Biography of mathematicians

Date

Name

Nationality

Major Achievements

BCE

African

First notched tally bones

BCE

Sumerian

Earliest documented counting and measuring system

BCE

Egyptian

Earliest fully-developed base 10 number system in use

BCE

Sumerian

Multiplication tables, geometrical exercises and division problems

BCE

Egyptian

Earliest papyri showing numeration system and basic arithmetic

BCE

Babylonian

Clay tablets dealing with fractions, algebra and equations

BCE

Egyptian

Rhind Papyrus (instruction manual in arithmetic, geometry, unit fractions, etc)

BCE

Chinese

First decimal numeration system with place value concept

BCE

Indian

Early Vedic mantras invoke powers of ten from a hundred all the way up to a trillion

BCE

Indian

“Sulba Sutra” lists several Pythagorean triples and simplified Pythagorean theorem for the sides of a square and a rectangle, quite accurate approximation to √2

BCE

Chinese

Lo Shu order three (3 x 3) “magic square” in which each row, column and diagonal sums to 15

BCE

Thales

Greek

Early developments in geometry, including work on similar and right triangles

BCE

Pythagoras

Greek

Expansion of geometry, rigorous approach building from first principles, square and triangular numbers, Pythagoras’ theorem

BCE

Hippasus

Greek

Discovered potential existence of irrational numbers while trying to calculate the value of √2

BCE

Zeno of Elea

Greek

Describes a series of paradoxes concerning infinity and infinitesimals

BCE

Hippocrates of Chios

Greek

First systematic compilation of geometrical knowledge, Lune of Hippocrates

BCE

Democritus

Greek

Developments in geometry and fractions, volume of a cone

BCE

Plato

Greek

Platonic solids, statement of the Three Classical Problems, influential teacher and popularizer of mathematics, insistence on rigorous proof and logical methods

BCE

Eudoxus of Cnidus

Greek

Method for rigorously proving statements about areas and volumes by successive approximations

BCE

Aristotle

Greek

Development and standardization of logic (although not then considered part of mathematics) and deductive reasoning

BCE

Euclid

Greek

Definitive statement of classical (Euclidean) geometry, use of axioms and postulates, many formulas, proofs and theorems including Euclid’s Theorem on infinitude of primes

BCE

Archimedes

Greek

Formulas for areas of regular shapes, “method of exhaustion” for approximating areas and value of π, comparison of infinities

BCE

Eratosthenes

Greek

“Sieve of Eratosthenes” method for identifying prime numbers

BCE

Apollonius of Perga

Greek

Work on geometry, especially on cones and conic sections (ellipse, parabola, hyperbola)

BCE

Chinese

“Nine Chapters on the Mathematical Art”, including guide to how to solve equations using sophisticated matrix-based methods

BCE

Hipparchus

Greek

Develop first detailed trigonometry tables

36 BCE

Mayan

Pre-classic Mayans developed the concept of zero by at least this time

CE

Heron (or Hero) of Alexandria

Greek

Heron’s Formula for finding the area of a triangle from its side lengths, Heron’s Method for iteratively computing a square root

CE

Ptolemy

Greek/Egyptian

Develop even more detailed trigonometry tables

CE

Sun Tzu

Chinese

First definitive statement of Chinese Remainder Theorem

CE

Indian

Refined and perfected decimal place value number system

CE

Diophantus

Greek

Diophantine Analysis of complex algebraic problems, to find rational solutions to equations with several unknowns

CE

Liu Hui

Chinese

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

CE

Indian

“Surya Siddhanta” contains roots of modern trigonometry, including first real use of sines, cosines, inverse sines, tangents and secants

CE

Aryabhata

Indian

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)

CE

Brahmagupta

Indian

Basic mathematical rules for dealing with zero (+, &#; and x), negative numbers, negative roots of quadratic equations, solution of quadratic equations with two unknowns

CE

Bhaskara I

Indian

First to write numbers in Hindu-Arabic decimal system with a circle for zero, remarkably accurate approximation of the sine function

CE

Muhammad Al-Khwarizmi

Persian

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

CE

Ibrahim ibn Sinan

Arabic

Continued Archimedes&#; investigations of areas and volumes, tangents to a circle

CE

Muhammad Al-Karaji

Persian

First use of proof by mathematical induction, including to prove the binomial theorem

CE

Ibn al-Haytham (Alhazen)

Persian/Arabic

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



Omar Khayyam

Persian

Generalized Indian methods for extracting square and cube roots to include fourth, fifth and higher roots, noted existence of different sorts of cubic equations



Bhaskara II

Indian

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



Leonardo of Pisa (Fibonacci)

Italian

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)



