1 JohannCarlFriedrichGauss(1777-1855) Germany
Gauss built the theory of complex numbers into its modern form. Gauss developed the arithmetic of congruences and became the premier number theoretician of all time. Other contributions of Gauss include hypergeometric series, foundations of statistics, and differential geometry. Much of Gauss's work wasn't published: unbeknownst to his colleagues it was Gauss who first discovered non-Euclidean geometry, doubly periodic elliptic functions, a prime distribution formula, quaternions, foundations of topology, the Law of Least Squares, Dirichlet's class number formula, the key Bonnet's Theorem of differential geometry (now usually called Gauss-Bonnet Theorem), the butterfly procedure for rapid calculation of Fourier series, and even the rudiments of knot theory. Gauss was first to prove the Fundamental Theorem of Functions of a Complex Variable (that the line-integral over a closed curve of a monogenic function is zero).
Gauss also did important work in several areas of physics, developed an important modification to Mercator's map projection, invented the heliotrope, and co-invented the telegraph.
Gauss once wrote "It is not knowledge, but the act of learning, ... which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again ..."
2 LeonhardEuler(1707-1783) Switzerland
Laplace once said "Read Euler: he is our master in everything." Euler took marvelous advantage of the analysis of Newton and Leibniz. He also gave the world modern trigonometry; pioneered (along with Lagrange) the calculus of variations; generalized and proved the Newton-Giraud formulae; and made important contributions to algebra, e.g. his study of hypergeometric series. He was also supreme at discrete mathematics, inventing graph theory.Euler was a very major figure in number theory.
Euler was also first to prove several interesting theorems of geometry, including facts about the9-point Feuerbach circle; relationships among a triangle's altitudes, medians, and circumscribing and inscribing circles; the famous Intersecting Chords Theorem; and an expression for a tetrahedron's volume in terms of its edge lengths. Euler was first to explore topology. Four of the most important constant symbols in mathematics (π,e,i= √-1, andγ= 0.57721566...) were all introduced or popularized by Euler, along with operators likeΣ.
3 Jules HenriPoincaré(1854-1912) France
Poincaré founded the theory of algebraic (combinatorial) topology, and is sometimes called the "Father of Topology". He was one of the most creative mathematicians ever. In addition to his topology, Poincaré laid the foundations of homology; he discovered automorphic functions (a unifying foundation for the trigonometric and elliptic functions), and essentially founded the theory of periodic orbits; he made major advances in the theory of differential equations..
As were most of the greatest mathematicians, Poincaré was intensely interested in physics. He made revolutionary advances in fluid dynamics and celestial motions; he anticipated Minkowski space and much of Einstein's Special Theory of Relativity (including the famous equationE = mc2).
Poincaré also found time to become a famous popular:
"A [worthy] mathematician experiences in his work the same impression as an artist; his pleasure is as great and of the same nature;"
"If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living."
4 Andrei AndreyevichMarkov(1856-1922) Russia
Markov did excellent work in a broad range of mathematics including analysis, number theory, algebra, continued fractions, approximation theory, and especially probability theory: it has been said that his accuracy and clarity transformed probability theory into one of the most perfected areas of mathematics. Markov is best known as the founder of the theory of stochastic processes. In addition to his Ergodic Theorem about such processes, theorems named after him include the Gauss-Markov Theorem of statistics, the Riesz-Markov Theorem of functional analysis, and the Markov Brothers' Inequality in the theory of equations.
5 Gottfried Wilhelm vonLeibniz(1646-1716) Germany
Leibniz was one of the most brilliant and prolific intellectuals ever. He predicted the Earth's molten core, introduced the notion of subconscious mind, and built the first calculator that could do multiplication.
Leibniz pioneered the common discourse of mathematics, including its continuous, discrete, and symbolic aspects. (His ideas on symbolic logic weren't pursued and it was left to Boole to reinvent this almost two centuries later.) Mathematical innovations attributed to Leibniz include the notations ∫f(x)dx,df(x)/dx,∛x, and even the use ofa·b(instead ofaX b) for multiplication; the concepts of matrix determinant and Gaussian elimination; the theory of geometric envelopes; and the binary number system. He worked in number theory, conjecturing Wilson's Theorem. He invented more mathematical terms than anyone, includingfunction,analysis situ,variable,abscissa,parameterandcoordinate.
