Math glossary at ENCYCLOPÆDIA BRITANNICA
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Abraham bar Hiyya, Spanish Jewish philosopher, astronomer, astrologer, and mathematician whose writings were among the first scientific and philosophical works to be written in Hebrew. He is sometimes known as Savasorda, a corruption of an Arabic term indicating that he held some civic office.
Abraham de Moivre, French mathematician who was a pioneer in the development of analytic trigonometry and in the theory of probability. A French Huguenot, de Moivre was jailed as a Protestant upon the revocation of the Edict of Nantes in 1685. When he was released shortly thereafter, he fled to...
Abū al-Wafāʾ, a distinguished Muslim astronomer and mathematician, who made important contributions to the development of trigonometry. Abū al-Wafāʾ worked in a private observatory in Baghdad, where he made observations to determine, among other astronomical parameters, the obliquity of the...
Ada Lovelace, English mathematician, an associate of Charles Babbage, for whose prototype of a digital computer she created a program. She has been called the first computer programmer. Lovelace was the daughter of famed poet Lord Byron and Annabella Milbanke Byron, who legally separated two months...
Adi Shamir, Israeli cryptographer and computer scientist and cowinner, with American computer scientists Leonard M. Adleman and Ronald L. Rivest, of the 2002 A.M. Turing Award, the highest honour in computer science, for their "ingenious contribution for making public-key cryptography useful in...
Adolphe Quetelet, Belgian mathematician, astronomer, statistician, and sociologist known for his application of statistics and probability theory to social phenomena. From 1819 Quetelet lectured at the Brussels Athenaeum, military college, and museum. In 1823 he went to Paris to study astronomy,...
Adrien-Marie Legendre, French mathematician whose distinguished work on elliptic integrals provided basic analytic tools for mathematical physics. Little is known about Legendre's early life except that his family wealth allowed him to study physics and mathematics, beginning in 1770, at the...
Ahn Cheol-Soo, physician, educator, politician, and computer entrepreneur who founded AhnLab, Inc., South Korea's largest Internet security firm. He later entered politics, establishing the People's Party (later reformed as Bareunmirae) and staging several unsuccessful bids for the presidency. Ahn,...
Al-Battānī, Arab astronomer and mathematician who refined existing values for the length of the year and of the seasons, for the annual precession of the equinoxes, and for the inclination of the ecliptic. He showed that the position of the Sun's apogee, or farthest point from the Earth, is...
Al-Karajī, mathematician and engineer who held an official position in Baghdad (c. 1010–1015), perhaps culminating in the position of vizier, during which time he wrote his three main works, al-Fakhrī fīʾl-jabr wa'l-muqābala ("Glorious on algebra"), al-Badī' fī'l-hisāb ("Wonderful on calculation"),...
Al-Khwārizmī, Muslim mathematician and astronomer whose major works introduced Hindu-Arabic numerals and the concepts of algebra into European mathematics. Latinized versions of his name and of his most famous book title live on in the terms algorithm and algebra. Al-Khwārizmī lived in Baghdad,...
Al-Kāshī, ranks among the greatest mathematicians and astronomers in the Islamic world. The first event known with certainty in al-Kāshī's life is his observation of a lunar eclipse on June 2, 1406, from Kāshān. His earliest surviving work is Sullam al-samāʾ (1407; "The Stairway of Heaven"), an...
Alain Connes, French mathematician who won the Fields Medal in 1982 for his work in operator theory. Connes received a bachelor's degree (1970) and a doctorate (1973) from the École Normale Supérieure (now part of the University of Paris). He held appointments at the National Centre for Scientific...
Alan Baker, British mathematician who was awarded the Fields Medal in 1970 for his work in number theory. Baker attended University College, London (B.S., 1961), and Trinity College, Cambridge (M.A. and Ph.D., 1964). He held an appointment at University College (1964–65) and then joined the faculty...
Alan Jay Perlis, American mathematician and computer scientist. He was the first winner, in 1966, of the A.M. Turing Award, given by the Association of Computing Machinery (ACM) and recognized internationally as the highest honour in computer science. In particular, Perlis was cited for "his...
