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Muhammed `Abu Jafar' ibn Musâ Al-Khowârizmi (ca 780-850) Persia, Iraq :

Al-Khowarizmi 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. He wrote the book Al-Jabr, which demonstrated simple algebra and geometry, and several other influential books including ones on calculating with the decimal system, trigonometry, geography, astronomy, the Hebrew calendar, etc. The word algorithm is borrowed from Al-Khowarizmi's name.

Maria Agnesi (1718-1799) Italian :

When Maria's mother died, Maria took it upon herself to educate her brothers. As a result of tutoring and teaching her brothers, she developed a text for them which became the publication that made her famous. Although it took over ten years for her work to be published, it was the first surviving works to be done by a female. The two volume text was over 1000 pages in elementary and advanced mathematics. The first volume focuses on arithmetic, algebra, trig, analytic geometry and calculus. The second volume focused on more advanced topics: infinite series and differential equations. Maria Agnesi also became the first professor of mathematics in a university. Maria continued at the university until the death of her father in 1752. He was her inspiration in her pursuit of mathematics, after he died, she left math (perhaps retained it as a hobby) and devoted the rest of her life to the poor, homeless and sick.

Aryabhata I (476 – 550 AD) :

Aryabhata I was an Indian mathematician who wrote the Aryabhatiya which summarizes Hindu mathematics up to that 6th Century. The Aryabhatiya which is a small astronomical treatise written in 118 versus giving a summary of Hindu mathematics up to that time. Its mathematical section contains 33 versus giving 66 mathematical rules without proof. The Aryabhatiya contains an introduction of 10 verses, followed by a section on mathematics with, as we just mentioned, 33 verses, then a section of 25 verses on the reckoning of time and planetary models, with the final section of 50 verses being on the sphere and eclipses.

Archimedes of Syracuse (287-212 BC) Greece :

Archimedes studied at Euclid's school (probably after Euclid's death), but his work far surpassed the works of Euclid. Archimedes made advances in number theory and algebra, but his greatest contributions were in geometry. His most remarkable construction was the "squaring of the parabola", much more difficult than the angle trisection for which he is also famous. (His angle trisection requires a markable straightedge.) His achievements are particularly impressive given the lack of good mathematical notation in his day. Archimedes' methods anticipated both the integral and differential calculus. His original achievements in physics include the principles of leverage, the first law of hydrostatics, and inventions like the compound pulley, the hydraulic screw, and war machines. Archimedes proved that the volume of a sphere is two-thirds the volume of a circumscribing cylinder. He requested that a representation of such a sphere and cylinder be inscribed on his tomb.

Apollonius of Perga (262-190 BC) Greece :

Apollonius, called "The Great Geometer," is widely considered to be the second-greatest of ancient mathematicians, behind only Archimedes. His writings on conic sections have been studied until modern times; he invented the names for parabola, hyperbola and ellipse; he developed methods for normal and curvature. Although astronomers eventually concluded it was not physically correct, Apollonius developed the ancient model of planetary orbits, and proved important theorems in this area. Since many of his works have survived only in a fragmentary form, several great modern mathematicians, including Fermat, have enjoyed reconstructing and reproving his "lost" theorems. (Among these, the most famous is to construct a circle tangent to three other circles.)

Niels Henrik Abel (1802-1829) Norway :

His most famous achievement was the (deceptively simple) Abel's Theorem of Convergence (published posthumously), one of the most important theorems in analysis; but there are several other Theorems which bear his name. Abel also made contributions in algebraic geometry and the theory of equations. Inversion (replacing y = f(x) with x = f-1(y)) is a key idea in mathematics (consider Newton's Fundamental Theorem of Calculus); Finding the roots of polynomials is a key mathematical problem: the general solution of the quadratic equation was known by ancients; the discovery of general methods for solving polynomials of degree three and four is usually treated as the major math achievement of the 16th century; so for over two centuries an algebraic solution for the general 5th-degree polynomial (quintic) was a Holy Grail sought by most of the greatest mathematicians. Abel proved that most quintics did not have such solutions. This discovery, at the age of only nineteen, would have quickly awed the world, but Abel was impoverished, had few contacts, and spoke no German.

Peter Barlow (1776-1862), England :

Peter Barlow was born in Norwich, England in 1776 and died on March 1, 1862 in Kent, England. Peter's area of math focus tended to be in Number Theory. In 1806 he became the Mathematical Master at Woolwich Academy, a position he held for forty-one years. His early works were published in the 'Ladies Diary' in 1801; he soon became a well known mathematician and began publishing his articles for encyclopedias. The Barlow Tables were published in 1814. When initially released, these tables were called 'New Mathematical Tables. The tables give factors, squares, cubes, square roots, reciprocals and hyperbolic logarithms of all numbers from 1 to 10 000. Peter Barlow was known for his attention to detail and accuracy. The tables were used repeatedly and were reprinted several times. If not for computers and calculators, they would be used today due to their accuracy. In fact, they are still available!

Bháscara Áchárya (1114-1185) India :

Bhascara (often called Bhaskara II) may have been the greatest of the Hindu mathematicians. He made achievements in several fields of mathematics including some Europe wouldn't learn until the time of Euler. His textbooks dealt with many matters, including solid geometry, combinations, and advanced arithmetic methods. He was also an astronomer, and wrote a comprehensive textbook on planetary motions that would not be equaled in Europe for many centuries. In algebra, he solved various equations including 2nd-order Diophantine, quadratic, Brouncker's and Pell's equations; some of his solutions were superior to those discovered in 17th century Europe. In analysis he took derivatives of polynomial and trigonometric functions, used Rolle's Theorem, and anticipated integral calculus. In trigonometry, which he valued for its own beauty as well as practical applications, he developed spherical trig and was first to present the identity sin(a + b) = sin(a)cos(b) + sin(b)cos(a)

George Boole (1815-1864) England :

Although it was his work in analysis that brought him fame in his own lifetime, it is his work in Boolean algebra and symbolic logic that eventually influenced computer scientists like Claude Shannon. Boole's book An Investigation of the Laws of Thought prompted Bertrand Russell to label him the "discoverer of pure mathematics."