Nasir al-Din al-Tusi

Persian

Developed field of spherical trigonometry, formulated law of sines for plane triangles



Qin Jiushao

Chinese

Solutions to quadratic, cubic and higher power equations using a method of repeated approximations



Yang Hui

Chinese

Culmination of Chinese “magic” squares, circles and triangles, Yang Hui’s Triangle (earlier version of Pascal’s Triangle of binomial co-efficients)



Kamal al-Din al-Farisi

Persian

Applied theory of conic sections to solve optical problems, explored amicable numbers, factorization and combinatorial methods



Madhava

Indian

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



Nicole Oresme

French

System of rectangular coordinates, such as for a time-speed-distance graph, first to use fractional exponents, also worked on infinite series



Luca Pacioli

Italian

Influential book on arithmetic, geometry and book-keeping, also introduced standard symbols for plus and minus



Niccolò Fontana Tartaglia

Italian

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)



Gerolamo Cardano

Italian

Published solution of cubic and quartic equations (by Tartaglia and Ferrari), acknowledged existence of imaginary numbers (based on √-1)



Lodovico Ferrari

Italian

Devised formula for solution of quartic equations



John Napier

British

Invention of natural logarithms, popularized the use of the decimal point, Napier’s Bones tool for lattice multiplication



Marin Mersenne

French

Clearing house for mathematical thought during 17th Century, Mersenne primes (prime numbers that are one less than a power of 2)



Girard Desargues

French

Early development of projective geometry and “point at infinity”, perspective theorem



René Descartes

French

Development of Cartesian coordinates and analytic geometry (synthesis of geometry and algebra), also credited with the first use of superscripts for powers or exponents



Bonaventura Cavalieri

Italian

“Method of indivisibles” paved way for the later development of infinitesimal calculus



Pierre de Fermat

French

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

British

Contributed towards development of calculus, originated idea of number line, introduced symbol ∞ for infinity, developed standard notation for powers



Blaise Pascal

French

Pioneer (with Fermat) of probability theory, Pascal’s Triangle of binomial coefficients



Isaac Newton

British

Development of infinitesimal calculus (differentiation and integration), laid ground work for almost all of classical mechanics, generalized binomial theorem, infinite power series



Gottfried Leibniz

German

Independently 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



Jacob Bernoulli

Swiss

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



Johann Bernoulli

Swiss

Further developed infinitesimal calculus, including the “calculus of variation”, functions for curve of fastest descent (brachistochrone) and catenary curve



Abraham de Moivre

French

De Moivre&#;s formula, development of analytic geometry, first statement of the formula for the normal distribution curve, probability theory



Christian Goldbach

German

Goldbach Conjecture, Goldbach-Euler Theorem on perfect powers



Leonhard Euler

Swiss

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



Johann Lambert

Swiss

Rigorous proof that π is irrational, introduced hyperbolic functions into trigonometry, made conjectures on non-Euclidean space and hyperbolic triangles



Joseph Louis Lagrange

Italian/French

Comprehensive treatment of classical and celestial mechanics, calculus of variations, Lagrange’s theorem of finite groups, four-square theorem, mean value theorem



Gaspard Monge

French

Inventor of descriptive geometry, orthographic projection



Pierre-Simon Laplace

French

Celestial mechanics translated geometric study of classical mechanics to one based on calculus, Bayesian interpretation of probability, belief in scientific determinism



Adrien-Marie Legendre

French

Abstract algebra, mathematical analysis, least squares method for curve-fitting and linear regression, quadratic reciprocity law, prime number theorem, elliptic functions



Joseph Fourier

French

Studied periodic functions and infinite sums in which the terms are trigonometric functions (Fourier series)



Carl Friedrich Gauss

German

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



Augustin-Louis Cauchy

French

Early pioneer of mathematical analysis, reformulated and proved theorems of calculus in a rigorous manner, Cauchy&#;s theorem (a fundamental theorem of group theory)



August Ferdinand Möbius

German

Möbius strip (a two-dimensional surface with only one side), Möbius configuration, Möbius transformations, Möbius transform (number theory), Möbius function, Möbius inversion formula



George Peacock

British

Inventor of symbolic algebra (early attempt to place algebra on a strictly logical basis)



Charles Babbage

British

Designed a &#;difference engine&#; that could automatically perform computations based on instructions stored on cards or tape, forerunner of programmable computer.