Alan Kay, American computer scientist and winner of the 2003 A.M. Turing Award, the highest honour in computer science, for his contributions to object-oriented programming languages, including Smalltalk. Kay received a doctorate in computer science from the University of Utah in 1969. In 1972 he...
Alan Turing, British mathematician and logician, who made major contributions to mathematics, cryptanalysis, logic, philosophy, and mathematical biology and also to the new areas later named computer science, cognitive science, artificial intelligence, and artificial life. The son of a civil...
Aleksandr Aleksandrovich Friedmann, Russian mathematician and physical scientist. After graduating from the University of St. Petersburg in 1910, Friedmann joined the Pavlovsk Aerological Observatory and, during World War I, did aerological work for the Russian army. After the war he was on the...
Aleksandr Osipovich Gelfond, Russian mathematician who originated basic techniques in the study of transcendental numbers (numbers that cannot be expressed as the root or solution of an algebraic equation with rational coefficients). He profoundly advanced transcendental number theory and the...
Alexandre Grothendieck, German French mathematician who was awarded the Fields Medal in 1966 for his work in algebraic geometry. After studies at the University of Montpellier (France) and a year at the École Normale Supérieure in Paris, Grothendieck received his doctorate from the University of...
Alfred North Whitehead, English mathematician and philosopher who collaborated with Bertrand Russell on Principia Mathematica (1910–13) and, from the mid-1920s, taught at Harvard University and developed a comprehensive metaphysical theory. Whitehead's grandfather Thomas Whitehead was a self-made...
Alfred Tarski, Polish-born American mathematician and logician who made important studies of general algebra, measure theory, mathematical logic, set theory, and metamathematics. Tarski completed his education at the University of Warsaw (Ph.D., 1923). He taught in Warsaw until 1939, when he moved...
Allen Newell, American computer scientist and one of the pioneers of the science of artificial intelligence (AI). Newell and his longtime collaborator Herbert A. Simon won the 1975 A.M. Turing Award, the highest honour in computer science, for their "basic contributions to artificial intelligence,...
Alonzo Church, U.S. mathematician. He earned a Ph.D. from Princeton University. His contributions to number theory and the theories of algorithms and computability laid the foundations of computer science. The rule known as Church's theorem or Church's thesis (proposed independently by Alan M...
Amedeo Avogadro, Italian mathematical physicist who showed in what became known as Avogadro's law that, under controlled conditions of temperature and pressure, equal volumes of gases contain an equal number of molecules. Avogadro was the son of Filippo Avogadro, conte di Quaregna e Cerreto, a...
American Philosophical Society, oldest extant learned society in the United States, founded under the impetus of Benjamin Franklin in 1743. At the beginning of the 21st century, it had more than 850 members, elected for their scholarly and scientific accomplishments in any of five areas—the...
Amir Pnueli, Israeli computer scientist and winner of the 1996 A.M. Turing Award, the highest honour in computer science, for "seminal work introducing temporal logic into computing science and for outstanding contributions to program and system verification." Pnueli received a bachelor's degree in...
Andrei Okounkov, Russian mathematician awarded a Fields Medal in 2006 "for his contributions bridging probability, representation theory and algebraic geometry." Okounkov received a doctorate in mathematics from Moscow State University (1995) and held positions at the Russian Academy of Sciences,...
Andrew Chi-Chih Yao, Chinese American computer scientist and winner of the 2000 A.M. Turing Award, the highest honour in computer science, for his "fundamental contributions to the theory of computation [computational complexity], including the complexity-based theory of pseudorandom number...
Andrew Russell Forsyth, British mathematician, best known for his mathematical textbooks. In 1877 Forsyth entered Trinity College, Cambridge, where he studied mathematics under Arthur Cayley. Forsyth graduated in 1881 as first wrangler (first place in the annual Mathematical Tripos contest) and was...
Andrew Wiles, British mathematician who proved Fermat's last theorem. In recognition he was awarded a special silver plaque—he was beyond the traditional age limit of 40 years for receiving the gold Fields Medal—by the International Mathematical Union in 1998. He also received the Wolf Prize...
Andrey Andreyevich Markov, Russian mathematician who helped to develop the theory of stochastic processes, especially those called Markov chains. Based on the study of the probability of mutually dependent events, his work has been developed and widely applied in the biological and social sciences...