Brahmagupta `Bhillamalacarya' (589-668) Rajasthan (India) :

No one person gets unique credit for the invention of the decimal system but Brahmagupta's textbook Brahmasphutasiddhanta was most influential, and is sometimes considered the first textbook "to treat zero as a number in its own right." It also treated negative numbers. Brahmagupta Bhillamalacarya (`The Teacher from Bhillamala') made great advances in arithmetic, algebra, numeric analysis, and geometry. Several theorems bear his name, including the formula for the area of a cyclic quadrilateral, which can be written: 16 A2 = (a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d) Another famous Brahmagupta theorem dealing with such quadrilaterals can be phrased "In a circle, if the chords AB and CD are perpendicular and intersect at E, then the line from E which bisects AC will be perpendicular to BD." Proving Brahmagupta's theorems are good challenges even today. In addition to his famous writings on practical mathematics and his ingenious theorems of geometry, Brahmagupta solved the general quadratic equation, and worked on Diophantine and Pell's equations. He proved Brahmagupta's identity (the set of sums of two squares is closed under multiplication). He applied mathematics to astronomy, predicting eclipses, etc.

Luitzen Egbertus Jan Brouwer (1881-1966) Holland :

Brouwer is often considered the Father of Topology; his two most important theorems being the Fixed Point Theorem, and the Invariance of Dimension. He developed the method of simplified approximations, important to algebraic topology; he also did work in geometry, set theory, measure theory, complex analysis and the foundations of mathematics. Brouwer is most famous as the founder of Intuitionism, a philosophy of mathematics in sharp contrast to Hilbert's Formalism, but Brouwer's philosophy also involved ethics and aesthetics and has been compared with that of Schopenhauer and Nietzsche. Part of his mathematics thesis was rejected as "... interwoven with some kind of pessimism and mystical attitude to life which is not mathematics ..." As a young man, Brouwer spent a few years to develop topology, but once his great talent was demonstrated and he was offered prestigious professorships, he devoted himself to Intuitionism, and acquired a reputation as eccentric and self-righteous. Intuitionism has had an important influence, although few strict adherents; since only constructive proofs are permitted, strict adherence would slow mathematical work. This didn't worry Brouwer who once wrote: "The construction itself is an art, its application to the world an evil parasite."

George Cantor (1845-1918) Russia, Germany :

Cantor single-handedly created modern set theory, defining cardinal numbers, well-ordering, ordinal numbers, and discovering the Theory of Transfinite Numbers. He defined equality between cardinal numbers based on the existence of a bijection, and was the first to demonstrate that the real numbers have a higher cardinal number than the integers. (The rationals have the same cardinality as the integers; the reals have the same cardinality as the points of N-space.) Although there are infinitely many distinct transfinite numbers, Cantor conjectured that C, the cardinality of the reals, was the second smallest transfinite number. This "Continuum Hypothesis" was included in Hilbert's famous List of Problems, and was finally resolved many years later: Cantor's Continuum Hypothesis is an "Undecidable Statement" of Set Theory. Cantor's invention of modern set theory is now considered one of the most important and creative achievements in modern mathematics. Cantor also made advances in number theory and trigonometric series. He gave the modern definition of irrational numbers, and anticipated the theory of fractals. Cantor once wrote "In mathematics the art of proposing a question must be held of higher value than solving it."

Girolamo Cardano (1501-1576) Italy :

Cardano was a highly respected physician and was first to describe typhoid fever. He was also a remarkable inventor: the combination lock, the gimbal, a ciphering tool, and the Cardan shaft with universal joints are all his inventions and are in use to this day. He made contributions to physics; he noted that projectile trajectories are parabolas, and may have been first to note the impossibility of perpetual motion machines. He did work in philosophy, geology, hydrodynamics, music; he wrote books on medicine and an encyclopedia of natural science. He was first to publish general solutions to cubic and quadratic equations (though these were largely based on others' work). He introduced complex numbers, although he did not develop their theory further. He introduced binomial coefficients and the binomial theorem, and introduced and solved the geometric hypocycloid problem.

Augustin-Louis Cauchy (1789-1857) France :

Cauchy did important work in analysis, algebra and number theory. One of his important contributions was the "theory of substitutions" (permutation group theory). Cauchy's research also included convergence of infinite series, differential equations, determinants, and probability. He invented the calculus of residues. Although he was one of the first great mathematicians to focus on abstract mathematics (another was Euler), he also made important contributions to mathematical physics, e.g. the theory of elasticity. He was the first to prove Fermat's conjecture that every positive integer can be expressed as the sum of k k-gon numbers for any k, and also refined Euler's results in discrete topology. Another of Cauchy's contributions was his insistence on rigorous proofs.

Arthur Cayley (1821-1895) England :

Cayley was the third most prolific mathematician in history, behind only Euler and Erdos. A list of the branches of mathematics Cayley pioneered will seem like an exaggeration: he was the essential founder of modern group theory, matrix algebra, and higher dimensional geometry, as well as the theory of invariants. He stated and proved the Cayley-Hamilton Theorem. He also did original research in combinatorics, elliptic and Abelian functions, and projective geometry (one of his many famous theorems is a generalization of Pascal's Mystic Hexagram result). Cayley once wrote: "As for everything else, so for a mathematical theory: beauty can be perceived but not explained."

René Déscartes (1596-1650) France :

He invented analytic geometry and is therefore called the "Father of Modern Mathematics." Because of his famous philosophical writings ("Cogito ergo sum") he is considered, along with Aristotle, to be one of the most influential thinkers in history. Descartes made important contributions to physics (e.g. the law of conservation of momentum), and mathematical notation (e.g. the use of superscripts to denote exponents). His famous mathematical theorems include the Rule of Signs (for determining the signs of polynomial roots), the elegant formula relating the radii of Soddy kissing circles, and an improvement on the ancient construction method for cube-doubling.

