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In Africa, first notched tally bones
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The Sumerians documented the system they used for counting and measuring
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Earliest fully-developed base 10 number system in use
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The Sumerians use multiplication tables, geometrical exercises and division problems.
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In Egypt, the earliest papyri showing the numeration system and basic arithmetic
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In Egypt, the Rhind Papyrus (instruction manual in arithmetic, geometry and fractions) is made.
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In Babylon, clay tablets dealing with fractions, algebra and equations
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In China, the first decimal numeration system with place value concept is used.
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Early Vedic mantras invoke powers of ten from a hundred all the way up to a trillion
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Lo Shu order three (3 x 3) “magic square” in which each row, column and diagonal sums to 15
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“Sulba Sutra” lists several Pythagorean triples and simplified Pythagorean theorem for the sides of a square and a rectangle, quite accurate approximation to √2
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Early developments in geometry, including work on similar and right triangles
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Discovered potential existence of irrational numbers while trying to calculate the value of √2
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Expansion of geometry, rigorous approach building from first principles, square and triangular numbers, Pythagoras’ theorem
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Describes a series of paradoxes concerning infinity and infinitesimals
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First systematic compilation of geometrical knowledge, Lune of Hippocrates
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Developments in geometry and fractions, volume of a cone
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Method for rigorously proving statements about areas and volumes by successive approximations
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Platonic solids, statement of the Three Classical Problems, influential teacher and popularizer of mathematics, insistence on rigorous proof and logical methods
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Development and standardization of logic (although not then considered part of mathematics) and deductive reasoning
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Definitive statement of classical (Euclidean) geometry, use of axioms and postulates, many formulas, proofs and theorems including Euclid’s Theorem on infinitude of primes
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Formulas for areas of regular shapes, “method of exhaustion” for approximating areas and value of π, comparison of infinities
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“Nine Chapters on the Mathematical Art”, including guide to how to solve equations using sophisticated matrix-based methods
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Work on geometry, especially on cones and conic sections (ellipse, parabola, hyperbola)
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Develop first detailed trigonometry tables
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Pre-classic Mayans developed the concept of zero by at least this time
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Heron’s Formula for finding the area of a triangle from its side lengths, Heron’s Method for iteratively computing a square root
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Develop even more detailed trigonometry tables
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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
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Diophantine Analysis of complex algebraic problems, to find rational solutions to equations with several unknowns
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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)
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Basic mathematical rules for dealing with zero (+, - and x), negative numbers, negative roots of quadratic equations, solution of quadratic equations with two unknowns
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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
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First use of proof by mathematical induction, including to prove the binomial theorem
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Established that dividing by zero yields infinity, found solutions to quadratic, cubic and quartic equations (including negative and irrational solutions) and to second order Diophantine equations, introduced some preliminary concepts of calculus
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Fibonacci Sequence of numbers, advocacy of the use of the Hindu-Arabic numeral system in Europe, Fibonacci's identity (product of two sums of two squares is itself a sum of two squares)
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Developed field of spherical trigonometry, formulated law of sines for plane triangles
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System of rectangular coordinates, such as for a time-speed-distance graph, first to use fractional exponents, also worked on infinite series
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Use of infinite series of fractions to give an exact formula for π, sine formula and other trigonometric functions, important step towards development of calculus
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Influential book on arithmetic, geometry and book-keeping, also introduced standard symbols for plus and minus
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Formula for solving all types of cubic equations, involving first real use of complex numbers (combinations of real and imaginary numbers), Tartaglia’s Triangle (earlier version of Pascal’s Triangle)
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Published solution of cubic and quartic equations (by Tartaglia and Ferrari), acknowledged existence of imaginary numbers (based on √-1)
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Invention of natural logarithms, popularized the use of the decimal point, Napier’s Bones tool for lattice multiplication
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Development of Cartesian coordinates and analytic geometry (synthesis of geometry and algebra), also credited with the first use of superscripts for powers or exponents
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Pioneer (with Fermat) of probability theory, Pascal’s Triangle of binomial coefficients
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Discovered many new numbers patterns and theorems (including Little Theorem, Two-Square Thereom and Last Theorem), greatly extending knowlege of number theory, also contributed to probability theory
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Contributed towards development of calculus, originated idea of number line, introduced symbol ∞ for infinity, developed standard notation for powers
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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
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ndependently developed infinitesimal calculus (his calculus notation is still used), also practical calculating machine using binary system (forerunner of the computer), solved linear equations using a matrix
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Development of infinitesimal calculus (differentiation and integration), laid ground work for almost all of classical mechanics, generalized binomial theorem, infinite power series
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Further developed infinitesimal calculus, including the “calculus of variation”, functions for curve of fastest descent (brachistochrone) and catenary curve
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Made important contributions in almost all fields and found unexpected links between different fields, proved numerous theorems, pioneered new methods, standardized mathematical notation and wrote many influential textbooks
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Comprehensive treatment of classical and celestial mechanics, calculus of variations, Lagrange’s theorem of finite groups, four-square theorem, mean value theorem
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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
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Celestial mechanics translated geometric study of classical mechanics to one based on calculus, Bayesian interpretation of probability, belief in scientific determinism
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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
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Developed theory of hyperbolic geometry and curved spaces
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Early pioneer of mathematical analysis, reformulated and proved theorems of calculus in a rigorous manner, Cauchy's theorem (a fundamental theorem of group theory)
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Devised Boolean algebra (using operators AND, OR and NOT), starting point of modern mathematical logic, led to the development of computer science
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Non-Euclidean elliptic geometry, Riemann surfaces, Riemannian geometry (differential geometry in multiple dimensions), complex manifold theory, zeta function, Riemann Hypothesis
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Möbius strip (a two-dimensional surface with only one side), Möbius configuration, Möbius transformations, Möbius transform (number theory), Möbius function, Möbius inversion formula
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Designed a "difference engine" that could automatically perform computations based on instructions stored on cards or tape, forerunner of programmable computer.
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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)
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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”)
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Proved over 3,000 theorems, identities and equations, including on highly composite numbers, partition function and its asymptotics, and mock theta functions
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Introduced Venn diagrams into set theory (now a ubiquitous tool in probability, logic and statistics)
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Breaking of the German enigma code, Turing machine (logical forerunner of computer), Turing test of artificial intelligence
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Incompleteness theorems (there can be solutions to mathematical problems which are true but which can never be proved), Gödel numbering, logic and set theory
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Finally proved Fermat’s Last Theorem for all numbers (by proving the Taniyama-Shimura conjecture for semistable elliptic curves)
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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
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Proved that continuum hypothesis could be both true and not true (i.e. independent from Zermelo-Fraenkel set theory)
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Work in game theory, differential geometry and partial differential equations, provided insight into complex systems in daily life such as economics, computing and military
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Calculated pi to 3 trillion digits