Leibniz' thoughts on mathematical physics had some influence. He was one of the first to articulate the law of energy conservation. He developed laws of motion that gave different insights from those of Newton; his views on cosmology anticipated theories of Mach and Einstein and are more in accord with modern physics than are Newton's views.
Although others found it, Leibniz discovered and proved a striking identity forπ:
π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...
6 Jacob Bernoulli(1654-1705) Switzerland
Jacob Bernoulli studied the works of Wallis and Barrow; he and Leibniz became friends and tutored each other. Jacob developed important methods for integral and differential equations, coining the wordintegral. He and his brother were the key pioneers in mathematics during the generations between the era of Newton-Leibniz and the rise of Leonhard Euler.
Jacob liked to pose and solve physical optimization problems. His most famous work was theArt of Conjecture, a textbook on probability and combinatorics which proves the Law of Large Numbers, the Power Series Equation, and introduces the Bernoulli numbers. He is credited with the invention of polar coordinates (though Newton and Alberuni had also discovered them). Jacob also did outstanding work in geometry, for example constructing perpendicular lines which quadrisect a triangle.
7 JohannBernoulli(1667-1748) Switzerland
Johann Bernoulli learned from his older brother and Leibniz, and went on to become principal teacher to Leonhard Euler. He developed exponential calculus; together with his brother Jacob, he founded the calculus of variations. Johann contributed more to calculus, discovered L'Hôpital's Rule before L'Hôpital did, and made important contributions in physics, e.g. about vibrations, elastic bodies, optics, tides, and ship sails.
8 Pierre deFermat(1601-1665) France
Pierre de Fermat was the most brilliant mathematician of his era and. Although mathematics was just his hobby (Fermat was a government lawyer), Fermat practically founded Number Theory, and also played key roles in the discoveries of Analytic Geometry and Calculus. Lagrange considered Fermat, rather than Newton or Leibniz, to be the inventor of calculus.
Fermat's most famous discoveries in number theory include the ubiquitously-usedFermat's Little Theorem; then = 4case of his conjecturedFermat's Last Theorem; andFermat's Christmas Theoremwhich may be considered the most difficult theorem of arithmetic which had been proved up to that date. Another famous conjecture by Fermat is that every natural number is the sum of three triangle numbers, or more generally the sum of k k-gonal numbers.
Fermat developed methods of differential and integral calculus. Although Kepler anticipated it, Fermat is credited with Fermat's Theorem on Stationary Points (df(x)/dx = 0at function extrema), the key to many problems in applied analysis. Fermat was also the first European to find the integration formula for the general polynomial; he used his calculus to find centers of gravity, etc.
10 Johnvon Neumann(1903-1957) Hungary, U.S.A.
John von Neumann was a childhood prodigy who could do very complicated mental arithmetic at an early age. Von Neumann pioneered the use of models in set theory. He proved a generalized spectral theorem sometimes called the most important result in operator theory. He developed von Neumann Algebras. He was first to state and prove the Minimax Theorem and thus invented game theory. He also worked in analysis, matrix theory, measure theory, numerical analysis, ergodic theory, group representations, continuous geometry, statistics and topology.
Von Neumann did very important work in fields other than pure mathematics. By treating the universe as a very high-dimensional phase space, he constructed an elegant mathematical basis (now called von Neumann algebras) for the principles of quantum physics. He advanced philosophical questions about time and logic in modern physics. He played key roles in the design of conventional, nuclear and thermonuclear bombs; he also advanced the theory of hydrodynamics. He applied game theory and Brouwer's Fixed-Point Theorem to economics, becoming a major figure in that field. His contributions to computer science are many: in addition to co-inventing the stored-program computer, he was first to use pseudo-random number generation, finite element analysis, the merge-sort algorithm, a "biased coin" algorithm, and Monte Carlo simulation. At the time of his death, von Neumann was working on a theory of the human brain.
11 Leonardo `Bigollo'Pisano (Fibonacci)(ca 1170-1245) Italy
Fibonacci relayed the mathematics of the Hindus, Persians, and Arabs. His writings cover a very broad range including new theorems of geometry, methods to construct and convert Egyptian fractions, irrational numbers, the Chinese Remainder Theorem, theorems about Pythagorean triplets, and the series 1, 1, 2, 3, 5, 8, 13, ....which is now linked with the name Fibonacci.