Andrey Nikolayevich Kolmogorov, Russian mathematician whose work influenced many branches of modern mathematics, especially harmonic analysis, probability, set theory, information theory, and number theory. A man of broad culture, with interests in technology, history, and education, he played an...
André Weil, French mathematician who was one of the most influential figures in mathematics during the 20th century, particularly in number theory and algebraic geometry. André was the brother of the philosopher and mystic Simone Weil. He studied at the École Normale Supérieure (now part of the...
Anita Borg, American computer scientist who advocated for women's advancement in technology. Borg attended the University of Washington in Seattle for two years. She later studied at New York University, where she received a doctorate (1981) for her work on synchronization efficiency in operating...
Antoine-Augustin Cournot, French economist and mathematician. Cournot was the first economist who, with competent knowledge of both subjects, endeavoured to apply mathematics to the treatment of economics. His main work in economics is Recherches sur les principes mathématiques de la théorie des...
Antoni Zygmund, Polish-born mathematician who exerted a major influence on 20th-century mathematics, particularly in harmonic analysis, a field utilized in science and technology for the formulation of descriptions of periodic phenomena such as waves, vibrations, and regularly repeating structures...
Apollonius of Perga, mathematician, known by his contemporaries as "the Great Geometer," whose treatise Conics is one of the greatest scientific works from the ancient world. Most of his other treatises are now lost, although their titles and a general indication of their contents were passed on by...
Apse, in astronomy, either of the two points on an elliptical orbit that are nearest to, and farthest from, the focus, or centre of attraction. The line of apsides, connecting the two points, is the major axis of the orbit. The point nearest the focus is the pericentre, or periapsis, and that...
Archimedes, the most-famous mathematician and inventor in ancient Greece. Archimedes is especially important for his discovery of the relation between the surface and volume of a sphere and its circumscribing cylinder. He is known for his formulation of a hydrostatic principle (known as Archimedes'...
Archytas of Tarentum, Greek scientist, philosopher, and major Pythagorean mathematician. Plato, a close friend, made use of his work in mathematics, and there is evidence that Euclid borrowed from him for the treatment of number theory in Book VIII of his Elements. Archytas was also an influential...
Argand diagram, graphic portrayal of complex numbers, those of the form x + yi, in which x and y are real numbers and i is the square root of −1. It was devised by the Swiss mathematician Jean Robert Argand about 1806. A similar representation had been proposed by the Danish surveyor Caspar Wessel...
Arithmetic, branch of mathematics in which numbers, relations among numbers, and observations on numbers are studied and used to solve problems. Arithmetic (a term derived from the Greek word arithmos, "number") refers generally to the elementary aspects of the theory of numbers, arts of...
Arithmetic function, any mathematical function defined for integers (…, −3, −2, −1, 0, 1, 2, 3, …) and dependent upon those properties of the integer itself as a number, in contrast to functions that are defined for other values (real numbers, complex numbers, or even other functions) and that...
Arithmometer, early calculating machine, built in 1820 by Charles Xavier Thomas de Colmar of France. Whereas earlier calculating machines, such as Blaise Pascal's Pascaline in France and Gottfried Wilhelm von Leibniz's Step Reckoner in Germany, were mere curiosities, with the Industrial Revolution...
Arthur Cayley, English mathematician and leader of the British school of pure mathematics that emerged in the 19th century. The interested viewer may read an extract from the geometry article he wrote for the 9th edition of the Encyclopædia Britannica (1875–89). Although Cayley was born in England,...
Arthur Eddington, English astronomer, physicist, and mathematician who did his greatest work in astrophysics, investigating the motion, internal structure, and evolution of stars. He also was the first expositor of the theory of relativity in the English language. Eddington was the son of the...
Arthur Edwin Kennelly, U.S. electrical engineer who made innovations in analytic methods in electronics, particularly the definitive application of complex-number theory to alternating-current (ac) circuits. After working as an office boy for a London engineering society, as an electrician, and on...