Johann Peter Gustav Lejeune Dirichlet (1805-1859) Germany :

Fermat had proved the impossibility of (non-trivial) xk + yk = zk for k = 4; Euler proved it for k = 3; Dirichlet became famous by proving impossibility for k = 5 at the age of 20. Later he proved the case k = 14 and, later still, found the flaw in Kummer's proof of the general case (Kummer ignored that unique factorization does not hold for the quadratic fields Dirichlet had invented to address the problem). More important than his work with Fermat's Last Theorem was his Unit Theorem, considered one of the most important theorems of algebraic number theory; it is said that Dirichlet discovered the difficult proof while listening to music in the Sistine Chapel. A key step in the proof uses "Dirichlet's Pigeonhole Principle", a trivial idea but which Dirichlet applied with great ingenuity. Dirichlet also did important work in analysis and is considered the founder of analytic number theory. He invented a method of L-series to prove that any arithmetic series has infinity of primes. It was Dirichlet who proved the fundamental Theorem of Fourier series: that periodic analytic function can always be represented as a simple trigonometric series. Other fundamental results Dirichlet contributed to analysis are Dirichlet's Principle and his Class Number Formula.

Eudoxus of Cnidus (408-355 BC) Asia Minor, Greece :

He developed the earliest techniques of the infinitesimal calculus. Eudoxus (or Pythagoras?) was the first person known to have recognized that the Earth rotates around the Sun. Four of Eudoxus' most famous discoveries were the volume of a cone, extension of arithmetic to the irrationals, summing formula for geometric series, and viewing p as the limit of polygonal perimeters. Eudoxus has been quoted as saying "Willingly would I burn to death like Phaeton, was this the price for reaching the sun and learning its shape, its size and its substance."

Euclid of Megara & Alexandria (ca 322-275 BC) Greece/Egypt :

Euclid may have been a student of Aristotle. He founded the school of mathematics at the great university of Alexandria. He was the first to prove that there are infinitely many prime numbers, proved the unique factorization theorem ("Fundamental Theorem of Arithmetic"), and established the relationship between perfect numbers and Mersenne primes. Among several books attributed to him are The Division of the Scale (a mathematical discussion of music), The Optics, The Cartoptrics (a treatise on the theory of mirrors), and his comprehensive math textbook The Elements. Apparently Desargues' Homology Theorem (a pair of triangles is coaxial if and only if it is copolar) was proved in one of these lost works; this is the fundamental theorem which initiated the study of projective geometry. Euclid is also believed to have proved that there are only five "Platonic" solids. The Elements introduced the notions of axiom and theorem. It was used as a textbook for 2000 years and in fact is still the basis for high school geometry, making Euclid the leading mathematics teacher of all time. Some think his best inspiration was recognizing that the Parallel Postulate must be an axiom rather than a theorem.

Leonhard Euler (1707-1783) Switzerland :

Euler made decisive contributions in all areas of mathematics; he gave the world modern trigonometry. He invented graph theory and generating functions. Four of the most important constant symbols in mathematics (p, e, i = sqrt(-1), and ? = 0.57721566...) were all introduced or popularized by Euler. He is particularly famous for unifying the trigonometric and exponential functions with the equation: ei x = cos x + i sin x. He developed the first method to estimate the Moon's orbit (the three-body problem which had stumped Newton), and he settled an arithmetic dispute involving 50 decimal places of a long convergent series. Both these feats were accomplished when he was totally blind. As a young man, Euler discovered and proved the following: p2/6 = 1-2 + 2-2 + 3-2 + 4-2 + ... This striking identity catapulted Euler to instant fame, since the right-side infinite sum was a famous unsolved problem of the day.

Paul Erdös (1913-1996) Hungary, U.S.A., Israel :

He is best known for work in Ramsey Theory, but made contributions in many other fields of mathematics, including graph theory, analytic number theory, probabilistic methods, approximation theory, and combinatorics. He is regarded as the second most prolific mathematician in history, behind only Euler. (Euler actually published fewer papers than Erdos, but most of Erdos' papers had co-authors -- he was famous for collaboration.) Although he is widely regarded as an important and influential mathematician, Erdos founded no new field of mathematics: He was a "problem solver" rather than a "theory developer." Erdos liked to speak of "God's Book of Proofs" and discovered new, more elegant, proofs of several existing theorems, including the two most famous and important about prime numbers: that there is always a prime between n and 2n. and that (loosely) 1/ln(n) is the probability that n is prime.

Pierre de Fermat (1601-1665) France :

He made advances in both continuous and discrete mathematics, and practically founded modern number theory. Fermat is most remembered for Fermat's Little Theorem, ubiquitous in number theory, and for his conjectured Fermat's Last Theorem, but he did much other work as well. Much of his work was never published and there are several important theorems attributed to him for which proofs had to be rediscovered; among these is that any prime (4n+1) can be represented as the sum of two squares in exactly one way. (Fermat records that he proved this with difficulty using his method of "infinite descent." Because of the "Last Theorem", which was actually just a private scribble, many wrongly suppose that Fermat's work abounds with false or unproven conjectures.) Fermat developed a system of analytic geometry which both preceded and surpassed that of Descartes; he developed methods of differential and integral calculus which Newton acknowledged as an inspiration. Solving df(x)/dx = 0 to find extrema of f(x) is perhaps the most useful idea in applied mathematics; this technique originated with Fermat. In collaboration with Blaise Pascal, Fermat founded the theory of probability. Fermat anticipated the principle of least action, which Lagrange and Hamilton would later develop, and used it to establish basic principles of optics.

Évariste Galois (1811-1832) France :

Galois, died before the age of twenty-one. He applied group theory to the theory of equations, revolutionizing both fields. (Galois coined the mathematical term "group.") While Abel was the first to prove that some polynomial equations had no algebraic solutions, Galois established the necessary and sufficient condition for algebraic solutions to exist. His principle treatise was a letter he wrote the night before his fatal duel, of which Hermann Weyl wrote: "This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind."

