-
35,000 BCE
Use of tally bones
In Africa, first notched tally bones -
3100 BCE
Counting and Measuring
The Sumerians documented the system they used for counting and measuring -
2700 BCE
Egyptian Decimal System
Earliest fully-developed base 10 number system in use -
2600 BCE
Sumerian Math
The Sumerians use multiplication tables, geometrical exercises and division problems. -
1800 BCE
Egyptian Numeration
In Egypt, the earliest papyri showing the numeration system and basic arithmetic -
1650 BCE
The Rhind Papyrus
In Egypt, the Rhind Papyrus (instruction manual in arithmetic, geometry and fractions) is made. -
1600 BCE
Babylonian tablets
In Babylon, clay tablets dealing with fractions, algebra and equations -
1200 BCE
The Decimal System in China
In China, the first decimal numeration system with place value concept is used. -
900 BCE
The Vedas in India
Early Vedic mantras invoke powers of ten from a hundred all the way up to a trillion -
650 BCE
Magic Squares
Lo Shu order three (3 x 3) “magic square” in which each row, column and diagonal sums to 15 -
600 BCE
The Sulba Sutra
“Sulba Sutra” lists several Pythagorean triples and simplified Pythagorean theorem for the sides of a square and a rectangle, quite accurate approximation to √2 -
546 BCE
Thales
Early developments in geometry, including work on similar and right triangles -
500 BCE
Hippasus
Discovered potential existence of irrational numbers while trying to calculate the value of √2 -
495 BCE
Pythagoras
Expansion of geometry, rigorous approach building from first principles, square and triangular numbers, Pythagoras’ theorem -
430 BCE
Zeno of Elea
Describes a series of paradoxes concerning infinity and infinitesimals -
410 BCE
Hippocrates of Chios
First systematic compilation of geometrical knowledge, Lune of Hippocrates -
370 BCE
Democritus
Developments in geometry and fractions, volume of a cone -
355 BCE
Eudoxus of Cnidus
Method for rigorously proving statements about areas and volumes by successive approximations -
348 BCE
Plato
Platonic solids, statement of the Three Classical Problems, influential teacher and popularizer of mathematics, insistence on rigorous proof and logical methods -
322 BCE
Aristotle
Development and standardization of logic (although not then considered part of mathematics) and deductive reasoning -
300 BCE
Euclid
Definitive statement of classical (Euclidean) geometry, use of axioms and postulates, many formulas, proofs and theorems including Euclid’s Theorem on infinitude of primes -
212 BCE
Archimedes
Formulas for areas of regular shapes, “method of exhaustion” for approximating areas and value of π, comparison of infinities -
200 BCE
Nine Chapters on the Mathematical Art
“Nine Chapters on the Mathematical Art”, including guide to how to solve equations using sophisticated matrix-based methods -
190 BCE
Apollonius of Perga
Work on geometry, especially on cones and conic sections (ellipse, parabola, hyperbola) -
120 BCE
Hipparchus
Develop first detailed trigonometry tables -
36 BCE
The Mayas
Pre-classic Mayans developed the concept of zero by at least this time -
70
Heron of Alexandria (10-70 CE)
Heron’s Formula for finding the area of a triangle from its side lengths, Heron’s Method for iteratively computing a square root -
168
Ptolemy (90-168 CE)
Develop even more detailed trigonometry tables -
280
Lui Hui (220-280 CE)
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 -
284
Diophantus (200-284 CE)
Diophantine Analysis of complex algebraic problems, to find rational solutions to equations with several unknowns -
550
Aryabhata (476-550 CE)
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) -
668
Brahmagupta (598-668 CE)
Basic mathematical rules for dealing with zero (+, - and x), negative numbers, negative roots of quadratic equations, solution of quadratic equations with two unknowns -
850
Muhammad Al-Khwarizmi (780-850 CE)
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 -
1029
Muhammad Al-Karaji 953-1029 CE
First use of proof by mathematical induction, including to prove the binomial theorem -
1185
Bhaskara II (1114-1185)
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 -
1250
Leonardo of Pisa (Fibonacci) 1170-1250
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) -
1274
Nasir al-Din al-Tusi (1201-1274)
Developed field of spherical trigonometry, formulated law of sines for plane triangles -
1382
Nicole Oresme (1323-1382)
System of rectangular coordinates, such as for a time-speed-distance graph, first to use fractional exponents, also worked on infinite series -
1425
Madhava (1350-1425)
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 -
1517
Luca Pacioli (1446-1517)
Influential book on arithmetic, geometry and book-keeping, also introduced standard symbols for plus and minus -