Leonardo provided Europe with the decimal system, algebra and the 'lattice' method of multiplication. It seems hard to believe but before the decimal system, mathematicians had no notation for zero.
Some histories describe him as bringing Islamic mathematics to Europe, but in Fibonacci's own preface toLiber Abaci, he specifically credits the Hindus.
12 Muhammed `Abu Jafar' ibnMusâal-Khowârizmi(ca 780-850) Khorasan (Uzbekistan), Iraq
Al-Khowârizmi (aka Mahomet ibn Moses) was a Persian who worked as a mathematician, astronomer and geographer early in the Golden Age of Islamic science. He introduced the Hindu decimal system to the Islamic world and Europe; invented the horary quadrant; improved the sundial; developed trigonometry tables; and improved on Ptolemy's astronomy and geography. He wrote the bookAl-Jabr, which demonstrated simple algebra and geometry, and several other influential books. He is often called the "Father of Algebra."
He also coined the wordcipher, which became Englishzero(although this was just a translation from the Sanskrit word for zero introduced by Aryabhata). He was an essential pioneer for Islamic science, and for the many Arab and Persian mathematicians who followed. Al-Khowârizmi's texts on algebra and decimal arithmetic are considered to be among the most influential writings ever.
13 Ya'qub `Abu Yusuf' ibnIshaqal-Kindi(803-873) Iraq
In mathematics Al-Kindi popularized the use of the decimal system, developed spherical geometry, wrote on many other topics and was a pioneer of cryptography (code-breaking).
14 Al-SabiThabitibnQurraal-Harrani(836-901) Harran, Iraq
Thabit produced important books in philosophy (including perhaps the famous mystic workDe Imaginibus), medicine, mechanics, astronomy, and especially several mathematical fields: analysis, non-Euclidean geometry, trigonometry, arithmetic, number theory. As well as being an original thinker, Thabit was a key translator of ancient Greek writings; he translated Archimedes' otherwise-lostBook of Lemmasand applied one of its methods to construct a regular heptagon.
He worked in plane and spherical trigonometry, and with cubic equations. He was an earlier practitioner of calculus and seems to have been first to take the integral of√x. Like Archimedes, he was able to calculate the area of an ellipse, and to calculate the volume of a paraboloid.
Thabit also worked in number theory where he is especially famous for his theorem about amicable numbers. (ThabitibnQurra's Theorem was rediscovered by Fermat and Descartes, and later generalized by Euler.) While many of his discoveries in geometry, plane and spherical trigonometry, and analysis (parabola quadrature, trigonometric law, principle of lever) duplicated work by Archimedes and Pappus, Thabit's list of novel achievements is impressive. Among the several great and famous Baghdad geometers, Thabit may have had the greatest genius.
15 Ibrahim ibnSinanibnThabitibnQurra(908-946) Iraq
IbnSinan, grandson of ThabitibnQurra, was one of the greatest Islamic. He was an early pioneer of analytic geometry, advancing the theory of integration, applying algebra to synthetic geometry, and writing on the construction of conic sections. He produced a new proof of Archimedes' famous formula for the area of a parabolic section. He worked on the theory of area-preserving transformations, with applications to map-making. He also advanced astronomical theory, and wrote a treatise on sundials.
16 Mohammed ibn al-Hasn (Alhazen) `Abu Ali' ibn al-Haythamal-Basra (965-1039) Iraq, Egypt
Al-Hassan ibn al-Haytham (Alhazen) made contributions to math, optics, and astronomy. Alhazen has been called the "Father of Modern Optics," the "Founder of Experimental Psychology" (mainly for his work with optical illusions).
In number theory, Alhazen worked with perfect numbers, Mersenne primes, and the Chinese Remainder Theorem. He stated Wilson's Theorem (which is sometimes called Al-Haytham's Theorem). Alhazen introduced the Power Series Theorem (later attributed to Jacob Bernoulli). His best mathematical work was with plane and solid geometry, especially conic sections; he calculated the areas of lunes, volumes of paraboloids, and constructed a heptagon using intersecting parabolas. He solved Alhazen's Billiard Problem (originally posed as a problem in mirror design). To solve it, Alhazen needed to anticipate Descartes' analytic geometry, anticipate Bézout's Theorem, tackle quartic equations and develop a rudimentary integral calculus. Alhazen's attempts to prove the Parallel Postulate make him (along with ThabitibnQurra) one of the earliest mathematicians to investigate non-Euclidean geometry.