William Rowan (Sir) Hamilton (1805-1865) Ireland :

Hamilton also made revolutionary contributions to dynamics, differential equations, the theory of equations, numerical analysis, fluctuating functions, and graph theory (he marketed a puzzle based on his Hamiltonian paths). He invented the ingenious hodograph. Hamilton himself considered his greatest accomplishment to be the development of quaternions, a non-Abelian field to handle 3-D rotations. While there is no 3-D analog to the Gaussian complex-number plane (based on the equation i2 = -1 ), quaternions derive from a 4-D analog based on i2 = j2 = k2 = ijk = -1. (Despite their being "obsoleted" by more general matrix and tensor methods, quaternions are still in wide engineering use because of certain practical advantages.) Hamilton once wrote: "On earth there is nothing great but man; in man there is nothing great but mind."

Johann Carl Friedrich Gauss (1777-1855) Germany :

Carl Friedrich Gauss, the "Prince of Mathematics," exhibited his calculative powers when he corrected his father's arithmetic before the age of three. His revolutionary nature was demonstrated at age twelve, when he began questioning the axioms of Euclid. His genius was confirmed at the age of nineteen when he proved that the regular n-gon was constructible, for odd n, if and only if n is the product of distinct prime Fermat numbers. At age 24 he published Disquisitiones Arithmeticae, probably the greatest book of pure mathematics ever. Gauss built the theory of complex numbers into its modern form, including the notion of "monogenic" functions which are now ubiquitous in mathematical physics. The other contributions of Gauss are quite numerous and include the Fundamental Theorem of Algebra (that an n-th degree polynomial has n complex roots), hypergeometric series, foundations of statistics, and differential geometry.

Godfrey Harold Hardy (1877-1947) England :

Hardy was an extremely prolific research mathematician who did important work in analysis (especially the theory of integration), number theory, global analysis, and analytic number theory. He was also an excellent teacher and wrote several excellent textbooks, as well as a famous treatise on the mathematical mind. He wrote "I am interested in mathematics only as a creative art." Although he emphasized pure mathematics (actually abhorring applied mathematics), his work has found application in population genetics, cryptography, thermodynamics and particle physics. Hardy is especially famous (and important) for his encouragement of and collaboration with Ramanujan. Among many results of this collaboration was the Hardy-Ramanujan Formula for partition enumeration, which Hardy later used as a model to develop the Hardy-Littlewood Circle Method; Hardy first used this method to prove stronger versions of the Hilbert-Waring theorem, and in prime number theory; the method has continued to be a very productive tool in analytic number theory.

Charles Hermite (1822-1901) France :

Along with Cayley and Sylvester, he founded the important theory of invariants. He was a kindly modest man who inspired his colleagues. Although he and Abel had proved that the general quintic lacked algebraic solutions, Hermite introduced an elliptic analog to the circular trigonometric functions and used these to provide a general solution for the quintic equation. He developed the concept of complex conjugate which is now ubiquitous in mathematical physics and matrix theory. He was first to prove that the Stirling and Euler generalizations of the factorial function are equivalent. Perhaps Hermite's most famous result was the proof that a broad class of numbers, including e, is transcendental.

David Hilbert (1862-1943) Prussia, Germany :

Hilbert was preeminent in many fields of mathematics, including axiomatic theory, invariant theory, algebraic number theory, class field theory and functional analysis. His examination of calculus led him to the invention of "Hilbert space," considered one of the key concepts of functional analysis and modern mathematical physics. He was a founder of fields like metamathematics and modern logic, and is sometimes considered the founder of the "formalist" school. He developed a new system of definitions and axioms for geometry, replacing the 2200 year-old system of Euclid. As a young Professor he proved his "Finiteness Theorem," now regarded as one of the most important results of general algebra. The methods he used were so novel that, at first, the "Finiteness Theorem" was rejected for publication as being "theology" rather than mathematics! In number theory, he proved Waring's famous conjecture which is now known as the Hilbert-Waring theorem. Hilbert provided a famous List of 23 Unsolved Problems, which inspired and directed the development of 20th-century mathematics.

Kurt Gödel (1906-1978) Germany, U.S.A :

Godel, who had the nickname Herr Warum ("Mr. Why") as a child, was perhaps the foremost logic theorist ever, clarifying the relationships between various modes of logic. He partially resolved both Hilbert's 1st and 2nd Problems, the latter with a proof so remarkable that it was connected to the drawings of Escher and music of Bach in the title of a famous book. He was a close friend of Albert Einstein, and was first to discover "paradoxical" solutions (e.g. time travel) to Einstein's equations. Two of the major questions confronting mathematics are: (1) are its axioms consistent (its theorems all being true statements)?, and (2) are its axioms complete (its true statements all being theorems)? Godel turned his attention to these fundamental questions. He proved that first-order logic was indeed complete, but that the more powerful axiom systems needed for arithmetic (constructible set theory) were necessarily incomplete. He also proved that the Axioms of Choice (AC) and the Generalized Continuum Hypothesis (GCH) were consistent with set theory, but that set theory's own consistency could not be proven. In Godel's famous proof of Incompleteness, he exhibits a true statement (G) which cannot be proven, to wit "G (this statement itself) cannot be proven." If G could be proven it would be a contradictory true statement, so consistency dictates that it indeed cannot be proven. But that's what G says, so G is true! This sounds like mere word play, but building from ordinary logic and arithmetic Godel was able to construct statement G rigorously.

Alexander Grothendieck (1928-) Germany, France :

Grothendieck has done brilliant work in several areas of mathematics including number theory, functional (and topological) analysis, but especially in the fields of algebraic geometry and category theory, both of which he revolutionized. He is considered a master of abstraction, rigor and presentation. He has produced many important and deep results in homological algebra, most notably his etale cohomology. e developed the theory of sheafs, invented the theory of schemes, and much more. He is most famous for his methods to unify different branches of mathematics, for example using algebraic geometry in number theory.