1557
Niccolò Fontana Tartaglia (1499-1557)
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) -
1576
Gerolamo Cardano (1501-1576)
Published solution of cubic and quartic equations (by Tartaglia and Ferrari), acknowledged existence of imaginary numbers (based on √-1) -
John Napier (1550-1617)
Invention of natural logarithms, popularized the use of the decimal point, Napier’s Bones tool for lattice multiplication -
René Descartes (1596-1650)
Development of Cartesian coordinates and analytic geometry (synthesis of geometry and algebra), also credited with the first use of superscripts for powers or exponents -
Blaise Pascal (1623-1662)
Pioneer (with Fermat) of probability theory, Pascal’s Triangle of binomial coefficients -
Pierre de Fermat (1601-1665)
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 -
John Wallis (1616-1703)
Contributed towards development of calculus, originated idea of number line, introduced symbol ∞ for infinity, developed standard notation for powers -
Jacob Bernoulli (1654-1705)
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 -
Gottfried Leibniz (1646-1716)
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 -
Isaac Newton (1643-1727)
Development of infinitesimal calculus (differentiation and integration), laid ground work for almost all of classical mechanics, generalized binomial theorem, infinite power series -
Johann Bernoulli (1667-1748)
Further developed infinitesimal calculus, including the “calculus of variation”, functions for curve of fastest descent (brachistochrone) and catenary curve -
Leonhard Euler (1707-1783)
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 -
Joseph Louis Lagrange(1736-1813)
Comprehensive treatment of classical and celestial mechanics, calculus of variations, Lagrange’s theorem of finite groups, four-square theorem, mean value theorem -
Carl Friedrich Gauss (1777-1825)
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 -
Pierre-Simon Laplace (1749-1827)
Celestial mechanics translated geometric study of classical mechanics to one based on calculus, Bayesian interpretation of probability, belief in scientific determinism -
Évariste Galois (1811-1832)
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 -
Nikolai Lobachevsky (1792-1856)
Developed theory of hyperbolic geometry and curved spaces -
Augustin-Louis Cauchy (1789-1857)
Early pioneer of mathematical analysis, reformulated and proved theorems of calculus in a rigorous manner, Cauchy's theorem (a fundamental theorem of group theory) -
George Boole (1815-1864)
Devised Boolean algebra (using operators AND, OR and NOT), starting point of modern mathematical logic, led to the development of computer science -
Bernhard Riemann1826-1866
Non-Euclidean elliptic geometry, Riemann surfaces, Riemannian geometry (differential geometry in multiple dimensions), complex manifold theory, zeta function, Riemann Hypothesis -
August Ferdinand Möbius (1790-1868)
Möbius strip (a two-dimensional surface with only one side), Möbius configuration, Möbius transformations, Möbius transform (number theory), Möbius function, Möbius inversion formula -
Charles Babbage (1791-1871)
Designed a "difference engine" that could automatically perform computations based on instructions stored on cards or tape, forerunner of programmable computer. -
Richard Dedekind (1831-1916)
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) -
Georg Cantor (1845-1918)
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”) -
Srinivasa Ramanujan (1887-1920)
Proved over 3,000 theorems, identities and equations, including on highly composite numbers, partition function and its asymptotics, and mock theta functions -
John Venn (1834-1923)
Introduced Venn diagrams into set theory (now a ubiquitous tool in probability, logic and statistics) -
Alan Turing (1912-1954)
Breaking of the German enigma code, Turing machine (logical forerunner of computer), Turing test of artificial intelligence -
Kurt Gödel (1906-1978)
Incompleteness theorems (there can be solutions to mathematical problems which are true but which can never be proved), Gödel numbering, logic and set theory -
Andrew Wiles (1953- )
Finally proved Fermat’s Last Theorem for all numbers (by proving the Taniyama-Shimura conjecture for semistable elliptic curves) -
André Weil (1906-1998)
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 -
Paul Cohen (1934-2007)
Proved that continuum hypothesis could be both true and not true (i.e. independent from Zermelo-Fraenkel set theory) -
John Nash (1928-2015)
Work in game theory, differential geometry and partial differential equations, provided insight into complex systems in daily life such as economics, computing and military -
Peter Trub (1990-)
Calculated pi to 3 trillion digits