Nikolai Lobachevsky

Russian

Developed theory of hyperbolic geometry and curved spaces independendly of Bolyai



Niels Henrik Abel

Norwegian

Proved impossibility of solving quintic equations, group theory, abelian groups, abelian categories, abelian variety



János Bolyai

Hungarian

Explored hyperbolic geometry and curved spaces independently of Lobachevsky



Carl Jacobi

German

Important contributions to analysis, theory of periodic and elliptic functions, determinants and matrices



William Hamilton

Irish

Theory of quaternions (first example of a non-commutative algebra)



Évariste Galois

French

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



George Boole

British

Devised Boolean algebra (using operators AND, OR and NOT), starting point of modern mathematical logic, led to the development of computer science



Karl Weierstrass

German

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



Arthur Cayley

British

Pioneer of modern group theory, matrix algebra, theory of higher singularities, theory of invariants, higher dimensional geometry, extended Hamilton&#;s quaternions to create octonions



Bernhard Riemann

German

Non-Euclidean elliptic geometry, Riemann surfaces, Riemannian geometry (differential geometry in multiple dimensions), complex manifold theory, zeta function, Riemann Hypothesis



Richard Dedekind

German

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)



John Venn

British

Introduced Venn diagrams into set theory (now a ubiquitous tool in probability, logic and statistics)



Marius Sophus Lie

Norwegian

Applied algebra to geometric theory of differential equations, continuous symmetry, Lie groups of transformations



Georg Cantor

German

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”)



Gottlob Frege

German

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



Felix Klein

German

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



Henri Poincaré

French

Partial solution to “three body problem”, foundations of modern chaos theory, extended theory of mathematical topology, Poincaré conjecture



Giuseppe Peano

Italian

Peano axioms for natural numbers, developer of mathematical logic and set theory notation, contributed to modern method of mathematical induction



Alfred North Whitehead

British

Co-wrote “Principia Mathematica” (attempt to ground mathematics on logic)



David Hilbert

German

23 “Hilbert problems”, finiteness theorem, “Entscheidungsproblem“ (decision problem), Hilbert space, developed modern axiomatic approach to mathematics, formalism



Hermann Minkowski

German

Geometry of numbers (geometrical method in multi-dimensional space for solving number theory problems), Minkowski space-time



Bertrand Russell

British

Russell’s paradox, co-wrote “Principia Mathematica” (attempt to ground mathematics on logic), theory of types



G.H.

Hardy

British

Progress toward solving Riemann hypothesis (proved infinitely many zeroes on the critical line), encouraged new tradition of pure mathematics in Britain, taxicab numbers



Pierre Fatou

French

Pioneer in field of complex analytic dynamics, investigated iterative and recursive processes



L.E.J.

Brouwer

Dutch

Proved several theorems marking breakthroughs in topology (including fixed point theorem and topological invariance of dimension)



Srinivasa Ramanujan

Indian

Proved over 3, theorems, identities and equations, including on highly composite numbers, partition function and its asymptotics, and mock theta functions



Gaston Julia

French

Developed complex dynamics, Julia set formula



John von Neumann

Hungarian/
American

Pioneer of game theory, design model for modern computer architecture, work in quantum and nuclear physics



Kurt Gödel

Austria

Incompleteness theorems (there can be solutions to mathematical problems which are true but which can never be proved), Gödel numbering, logic and set theory



André Weil

French

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



Alan Turing

British

Breaking of the German enigma code, Turing machine (logical forerunner of computer), Turing test of artificial intelligence



Paul Erdös

Hungarian

Set and solved many problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory and probability theory



Edward Lorenz

American

Pioneer in modern chaos theory, Lorenz attractor, fractals, Lorenz oscillator, coined term “butterfly effect”



Julia Robinson

American

Work on decision problems and Hilbert&#;s tenth problem, Robinson hypothesis



Benoît Mandelbrot

French

Mandelbrot set fractal, computer plottings of Mandelbrot and Julia sets



Alexander Grothendieck

French

Mathematical structuralist, revolutionary advances in algebraic geometry, theory of schemes, contributions to algebraic topology, number theory, category theory, etc



John Nash

American

Work in game theory, differential geometry and partial differential equations, provided insight into complex systems in daily life such as economics, computing and military



Paul Cohen

American

Proved that continuum hypothesis could be both true and not true (i.e.

independent from Zermelo-Fraenkel set theory)



John Horton Conway

British

Important contributions to game theory, group theory, number theory, geometry and (especially) recreational mathematics, notably with the invention of the cellular automaton called the &#;Game of Life&#;



Yuri Matiyasevich

Russian

Final proof that Hilbert’s tenth problem is impossible (there is no general method for determining whether Diophantine equations have a solution)



Andrew Wiles

British

Finally proved Fermat’s Last Theorem for all numbers (by proving the Taniyama-Shimura conjecture for semistable elliptic curves)



Grigori Perelman

Russian

Finally proved Poincaré Conjecture (by proving Thurston&#;s geometrization conjecture), contributions to Riemannian geometry and geometric topology