Christiaan Huygens (1629-1695) Holland, France :

Christiaan Huygens (or Hugens or Huyghens) was one of the greatest scientists during Europe's Renaissance. He developed laws of motion before Newton, including the inverse-square law of gravitation, centripetal force, treatment of solid bodies rather than point approximations. He advanced the wave ("undulatory") theory of light, a key concept being Huygen's Principle that each point on a wave front acts as a new source of radiation. (Because of Newton's high reputation and corpuscular theory of light, Huygens' superior wave theory was largely ignored for almost two centuries.) His optical discoveries include explanations for polarization and phenomena like haloes. Huygens is famous for his inventions of clocks and lenses. He designed the first reliable pendulum clock, and the first balance spring watch. He invented superior lens grinding techniques, the achromatic eye-piece, and the best telescope of his day. As a mathematician, he was first to show that the cycloid solves the tautochrone problem; he used this fact to design an (impractical) compound pendulum clock that would be more accurate than an ordinary pendulum clock. He was first to find the flaw in Saint-Vincent's then-famous circle-squaring method; Huygens himself solved some related quadrature problems. He introduced the concepts of evolute and involute.

Carl G. J. Jacobi (1804-1851) Germany :

Jacobi was a prolific mathematician who did decisive work in the algebra and analysis of complex variables, and did work in number theory (e.g. cubic reciprocity) which excited Carl Gauss. Jacobi's most important early achievement was the theory of elliptic functions. He also made important advances in many other areas, including higher fields, number theory, algebraic geometry, differential equations, theta functions, q-series, determinants, Abelian functions, and physics. He devised the algorithms still used to calculate eigenvectors and for other important matrix manipulations. Jacobi was the first to apply elliptic functions to number theory, producing a new proof of Fermat's famous conjecture (Lagrange's theorem) that every integer is the sum of four squares.

Shri Dattathreya Ramachandra Kaprekar (1905-1986) India :

Shri Dattathreya Ramachandra Kaprekar was born on January 17, 1905 in Dahanu which is near Mumbai, India. Recreational math became his hobby as a child he enjoyed spending time solving math puzzles and problems. In 1946 he discovered Kaprekar's Constant which was named after him. The Constant is 6174. Here's how it works: You can take any four-digit number and re-arrange the digits in decreasing order. All digits MUST be different. We'll use 4521 - let's order the digits from highest to lowest which gives us 5421. Now take the number and order the digits from lowest to highest and subtract from the number you ordered from high to low.(Repeat the process until you come to the Constant of 6174) Original number: 4521 5421-1245 = 4176 7641-1467 = 6174 After going through the process twice, we reach 6174. Try another 4 digit number: 9472 9742-2479 = 7263 7632-2367 = 5265 6552-2556 = 3996 9963-3699 = 6264 6642-2466 = 4176 7641-1467 = 6174 What happens when you keep repeating the process? What did you notice when you end up getting 2 digits that are the same through the process? Can you find a number that requires the greatest amount of subtractions? What happens if you try this on a 3 digit number?

Omar al-Khayyám (ca 1048-1130) Persia :

Khayyam did clever work with geometry, developing an alternate to Euclid's Parallel Postulate and then deriving the parallel result using theorems based on the Khayyam-Saccheri quadrilateral. He derived solutions to cubic equations using the intersection of conic sections with circles. Remarkably, he stated that the cubic solution could not be achieved with straightedge and compass, a fact that wouldn't be proved until the 19th century. Khayyam did even more important work in algebra, writing an influential textbook, and developing new solutions for various higher-degree equations. He discovered the binomial coefficients. His symbol ('shay') for an unknown in an algebraic equation was transliterated to become our 'x'. Khayyam was also an important astronomer, measuring the year far more accurately than ever before, improving the Persian calendar, and building a famous star map. Today Omar al-Khayyam is most famous for his rich poetry (The Rubaiyat of Omar Khayyam).

Christian Felix Klein (1849-1925) Germany :

Klein's key contribution was an application of invariant theory to unify geometry with group theory. This radical new view of geometry inspired Sophus Lie's Lie groups, and also led to the remarkable unification of Euclidean and non-Euclidean geometries which is probably Klein's most famous result. Klein did other work in function theory, providing links between several areas of mathematics including number theory, group theory, hyperbolic geometry, and abstract algebra. His famous "Klein's bottle" was just one of many results from his new approach to higher-dimensional geometries and equations. Klein did important work in mathematical physics, e.g. writing about gyroscopes. Klein is also famous for his book on the icosahedron, reasoning from its symmetries to develop the elliptic modular and automorphic functions which he used to solve the general quintic equation. He formulated a "grand uniformization theorem" about automorphic functions but suffered a health collapse before completing the proof. His focus then changed to teaching; he devised a mathematics curriculum for secondary schools which had world-wide influence.

Andrey Nikolaevich Kolmogorov (1903-1987) Russia :

Kolmogorov had a powerful intellect and excelled in many fields. As a youth he dazzled his teachers by constructing toys that appeared to be "Perpetual Motion Machines." At the age of 19, he achieved fame by finding a Fourier series that diverges almost everywhere, and decided to devote himself to mathematics. He is considered the founder of the fields of intuitionistic logic, algorithmic complexity theory, and modern probability theory. He also excelled in topology, set theory, trigonometric series, and random processes. He (and his student) resolved Hilbert's 13th Problem. While Kolmogorov's work in probability theory had direct applications to physics, Kolmogorov also did work in physics directly, especially the study of turbulence. There are dozens of theorems or equations named after Kolmogorov, such as the "Kolmogorov backward equation" and the intriguing Zero-One Law of "tail events" among random variables.

Joseph-Louis (Comte de) Lagrange (1736-1813) Italy/France :

He 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 invented partial differential equations, and the calculus of variations. He proved a fundamental Theorem of Group Theory, as well as two number theory theorems of great historic interest: Wilson's prime-number theorem, and the fact that every positive integer is the sum of four squares. He laid the foundations for the theory of polynomial equations which Cauchy, Abel, Galois and Poincaré would later complete.