17 Omaral-Khayyám(1048-1123) Persia
Omar Khayyám (aka Ghiyasod-Din Abol-Fath Omar ibnEbrahim Khayyam Neyshaburi) was one of the greatest Islamic mathematicians. He did clever work with geometry, developing an alternate to Euclid's Parallel Postulate and then deriving the parallel result using theorems based on theKhayyam-Saccheri quadrilateral. He derived solutions to cubic equations using the intersection of conic sections with circles. Khayyám did even more important work in algebra, writing an influential textbook, and developing new solutions for various higher-degree equations. He may have been first to develop Pascal's Triangle (which is still called Khayyám's Triangle in Persia), along with the essential Binomial Theorem (Al-Khayyám's Formula): (x+y)n= n!Σxkyn-k/ k!(n-k)!
Khayyám was also an important astronomer; he measured the year far more accurately than ever before, improved the Persian calendar, built a famous star map, and believed that the Earth rotates on its axis. He was a polymath: in addition to being a philosopher of far-ranging scope, he also wrote treatises on music, mechanics and natural science. Despite his great achievements in algebra, geometry, astronomy, and philosophy, today Omar al-Khayyám is most famous for his rich poetry (The Rubaiyat of Omar Khayyám).
18 Joseph-Louis (Comte de)Lagrange(1736-1813) Italy, France
Lagrange excelled in all fields of analysis and number theory; he made key contributions to the theories of determinants, continued fractions, and many other fields. He developed partial differential equations far beyond those of D. Bernoulli and d'Alembert, developed the calculus of variations far beyond that of the Bernoullis, discovered the Laplace transform before Laplace did, and developed terminology and notation (e.g. the use off'(x)andf''(x)for a function's 1st and 2nd derivatives).
He laid the foundations for the theory of polynomial equations which Cauchy, Abel, Galois and Poincaré would later complete. Number theory was almost just a diversion for Lagrange, whose focus was analysis; nevertheless he was the master of that field as well, proving difficult and historic theorems including Wilson's Theorem; Lagrange's Four-Square Theorem; and thatn·x2+ 1 = y2has solutions for every positive non-square integern.
Lagrange once wrote "As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when these two sciences have been united, they have lent each mutual forces, and have marched together towards perfection."
19 Pierre-Simon (Marquis de)Laplace(1749-1827) France
Laplace was the preeminent mathematical astronomer, and is often called the "French Newton." He advanced the nebular hypothesis of solar system origin, and was first to conceive of black holes. (He also conceived of multiple galaxies, but this was Lambert's idea first.)
Laplace viewed mathematics as just a tool for developing his physical theories. Nevertheless, he made many important mathematical discoveries and inventions. He was the premier expert at differential and difference equations, and definite integrals. He developed spherical harmonics, potential theory, and the theory of determinants; anticipated Fourier's series; and advanced Euler's technique of generating functions. In the fields of probability and statistics he made key advances: he proved the Law of Least Squares, and introduced the controversial ("Bayesian") rule of succession. In the theory of equations, he was first to prove that any polynomial of even degree must have a real quadratic factor.
Laplace is famous for skipping difficult proof steps with the phrase "It is easy to see". He was surely one of the greatestappliedmathematicians ever.
20 Siméon DenisPoisson(1781-1840) France
Siméon Poisson was a protégé of Laplace and, like his mentor, is among the greatest applied mathematicians ever. Poisson was an extremely prolific researcher and also an excellent teacher. In addition to important advances in several areas of physics, Poisson made key contributions to Fourier analysis, definite integrals, path integrals, statistics, partial differential equations, calculus of variations and other fields of mathematics.
Dozens of discoveries are named after Poisson; for example the Poisson summation formula which has applications in analysis, number theory, lattice theory, etc. He was first to note the paradoxical properties of the Cauchy distribution. He made improvements to Lagrange's equations of celestial motions, which Lagrange himself found inspirational.