1.
Hugo Steinhaus
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Władysław Hugo Dionizy Steinhaus /ˈhjuːɡoʊ ˈstaɪnˌhaʊs/ was a Polish mathematician and educator. He is credited with discovering mathematician Stefan Banach, with whom he gave a contribution to functional analysis through the Banach–Steinhaus theorem. Notably he is regarded as one of the founders of game theory. Steinhaus was born on January 14,1887 in Jasło, Austria-Hungary to a family with Jewish roots and his father, Bogusław, was a local industrialist, owner of a brick factory and a merchant. His mother was Ewelina, née Lipschitz, hugos uncle, Ignacy Steinhaus, was an activist in the Koło Polskie, and a deputy to the Galician Diet, the regional assembly of the Kingdom of Galicia and Lodomeria. Hugo finished his studies at the gymnasium in Jasło in 1905 and his family wanted him to become an engineer but he was drawn to abstract mathematics and began to study the works of famous contemporary mathematicians on his own. In the same year he began studying philosophy and mathematics at the University of Lemberg, in 1906 he transferred to Göttingen University. At that University he received his Ph. D. in 1911, the title of his thesis was Neue Anwendungen des Dirichletschen Prinzips. At the start of World War I Steinhaus returned to Poland and served in Józef Piłsudskis Polish Legion, in 1917 he started to work at the University of Lemberg and acquired his habilitation qualification in 1920. In 1921 he became a profesor nadzwyczajny and in 1925 profesor zwyczajny at the same university, during this time he taught a course on the then cutting edge theory of Lebesgue integration, one of the first such courses offered outside France. Steinhaus considered escaping to Hungary but ultimately decided to remain in Lwów, the Soviets reorganized the university to give it a more Ukrainian character, but they did appoint Stefan Banach as the dean of the mathematics department and Steinhaus resumed teaching there. The faculty of the department at the school were also strengthened by several Polish refugees from German occupied Poland. Steinhaus, because of his Jewish background, spent the Nazi occupation in hiding, first among friends in Lwów, then in the towns of Osiczyna, near Zamość and Berdechów. The Polish anti-Nazi resistance provided him with documents of a forest ranger who had died sometime earlier. Under this name he taught clandestine classes, the method relied on the relative frequency with which the obituaries stated that the soldier who died was someones son, someones second son, someones third son and so on. Although initially he had doubts, he turned down offers for faculty positions in Łódź and Lublin and it was also most likely Steinhaus who preserved the original Scottish Book from Lwów throughout the war and subsequently sent it to Stanisław Ulam, who translated it into English. With Steinhaus help, Wrocław University became renowned for mathematics, much as the University of Lwów had been, later, in the 1960s, Steinhaus served as a visiting professor at the University of Notre Dame and the University of Sussex. He also wrote in the area of applied mathematics and enthusiastically collaborated with engineers, geologists, economists, physicians, biologists and, in Kacs words, even lawyers

2.
Real number
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In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing

3.
Elwyn Berlekamp
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Elwyn Ralph Berlekamp is an American mathematician. He is an emeritus of mathematics and EECS at the University of California. Berlekamp is known for his work in coding theory and combinatorial game theory, Berlekamp was born in Dover, Ohio. His family moved to Northern Kentucky, where Berlekamp graduated from Ft. Thomas Highlands high school in Ft. Thomas, Campbell county, while an undergraduate at the Massachusetts Institute of Technology, he was a Putnam Fellow in 1961. He completed his bachelors and masters degrees in engineering in 1962. Continuing his studies at MIT, he finished his Ph. D. in electrical engineering in 1964, his advisors were Robert G. Gallager, Peter Elias, Claude Shannon, and John Wozencraft. Berlekamp taught electrical engineering at the University of California, Berkeley from 1964 until 1966, in 1971, Berlekamp returned to Berkeley as professor of mathematics and EECS, where he served as the advisor for over twenty doctoral students. He is a member of the National Academy of Engineering and the National Academy of Sciences and he was elected a Fellow of the American Academy of Arts and Sciences in 1996, and became a fellow of the American Mathematical Society in 2012. In 1991, he received the IEEE Richard W. Hamming Medal, and in 1993, in 1998, he received a Golden Jubilee Award for Technological Innovation from the IEEE Information Theory Society. He is on the board of directors of Gathering 4 Gardner, in the mid-1980s, he was president of Cyclotomics, Inc. a corporation that developed error-correcting code technology. With John Horton Conway and Richard K. Guy, he co-authored Winning Ways for your Mathematical Plays and he has studied various games, including dots and boxes, Fox and Geese, and, especially, Go. With David Wolfe, Berlekamp co-authored the book Mathematical Go, which describes methods for analyzing certain classes of Go endgames, outside of mathematics and computer science, Berlekamp has also been active in money management. In 1986, he began studies of commodity and financial futures. In 1989, Berlekamp purchased the largest interest in a company named Axcom Trading Advisors. After the firms futures trading algorithms were rewritten, Axcoms Medallion Fund had a return of 55%, net of all management fees, the fund has subsequently continued to realize annualized returns exceeding 30% under management by James Harris Simons and his Renaissance Technologies Corporation. Berlekamp and his wife Jennifer have two daughters and a son and live in Piedmont, California, thesis, Massachusetts Institute of Technology, Dept. of Electrical Engineering,1964. Algebraic Coding Theory, New York, McGraw-Hill,1968, revised ed. Aegean Park Press,1984, ISBN 0-89412-063-8. Winning Ways for your Mathematical Plays, 1st edition, New York, Academic Press,2 vols

4.
Ronald Graham
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He has done important work in scheduling theory, computational geometry, Ramsey theory, and quasi-randomness. Graham was born in Taft, California, in 1962, he received his Ph. D. in mathematics from the University of California, Berkeley and began working at Bell Labs and later AT&T Labs. He was director of information sciences in AT&T Labs, but retired from AT&T in 1999 after 37 years and his 1977 paper considered a problem in Ramsey theory, and gave a large number as an upper bound for its solution. Graham popularized the concept of the Erdős number, named after the highly prolific Hungarian mathematician Paul Erdős, a scientists Erdős number is the minimum number of coauthored publications away from a publication with Erdős. He co-authored almost 30 papers with Erdős, and was also a good friend, Erdős often stayed with Graham, and allowed him to look after his mathematical papers and even his income. Graham and Erdős visited the young mathematician Jon Folkman when he was hospitalized with brain cancer, between 1993 and 1994 Graham served as the president of the American Mathematical Society. He has published about 320 papers and five books, including Concrete Mathematics with Donald Knuth and he is married to Fan Chung Graham, who is the Akamai Professor in Internet Mathematics at the University of California, San Diego. He has four children, daughters Ché, Laura and Christy, in 2003, Graham won the American Mathematical Societys annual Steele Prize for Lifetime Achievement. The prize was awarded on January 16 that year, at the Joint Mathematics Meetings in Baltimore, in 1999 he was inducted as a Fellow of the Association for Computing Machinery. Graham has won other prizes over the years, he was one of the laureates of the prestigious Pólya Prize the first year it was ever awarded. And the Carl Allendoerfer prize which was established in 1976 for the reasons, however for a different magazine. In 2012 he became a fellow of the American Mathematical Society, with Paul Erdős, Old and new results in combinatorial number theory. L’Enseignement Mathématique,1980 with Fan Chung, Erdős on Graphs, a. K. Peters,1998 with Jaroslav Nešetřil, The mathematics of Paul Erdős. Springer,1997 Rudiments of Ramsey Theory, American Mathematical Society,1981 with Donald E. Knuth & Oren Patashnik, Concrete Mathematics, a foundation for computer science. Addison-Wesley,1989,1994 with Joel H, spencer & Bruce L. Rothschild, Ramsey Theory. Wiley,1980,1990 with Martin Grötschel & László Lovász, Handbook of Combinatorics

5.
Journal of Number Theory
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The Journal of Number Theory is a mathematics journal that publishes a broad spectrum of original research in number theory. The journal was established in 1969 by R. P. Bambah, P. Roquette, A. Ross, A. Woods and it is currently published monthly by Elsevier, with 6 volumes per year