Aryabhata, astronomer and the earliest Indian mathematician whose work and history are available to modern scholars. He is also known as Aryabhata I or Aryabhata the Elder to distinguish him from a 10th-century Indian mathematician of the same name. He flourished in Kusumapura—near Patalipurta...
Associative law, in mathematics, either of two laws relating to number operations of addition and multiplication, stated symbolically: a + (b + c) = (a + b) + c, and a(bc) = (ab)c; that is, the terms or factors may be associated in any way desired. While associativity holds for ordinary arithmetic...
Asymptote, In mathematics, a line or curve that acts as the limit of another line or curve. For example, a descending curve that approaches but does not reach the horizontal axis is said to be asymptotic to that axis, which is the asymptote of the...
Athanasius Kircher, Jesuit priest and scholar, sometimes called the last Renaissance man, important for his prodigious activity in disseminating knowledge. Kircher learned Greek and Hebrew at the Jesuit school in Fulda, pursued scientific and humanistic studies at Paderborn, Cologne, and Koblenz,...
Atle Selberg, Norwegian-born American mathematician who was awarded the Fields Medal in 1950 for his work in number theory. In 1986 he shared (with Samuel Eilenberg) the Wolf Prize. Selberg attended the University of Oslo (Ph.D., 1943) and remained there as a research fellow until 1947. He then...
August Ferdinand Möbius, German mathematician and theoretical astronomer who is best known for his work in analytic geometry and in topology. In the latter field he is especially remembered as one of the discoverers of the Möbius strip. Möbius entered the University of Leipzig in 1809 and soon...
August Leopold Crelle, German mathematician and engineer who advanced the work and careers of many young mathematicians of his day and founded the Journal für die reine und angewandte Mathematik ("Journal for Pure and Applied Mathematics"), now known as Crelle's Journal. A civil engineer in the...
Augustin-Louis Cauchy, French mathematician who pioneered in analysis and the theory of substitution groups (groups whose elements are ordered sequences of a set of things). He was one of the greatest of modern mathematicians. At the onset of the Reign of Terror (1793–94) during the French...
Augustus De Morgan, English mathematician and logician whose major contributions to the study of logic include the formulation of De Morgan's laws and work leading to the development of the theory of relations and the rise of modern symbolic, or mathematical, logic. De Morgan was educated at...
Augustus Edward Hough Love, British geophysicist and mathematician who discovered a major type of seismic wave that was subsequently named for him. Love held the Sedleian professorship of natural philosophy at the University of Oxford from 1899 to 1940. In his analysis of earthquake waves, Love...
Austausch coefficient, in fluid mechanics, particularly in its applications to meteorology and oceanography, the proportionality between the rate of transport of a component of a turbulent fluid and the rate of change of density of the component. In this context, the term component signifies not...
Automata theory, body of physical and logical principles underlying the operation of any electromechanical device (an automaton) that converts information from one form into another according to a definite procedure. Real or hypothetical automata of varying complexity have become indispensable...
Automorphism, in mathematics, a correspondence that associates to every element in a set a unique element of the set (perhaps itself) and for which there is a companion correspondence, known as its inverse, such that one followed by the other produces the identity correspondence (i); i.e., the...
Avraham Trahtman, Russian-born Israeli mathematician who solved the road-colouring problem (a variant of the traveling salesman problem). Trahtman earned an undergraduate degree (1967) and a graduate degree (1973) in mathematics from Ural State University, in Sverdlovsk (now Yekaterinburg, Russia)...
Axiom, in logic, an indemonstrable first principle, rule, or maxim, that has found general acceptance or is thought worthy of common acceptance whether by virtue of a claim to intrinsic merit or on the basis of an appeal to self-evidence. An example would be: "Nothing can both be and not be at the ...
Axiom of choice, statement in the language of set theory that makes it possible to form sets by choosing an element simultaneously from each member of an infinite collection of sets even when no algorithm exists for the selection. The axiom of choice has many mathematically equivalent formulations,...
Bahāʾ ad-dīn Muḥammad ibn Ḥusayn al-ʿĀmilī, theologian, mathematician, jurist, and astronomer who was a major figure in the cultural revival of Ṣafavid Iran. Al-ʿĀmilī was educated by his father, Shaykh Ḥusayn, a Shīʿite theologian, and by excellent teachers of mathematics and medicine. After his...
