Date

Name

Nationality

Major Achievements

35000 BC


African

First notched tally bones

3100 BC


Sumerian

Earliest documented counting and measuring system

2700 BC


Egyptian

Earliest fullydeveloped base 10 number system in use

2600 BC


Sumerian

Multiplication tables, geometrical exercises and division problems

20001800 BC


Egyptian

Earliest papyri showing numeration system and basic arithmetic

18001600 BC


Babylonian

Clay tablets dealing with fractions, algebra and equations

1650 BC


Egyptian

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

1200 BC


Chinese

First decimal numeration system with place value concept

1200900 BC


Indian

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

800400 BC


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

650 BC


Chinese

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

624546 BC

Thales

Greek

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

570495 BC

Pythagoras

Greek

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

500 BC

Hippasus

Greek

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

490430 BC

Zeno of Elea

Greek

Describes a series of paradoxes concerning infinity and infinitesimals

470410 BC

Hippocrates of Chios

Greek

First systematic compilation of geometrical knowledge, Lune of Hippocrates

460370 BC

Democritus

Greek

Developments in geometry and fractions, volume of a cone

428348 BC

Plato

Greek

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

410355 BC

Eudoxus of Cnidus

Greek

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

384322 BC

Aristotle

Greek

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

300 BC

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

287212 BC

Archimedes

Greek

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

276195 BC

Eratosthenes

Greek

“Sieve of Eratosthenes” method for identifying prime numbers

262190 BC

Apollonius of Perga

Greek

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

200 BC


Chinese

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

190120 BC

Hipparchus

Greek

Develop first detailed trigonometry tables

36 BC


Mayan

Preclassic Mayans developed the concept of zero by at least this time

1070 AD

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

90168 AD

Ptolemy

Greek/Egyptian

Develop even more detailed trigonometry tables

200 AD

Sun Tzu

Chinese

First definitive statement of Chinese Remainder Theorem

200 AD


Indian

Refined and perfected decimal place value number system

200284 AD

Diophantus

Greek

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

220280 AD

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

400 AD


Indian

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

476550 AD

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)

598668 AD

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

600680 AD

Bhaskara I

Indian

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

780850 AD

Muhammad AlKhwarizmi

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

908946 AD

Ibrahim ibn Sinan

Arabic

Continued Archimedes' investigations of areas and volumes, tangents to a circle

9531029 AD

Muhammad AlKaraji

Persian

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

9661059 AD

Ibn alHaytham (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

10481131

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

11141185

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

11701250

Leonardo of Pisa (Fibonacci)

Italian

Fibonacci Sequence of numbers, advocacy of the use of the HinduArabic numeral system in Europe, Fibonacci's identity (product of two sums of two squares is itself a sum of two squares)

12011274

Nasir alDin alTusi

Persian

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

12021261

Qin Jiushao

Chinese

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

12381298

Yang Hui

Chinese

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

12671319

Kamal alDin alFarisi

Persian

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

13501425

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

13231382

Nicole Oresme

French

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

14461517

Luca Pacioli

Italian

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

14991557

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)

15011576

Gerolamo Cardano

Italian

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

15221565

Lodovico Ferrari

Italian

Devised formula for solution of quartic equations

15501617

John Napier

British

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

15881648

Marin Mersenne

French

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

15911661

Girard Desargues

French

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

15961650

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

15981647

Bonaventura Cavalieri

Italian

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

16011665

Pierre de Fermat

French

Discovered many new numbers patterns and theorems (including Little Theorem, TwoSquare Thereom and Last Theorem), greatly extending knowlege of number theory, also contributed to probability theory

16161703

John Wallis

British

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

16231662

Blaise Pascal

French

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

16431727

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

16461716

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

16541705

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

16671748

Johann Bernoulli

Swiss

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

16671754

Abraham de Moivre

French

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

16901764

Christian Goldbach

German

Goldbach Conjecture, GoldbachEuler Theorem on perfect powers

17071783

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

17281777

Johann Lambert

Swiss

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

17361813

Joseph Louis Lagrange

Italian/French

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

17461818

Gaspard Monge

French

Inventor of descriptive geometry, orthographic projection

17491827

PierreSimon Laplace

French

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

17521833

AdrienMarie Legendre

French

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

17681830

Joseph Fourier

French

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

17771825

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, nonEuclidean geometry, Gaussian curvature

17891857

AugustinLouis 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)

17901868

August Ferdinand Möbius

German

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

17911858

George Peacock

British

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

17911871

Charles Babbage

British

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

17921856

Nikolai Lobachevsky

Russian

Developed theory of hyperbolic geometry and curved spaces independendly of Bolyai

18021829

Niels Henrik Abel

Norwegian

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

18021860

János Bolyai

Hungarian

Explored hyperbolic geometry and curved spaces independently of Lobachevsky

18041851

Carl Jacobi

German

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

18051865

William Hamilton

Irish

Theory of quaternions (first example of a noncommutative algebra)

18111832

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

18151864

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

18151897

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

18211895

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

18261866

Bernhard Riemann

German

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

18311916

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)

18341923

John Venn

British

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

18421899

Marius Sophus Lie

Norwegian

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

18451918

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”)

18481925

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

18491925

Felix Klein

German

Klein bottle (a onesided closed surface in fourdimensional space), Erlangen Program to classify geometries by their underlying symmetry groups, work on group theory and function theory

18541912

Henri Poincaré

French

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

18581932

Giuseppe Peano

Italian

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

18611947

Alfred North Whitehead

British

Cowrote “Principia Mathematica” (attempt to ground mathematics on logic)

18621943

David Hilbert

German

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

18641909

Hermann Minkowski

German

Geometry of numbers (geometrical method in multidimensional space for solving number theory problems), Minkowski spacetime

18721970

Bertrand Russell

British

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

18771947

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

18781929

Pierre Fatou

French

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

18811966

L.E.J. Brouwer

Dutch

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

18871920

Srinivasa Ramanujan

Indian

Proved over 3,000 theorems, identities and equations, including on highly composite numbers, partition function and its asymptotics, and mock theta functions

18931978

Gaston Julia

French

Developed complex dynamics, Julia set formula

19031957

John von Neumann

Hungarian/ American

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

19061978

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

19061998

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

19121954

Alan Turing

British

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

19131996

Paul Erdös

Hungarian

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

19172008

Edward Lorenz

American

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

19191985

Julia Robinson

American

Work on decision problems and Hilbert's tenth problem, Robinson hypothesis

19242010

Benoît Mandelbrot

French

Mandelbrot set fractal, computer plottings of Mandelbrot and Julia sets

1928

Alexander Grothendieck

French

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

1928

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

19342007

Paul Cohen

American

Proved that continuum hypothesis could be both true and not true (i.e. independent from ZermeloFraenkel set theory)

1937

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"

1947

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)

1953

Andrew Wiles

British

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

1966

Grigori Perelman

Russian

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