Pierre-Simon (Marquis de) Laplace (1749-1827) France :

Laplace was the preeminent mathematical astronomer, and is often called the "French Newton." His masterpiece was Mecanique Celeste which redeveloped and improved Newton's work on planetary motions using calculus. Hemade many important mathematical discoveries and inventions, most notably the Laplace Transform. He developed spherical harmonics, potential theory, the theory of determinants, and advanced Euler's technique of generating functions. In the fields of probability and statistics he made important 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.

Gottfried Wilhelm Leibniz (1646-1716) Germany :

He invented the concepts of matrix determinant and geometric envelope; he designed the first calculator that could do multiplication; and he discovered and proved the striking identity: p/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ... (This identity was independently discovered by others including, perhaps, Madhava three centuries earlier.)

Leonardo `Bigollo' Pisano (Fibonacci) (ca 1170-1245) Italy :

Leonardo (known today as Fibonacci) introduced new methods of arithmetic to Europe, and relayed the mathematics of the Hindus, Persians, and Arabs. He re-introduced older Greek ideas like Mersenne numbers and Diophantine equations, and made original contributions in geometry and number theory. His writings cover a broad range including methods to construct and convert Egyptian fractions (which were still in wide use), irrational numbers, the Chinese Remainder Theorem, theorems about Pythagorean triples, 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, all far superior to the methods then in use. His clever notation for quantities like 5 yards, 2 feet, and 3 inches is more efficient than today's notation.

Madhava of Sangamagrama (1340-1425) India :

Madhava, also known as Irinjaatappilly Madhavan Namboodiri, is considered the founder of the important Kerala School of mathematics and astronomy. His analytic geometry preceded and surpassed Descartes', and included differentiation and integration. He also did work with continued fractions, trigonometry, and geometry. Madhava is most famous for his work with Taylor series, discovering identities like sin q = q - q3/3! + q5/5! - ... , and perhaps the formula for p attributed to Leibniz.

Gaspard Monge (1746-1818) France :

Gaspard Monge, son of a humble peddler, was an industrious and creative inventor who astounded early with his genius, becoming a professor of physics at age 16. Monge is most famous for laying the foundation for differential geometry. Monge's most famous theorems of geometry are the "Three Circles Theorem" and "Four Spheres Theorem."

Abraham de Moivre (1667-1754), France :

Abraham de Moivre, (a good friend of Issac Newton) was born on May 16th 1667 in Vitry (close to Paris), France and died November 27th 1754 in London, England. Although De Moivre attended college and studied privately, it doesn't appear that he received a degree. He apparently served time in prison for about a year for being protestant, after serving his term and with the expulsion of the Huguenots, he emingrated to England. During his late teens, he worked as a private math tutor. In 1697 he was elected a fellow of the Royal Society. By 1710 he was appointed to the Commission set up by the Royal Society to review the claims of Newton and Leibniz who eventually discovered calculus. de Moivre was a foreigner which makes it difficult to gain an appointment with the Commission, however, due to his friendship with Newton, he was appointed. . De Moivre pioneered the analytic trigonometry/geometry and the theory of probability. He is famous for De Moivre's Formula.

John Napier (1550- ) Scotland :

John Napier was a Scottish mathematician and inventor. Napier is famous for creating mathematical logarithms, creating the decimal point, and for inventing Napier's Bones, a calculating instrument. Napier a mathematician and an inventor. He proposed several military inventions including: burning mirrors that set enemy ships on fire, special artillery that destroyed everything within a radius of four miles, bulletproof clothing, a crude version of a tank, and a submarine-like device. John Napier invented a hydraulic screw with a revolving axle that lowered water levels in coal pits. Napier also worked on agricultural innovations to improve crops with manures and salt. As a Mathematician, the highlight of John Napier's life was the creation of logarithms and the decimal notation for fractions. His other mathematical contributions included: a mnemonic for formulas used in solving spherical triangles, two formulas known as Napier's analogies used in solving spherical triangles, and the exponential expressions for trigonometric functions. In 1621, English mathematician and clergyman, William Oughtred used Napier's logarithms when he invented the slide rule. Oughtred invented the standard rectilinear slide rule and circular slide rule. Napier's bones was multiplication tables written on strips of wood or bones. The invention was used for multiplying, dividing, and taking square roots and cube roots.

John von Neumann (1903-1957) Hungary, U.S.A. :

John von Neuman (born Neumann Janos Lajos) 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, thus improving the axiomatic basis of mathematics; he developed von Neumann Algebras; he invented and developed game theory; he invented cellular automata, famously constructing a self-reproducing automaton. He also worked in analysis, operator theory, matrix theory, statistics and topology. He inspired some of Godel's famous work. He is credited with (partial) solution to Hilbert's 5th Problem. 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 a key role in the design of conventional, nuclear and thermonuclear bombs. 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, and a "biased coin" algorithm. At the time of his death he was working on a theory of the human brain.

Isaac (Sir) Newton (1642-1727) England :

His genius seems to have blossomed at about age 22 when, on leave from University, he began revolutionary advances in mathematics, optics, dynamics, thermodynamics, acoustics and celestial mechanics. (Newton's other intellectual interests included chemistry, theology, astrology and alchemy.) Although others also developed the techniques independently, Newton is regarded as the Father of Calculus (what he called the "method of fluxions"); his most crucial insight being what is now called the Fundamental Theorem of Calculus (that integration and differentiation are each other's inverse operation). He applied calculus to solve a variety of problems: finding areas, tangents, the lengths of curves and the maxima and minima of functions. In addition to several other important advances in analytic geometry, his mathematical works include the Binomial Theorem, his eponymous numeric method, the idea of polar coordinates, and power series for exponential and trigonometric functions. (His equation ex = sum xk / k! has been called the "most important series in mathematics.") He contributed to algebra and the theory of equations, proving facts about cubic equations, attempting generalization of Descartes' rule of signs, etc. He proved that same-mass spheres of any radius have equal gravitational attraction, a key to celestial motions.