Barbara Jane Liskov, American winner of the 2008 A.M. Turing Award, the highest honour in computer science, for her "pioneering work in the design of computer programming languages." After she earned a bachelor's degree in mathematics in 1961 from the University of California, Berkeley, Liskov...
Bartlett's test, in statistics, a test to ascertain if multiple samples have the same variance (the square of the sample's standard deviation). The test, which is a standard tool in analysis of variance (ANOVA) computer programs, can be used when a single measurable variable is involved, such as...
Base, in mathematics, an arbitrarily chosen whole number greater than 1 in terms of which any number can be expressed as a sum of that base raised to various powers. See numerals and numeral...
Bayes's theorem, in probability theory, a means for revising predictions in light of relevant evidence, also known as conditional probability or inverse probability. The theorem was discovered among the papers of the English Presbyterian minister and mathematician Thomas Bayes and published...
Bayesian analysis, a method of statistical inference (named for English mathematician Thomas Bayes) that allows one to combine prior information about a population parameter with evidence from information contained in a sample to guide the statistical inference process. A prior probability...
Beal's conjecture, in number theory, a generalization of Fermat's last theorem. Fermat's last theorem, which was proposed in 1637 by the French mathematician Pierre de Fermat and proved in 1995 by the English mathematician Andrew Wiles, states that for positive integers x, y, z, and n, xn + yn = zn...
Benjamin Banneker, mathematician, astronomer, compiler of almanacs, inventor, and writer, one of the first important African American intellectuals. Banneker, a freeman, was raised on a farm near Baltimore that he would eventually inherit from his father. Although he periodically attended a...
Benjamin Peirce, American mathematician, astronomer, and educator who computed the general perturbations of the planets Uranus and Neptune. Peirce graduated from Harvard University in 1829 and accepted a teaching position with George Bancroft at his Round Hill School in Northampton, Massachusetts...
Benjamin Robins, British mathematician and military engineer who laid the groundwork for modern ordnance (field-artillery) theory and practice with his New Principles of Gunnery (1742), which invalidated old suppositions about the nature and action of gunpowder and the flight of projectiles and...
Benoit Mandelbrot, Polish-born French American mathematician universally known as the father of fractals. Fractals have been employed to describe diverse behaviour in economics, finance, the stock market, astronomy, and computer science. Mandelbrot was educated at the École Polytechnique (1945–47)...
Bernhard Bolzano, Bohemian mathematician and theologian who provided a more detailed proof for the binomial theorem in 1816 and suggested the means of distinguishing between finite and infinite classes. Bolzano graduated from the University of Prague as an ordained priest in 1805 and was...
Bernhard Riemann, German mathematician whose profound and novel approaches to the study of geometry laid the mathematical foundation for Albert Einstein's theory of relativity. He also made important contributions to the theory of functions, complex analysis, and number theory. Riemann was born...
Bernoulli family, Two generations of distinguished Swiss mathematicians. Jakob (1655–1705) and Johann (1667–1748) were the sons of a pharmacist who wanted one boy to study theology and the other medicine. Over his objections, both pursued careers in mathematics, making important discoveries in...
Bertrand Russell, British philosopher, logician, and social reformer, founding figure in the analytic movement in Anglo-American philosophy, and recipient of the Nobel Prize for Literature in 1950. Russell's contributions to logic, epistemology, and the philosophy of mathematics established him as...
Bessel function, any of a set of mathematical functions systematically derived around 1817 by the German astronomer Friedrich Wilhelm Bessel during an investigation of solutions of one of Kepler's equations of planetary motion. Particular functions of the set had been formulated earlier by the ...
Bhaskara I, Indian astronomer and mathematician who helped to disseminate the mathematical work of Aryabhata (born 476). Little is known about the life of Bhaskara; I is appended to his name to distinguish him from a 12th-century Indian astronomer of the same name. In his writings there are clues...
Bhāskara II, the leading mathematician of the 12th century, who wrote the first work with full and systematic use of the decimal number system. Bhāskara II was the lineal successor of the noted Indian mathematician Brahmagupta (598–c. 665) as head of an astronomical observatory at Ujjain, the...