's problem�, established beginnings of link between algebra and geometry Generalized Indian methods for extracting square and cube roots to include fourth, fifth and higher roots, noted existence of different sorts of cubic equations Established that dividing by zero yields infinity, found solutions to quadratic, cubic 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coefficients Development of infinitesimal calculus (differentiation and integration), laid ground work for almost all of classical mechanics, generalized binomial theorem, infinite power series Helped to consolidate infinitesimal calculus, developed a technique for solving separable differential equations, added a theory of permutations and combinations to probability theory, Bernoulli Numbers sequence, transcendental curves De Moivre's formula, development of analytic geometry, first statement of the formula for the normal distribution curve, probability theory Pattern in occurrence of prime numbers, construction of heptadecagon, Fundamental Theorem of Algebra, exposition of complex numbers, least squares approximation method, Gaussian distribution, Gaussian function, Gaussian error curve, non-Euclidean geometry, Gaussian curvature Early pioneer of mathematical analysis, reformulated and proved theorems of calculus in a rigorous manner, Cauchy's theorem (a fundamental theorem of group theory) M�bius strip (a two-dimensional surface with only one side), M�bius configuration, M�bius transformations, M�bius transform (number theory), M�bius function, M�bius inversion formula Proved that there is no general algebraic method for solving polynomial equations of degree greater than four, laid groundwork for abstract algebra, Galois theory, group theory, ring theory, etc Devised Boolean algebra (using operators AND, OR and NOT), starting point of modern mathematical logic, led to the development of computer science Discovered a continuous function with no derivative, advancements in calculus of variations, reformulated calculus in a more rigorous fashion, pioneer in development of mathematical analysis Pioneer of modern group theory, matrix algebra, theory of higher singularities, theory of invariants, higher dimensional geometry, extended Hamilton's quaternions to create octonions Non-Euclidean elliptic geometry, Riemann surfaces, Riemannian geometry (differential geometry in multiple dimensions), complex manifold theory, zeta function, Riemann Hypothesis Defined some important concepts of set theory such as similar sets and infinite sets, proposed Dedekind cut (now a standard definition of the real numbers) Applied algebra to geometric theory of differential equations, continuous symmetry, Lie groups of transformations Creator of set theory, rigorous treatment of the notion of infinity and transfinite numbers, Cantor's theorem (which implies the existence of an �infinity of infinities�) One of the founders of modern logic, first rigorous treatment of the ideas of functions and variables in logic, major contributor to study of the foundations of mathematics Klein bottle (a one-sided closed surface in four-dimensional space), Erlangen Program to classify geometries by their underlying symmetry groups, work on group theory and function theory Partial solution to �three body problem�, foundations of modern chaos theory, extended theory of mathematical topology, Poincar� conjecture Peano axioms for natural numbers, developer of mathematical logic and set theory notation, contributed to modern method of mathematical induction Pioneer of game theory, design model for modern computer architecture, work in quantum and nuclear physics Incompleteness theorems (there can be solutions to mathematical problems which are true but which can never be proved), G�del numbering, logic and set theory Theorems allowed connections between algebraic geometry and number theory, Weil conjectures (partial proof of Riemann hypothesis for local zeta functions), founding member of influential Bourbaki group Breaking of the German enigma code, Turing machine (logical forerunner of computer), Turing test of artificial intelligence Set and solved many problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory and probability theory Mathematical structuralist, revolutionary advances in algebraic geometry, theory 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