Amalie Emma Noether (1882-1935) Germany :

Noether was an innovative researcher who made several major advances in abstract algebra, including a new theory of ideals, the inverse Galois problem, and the general theory of commutative rings. She originated novel reasoning methods, especially one based on "chain conditions," which advanced invariant theory and abstract algebra; her insistence on generalization led to a unified theory of modules and Noetherian rings; some of her work anticipated modern category theory. Her invention of homology groups revolutionized topology. Noether also made advances in mathematical physics; Noether's Theorem establishing that certain symmetries imply conservation laws has been called the most important Theorem in physics since the Pythagorean Theorem. Noether was an unusual and inspiring teacher; and generously helped students and colleagues, even allowing them to claim her work as their own. It is widely agreed that she was the greatest female mathematician ever.

Jules Henri Poincaré (1854-1912) France :

Poincaré was clumsy and frail and supposedly flunked an IQ test, but he was one of the most creative mathematicians ever, and surely the greatest mathematician of the Constructivist ("intuitionist") style. He produced a large amount of brilliant work in all areas of mathematics, but also found time to become a famous popular writer of philosophy, saying, for example, "Mathematics is the art of giving the same name to different things." Poincare's masterpieces include combinatorial (or algebraic) topology, the theory of differential equations, foundations of homology, the theory of periodic orbits, and the discovery of automorphic functions (a unifying foundation for the trigonometric and elliptic functions). He anticipated modern chaos theory. He posed "Poincare's conjecture," which for an entire century was one of the most famous unsolved problems in mathematics and which can be explained without equations to a layman (provided the layman can visualize 3-D surfaces in 4-space). Recently Poincare's conjecture was settled and the first Million Dollar math prize in history is likely to be awarded. As with all the greatest mathematicians, Poincaré was interested in physics. He made revolutionary advances in fluid dynamics and celestial motions. He is sometimes considered to be a co-inventor of the Special Theory of Relativity. With his fame, he helped the world recognize the importance of the new physical theories of Einstein and Planck.

Pappus of Alexandria (ca 300) Egypt, Greece :

Pappus may have been the greatest Western mathematician during the 14 centuries that separated Apollonius and Fibonacci. He wrote about arithmetic methods, plane and solid geometry, the axiomatic method, celestial motions and mechanics. Pappus presents several ingenious geometric theorems including Desargues' Homology Theorem, a special case of Pascal's Hexagram Theorem, and Pappus' Theorem itself (two projective pencils can always be brought into a perspective position). For these theorems, Pappus is sometimes called the "Father of Projective Geometry." Other ingenious theorems include an angle trisection method using a fixed hyperbola. He stated (but didn't prove) the Isoperimetric Theorem, also writing "Bees know this fact which is useful to them, that the hexagon ... will hold more honey for the same material than [a square or triangle]."

Blaise Pascal (1623-1662) France :

At the age of sixteen he stated and proved Pascal's Theorem, a fact relating any six points on any conic section. The Theorem is sometimes called the "Cat's Cradle" or the "Mystic Hexagram." Pascal followed up this result by showing that each of Apollonius' famous theorems about conic sections was a corollary of the Mystic Hexagram; along with Gerard Desargues (1591-1661), he was a key pioneer of projective geometry. Returning to geometry late in life, Pascal advanced the theory of the cycloid. In addition to his work in classic and projective geometry, he founded probability theory, made contributions to axiomatic theory, and the invention of calculus. His name is associated with the Pascal's Triangle of combinatorics and Pascal's Wager in theology. At the age of eighteen he designed and built the world's first automatic adding machine. (Although he continued to refine this invention, it was never a commercial success.)

Jean-Victor Poncelet (1788-1867) France :

• Poncelet became one of the most influential geometers ever; he is especially noted for his Principle of Continuity, an intuition with broad application. • His theorems of geometry include his Closure Theorem about Poncelet Traverses and the Poncelet-Brianchon Hyperbola Theorem. Perhaps his most famous theorem, although it was left to Steiner to complete a proof, is the beautiful Poncelet-Steiner Theorem about straight-edge constructions.

Pythagoras of Samos (ca 575-505 BC) Greece :

He became the most influential of early Greek mathematicians. He is credited with being first to use axioms and deductive proofs. Pythagoras is sometimes called the "First Philosopher." It is reported that Pythagoras proposed that the Earth is round (and perhaps that it rotates around the Sun), and believed thinking was located in the brain rather than heart. Pythagoras discovered that harmonious intervals in music are based on simple rational numbers. This led to a fascination with integers and mystic numerology; he is sometimes called the "Father of Numbers" and once said "Number rules the universe." The Pythagorean Theorem was known long before Pythagoras, but he is often credited with the first proof. (But Apastamba proved it in India at about the same time, and some historians believe Pythagoras journeyed to India and learned of the proof there.) He also discovered the simple parametric form of Pythagorean triplets (xx-yy, 2xy, xx+yy). Other discoveries of the Pythagorean school include the golden ratio (attributed to Theano), and irrational numbers (attributed to Hippasus). It is said that the discovery of irrational numbers upset the Pythagoreans so much they tossed Hippasus into the ocean!

Srinivasa Ramanujan Iyengar (1887-1920) India :

Like Abel, Ramanujan was a self-taught prodigy who lived in a country distant from his mathematical peers, and suffered from poverty: childhood dysentery and vitamin deficiencies probably led to his early death. Yet he produced 4000 theorems or conjectures in number theory, algebra, and combinatorics. His specialties included infinite series, elliptic functions, continued fractions, partition enumeration, definite integrals, modular equations, gamma functions, "mock theta" functions, hypergeometric series, and "highly composite" numbers. Much of his methodology, including unusual ideas about divergent series, was his own invention. (As a young man he made the absurd claim that 1+2+3+4+... = -1/12. Later it was noticed that this claim translates to a true statement about the Riemann zeta function, with which Ramanujan was unfamiliar.) Ramanujan's innate ability for algebraic manipulations equaled or surpassed that of Euler and Jacobi. Although many formulae have been discovered to calculate p, a bizarre formula of Ramanujan is often used, because of its fast convergence. Many of Ramanujan's results would probably never have been discovered without him, and are so inspirational that there is a periodical dedicated to them. The theories of strings and crystals have benefited from Ramanujan's work. (Today some professors achieve fame just by finding a new proof for one of Ramanujan's many results.)

Georg Friedrich Bernhard Riemann (1826-1866) Germany :

Riemann was a fantastic genius whose work was both novel and rigorous. He had poor physical health and died at an early age, but still made revolutionary contributions in many areas of mathematics. He applied topology to analysis, and applied analysis to number theory. His single paper on the Prime Number distribution conjecture is considered the most important ever on that frequently studied topic. He introduced the clarifying notion of the Riemann integral. He posed the "Hypothesis of Riemann's zeta function," which is certainly the most important and famous unsolved problem in mathematics. Riemann's masterpieces were differential geometry, tensor analysis, non-Euclidean geometry, the theory of functions, and, especially, the theory of manifolds. He generalized the notions of distance and curvature and, therefore, described new possibilities for the geometry of space itself. Despite his great creativity (Gauss praised Riemann's "gloriously fertile originality") Riemann once said: "If only I had the theorems! Then I should find the proofs easily enough."

Jakob Steiner (1796-1863) Switzerland :

Jakob Steiner made many major advances in synthetic geometry, hoping that classical methods could avoid any need for analysis; and indeed he was able to equal or surpass methods of the calculus of variations using just pure geometry. Steiner developed several famous construction methods, e.g. for a triangle's smallest circumscribing and largest inscribing ellipses. Perhaps his three most famous theorems are the Isoperimetric Theorem (among solids of equal volume the sphere will have minimum area, etc.); the Poncelet-Steiner Theorem (lengths constructible with straightedge and compass can be constructed with straightedge alone as long as the picture plane contains the center and circumference of some circle); and his theorem about self-homologous elements in projective geometry. Along with Apollonius of Perga (who lived 2000 years earlier) Steiner is considered one of the two greatest geometers ever.

J.J Sylvester:( 1814-1897) :

James Joseph Sylvester was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory and combinatorics. He played a leadership role in American mathematics in the latter half of the 19th century as a professor at the Johns Hopkins University and as founder of the American Journal of Mathematics. Sylvester invented a great number of mathematical terms such as discriminant. He has given a name to Euler's totient function f(n). His collected scientific work fills four volumes. In 1880, the Royal Society of London awarded Sylvester the Copley Medal, its highest award for scientific achievement; in 1901, it instituted the Sylvester Medal in his memory, to encourage mathematical research after his death in Oxford, Oxfordshire, England. Sylvester (1870) published a book entitled The Laws of Verse in which he attempted to codify a set of laws for prosody in poetry.

Thales of Miletus (ca 625 - 546 BC) Greece :

Thales was the "Chief" of the "Seven Sages" of ancient Greece, and he is often called the "Father of Science" or the "First Philosopher." Several fundamental theorems about triangles are attributed to Thales, including the law of similar triangles (which Thales used famously to calculate the height of the Great Pyramid) and the fact that any angle inscribed in a semicircle is a right angle. Thales was also an astronomer; he invented the 365-day calendar and is the first person known to have correctly predicted a solar eclipse. His theories of physics would seem quaint today, but he seems to have been the first to describe magnetism and static electricity. Thales was also a famous politician, ethicist, and military strategist. Aristotle said, "To Thales the primary question was not what do we know, but how do we know it."

Karl Wilhelm Theodor Weierstrass (1815-1897) Germany :

Weierstrass devised new definitions for the primitives of calculus and was then able to prove several fundamental but hitherto unproven theorems. He advanced the field of calculus of variations. Weierstrass shocked his colleagues when he demonstrated a continuous function which is differentiable nowhere. He found simpler proofs of many existing theorems, including Gauss' Fundamental Theorem of Algebra and the fundamental Hermite-Lindemann Transcendence Theorem. He found a fundamental flaw in Steiner's proof of the Isoperimetric Theorem, and became the first to supply a fully rigorous proof of that famous and ancient result. Starting strictly from the integers, he also applied his axiomatic methods to a definition of irrational numbers. Weierstrass is now called the "Father of Modern Analysis." Weierstrass once wrote: "A mathematician who is not also something of a poet will never be a complete mathematician."

André Weil (1906-1998) France, U.S.A. :

Weil made profound contributions to many areas of mathematics, especially algebraic geometry, which he connected with number theory. His "Weil conjectures" were very influential; these and other works laid the groundwork for many of Grothendieck's achievements. Weil proved a special case of the Riemann hypothesis; he contributed to the recent proof of "Fermat's last Theorem;" he also worked in group theory, general and algebraic topology, differential geometry, sheaf theory, representation theory, and theta functions. He invented several new concepts including vector bundles, and uniform space. His work has found applications in particle physics and string theory. He is considered to be one of the most influential of modern mathematicians. Weil's biography is interesting. He studied Sanskrit as a child, loved to travel, taught at a Muslim university in India for two years (intending to teach French civilization), wrote as a young man under the famous pseudonym Nicolas He once wrote: "Every mathematician worthy of the name has experienced [a] lucid exaltation in which one thought succeeds another as if miraculously."

Hermann Klaus Hugo Weyl (1885-1955) Germany, U.S.A. :

Weyl studied under Hilbert and became one of the premier mathematicians of the 20th century. He excelled at many fields including integral equations, harmonic analysis, analytic number theory, and the foundations of mathematics, but he is most respected for his revolutionary advances in geometric function theory (e.g., differentiable manifolds), the theory of compact groups (incl. representation theory), and theoretical physics (e.g., Weyl tensor, gauge field theory and invariance). Weyl was also a very influential figure in all three major fields of 20th century physics: relativity, unified field theory and quantum mechanics. Weyl once wrote: "My work always tried to unite the Truth with the Beautiful, but when I had to choose one or the other, I usually chose the Beautiful."


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