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	Julius Wihelm Richard Dedekind (Site not responding. Last check: 2007-10-26)
		Richard Dedekind attended school in Brunswick from the age of 7, and at this stage mathematics was not his main interest.
		Dedekind made a number of highly significant contributions to mathematics and his work would change the style of mathematics into what is familiar to us today.
		In the book Dedekind presented a logical theory of number and of complete induction, presented his principal conception of the essence of arithmetic, and dealt with the role of the complete system of real numbers in geometry in the problem of the continuity of space.
	www.stetson.edu /~efriedma/periodictable/html/Db.html (695 words)

	Richard Dedekind - Wikipedia, the free encyclopedia
		Dedekind was elected to the Academies of Berlin (1880) and Rome, and to the Paris Académie des Sciences (1900).
		If there existed a one-to-one correspondence between two sets, Dedekind said that the two sets were "similar." He invoked similarity to give the first precise definition of an infinite set: a set is infinite when it is "similar to a proper part of itself," in modern terminology, is equinumerous to one of its proper subsets.
		Dedekind's study of Dirichlet's work was what led him to his later study of algebraic number fields and ideals.
	en.wikipedia.org /wiki/Richard_Dedekind (1165 words)

	Richard Julius Wilhelm Dedekind
		Richard Dedekind was a German mathematician who was born in 1831 in Brunswick.
		Dedekind made many original and important contributions to the theory of algebraic numbers.
		Dedekind's accomplishment was to define irrational numbers in terms of rationals.
	www.engr.iupui.edu /~orr/webpages/cpt120/mathbios/rdedek.htm (843 words)

	DEDEKIND, JULIUS WILHELM RICHARD. The Columbia Encyclopedia: Sixth Edition. 2000 (Site not responding. Last check: 2007-10-26)
		Dedekind studied at Göttingen under the German mathematician Carl Gauss and in 1852 received his doctorate there for a thesis on Eulerian integrals.
		Dedekind led the effort to formulate rigorous definitions of basic mathematical concepts.
		Perhaps his best-known contribution is the “Dedekind cut,” whereby real numbers can be defined in terms of rational numbers.
	www.bartleby.com /aol/65/de/Dedekind.html (94 words)

	Encyclopaedia Britannica entry (Site not responding. Last check: 2007-10-26)
		Dedekind perceived that the character of the continuum need not depend on the quantity of points on a line segment (or continuum) but rather on how the line submits to being divided.
		His method, now called the Dedekind cut, consisted in separating all the real numbers in a series into two parts such that each real number in one part is less than every real number in the other.
		Dedekind gave a sympathetic hearing to an exposition of the revolutionary idea of sets that Cantor had just published, which later became prominent in the teaching of modern mathematics.
	www.aam314.vzz.net /EB/Dedekind.html (761 words)

	Dedekind, (Julius Wilhelm) Richard - Hutchinson encyclopedia article about Dedekind, (Julius Wilhelm) Richard
		In 1872 he introduced the Dedekind cut (which divides a line of infinite length representing all real numbers) to define irrational numbers in terms of pairs of sequences of rational numbers.
		Dedekind was born in Brunswick and studied at Göttingen.
		In 1858 he succeeded in producing a purely arithmetic definition of continuity and an exact formulation of the concept of the irrational number.
	encyclopedia.farlex.com /Dedekind,+(Julius+Wilhelm)+Richard (217 words)

	Julius Wilhelm Richard Dedekind Biography
		Dedekind was born in Braunschweig (Brunswick) the youngest of four children of Julius Levin Ulrich Dedekind.
		Among Dedekind's main professors was Moritz Abraham Stern who at that time wrote many works on number theory.
		Dedekind was among the first mathematicians who had accepted Cantor's work on the theory of infinite sets; other mathematicians didn't yet understand their ideas.
	www.biographybase.com /biography/Dedekind_Julius_Wilhelm_Richard.html (826 words)

	Dedekind, (Julius Wilhelm) Richard (1831-1916)
		A German mathematician whose most important contribution was the discovery of what became known as the Dedekind cut.
		Dedekind's brilliant idea was to represent the real numbers by such divisions of the rationals.
		He also provided important support for Georg Cantor's set theory, which was highly controversial at the time.
	www.daviddarling.info /encyclopedia/D/Dedekind.html (99 words)

	Program Files\Netscape\Communicator\Program\dedexxx
		Dedekind was then qualified as a university teacher and he began teaching at Gottingin giving courses on probability and geometry.
		It was in the third and fourth editions of Vorlesungen über Zahlentheorie, published in 1879 and 1894, that Dedekind wrote supplements in which he introduced the notion of an ideal which is fundamental to ring theory.
		Dedekind formulated his theory in the ring of integers of an algebraic number field.
	www.andrews.edu /~calkins/math/biograph/199900/biodedek.htm (1409 words)

	Mathematicians D
		Dedekind was born in Brunswick, the birthplace of Gauss, and received his degree under Gauss at Göttingen.
		Dedekind's construction of the real numbers using `Dedekind cuts' was part of the effort of Dedekind, Cantor, and Weierstrass, and others to bring a rigor to analysis.
		In algebraic number theory Dedekind introduced his theory of ideals to restore unique factorization; today integral domains in which every ideal is a unique product of prime ideals are called Dedekind domains.
	www.mlahanas.de /Stamps/Data/Mathematician/D.htm (138 words)

	Columbia Encyclopedia - Dedekind Julius Wilhelm Richard - AOL Research & Learn
		Dedekind studied at Göttingen under the German mathematician Carl Gauss and in 1852 received his doctorate there for a thesis on Eulerian integrals.
		Dedekind led the effort to formulate rigorous definitions of basic mathematical concepts.
		Perhaps his best-known contribution is the Dedekind cut, whereby real numbers can be defined in terms of rational numbers.
	reference.aol.com /columbia/_a/dedekind-julius-wilhelm-richard/20051205230609990005 (146 words)

	Dedekind, Julius Wilhelm Richard (Site not responding. Last check: 2007-10-26)
		Julius Wilhelm Richard Dedekind (6.11.1831 - 12.02.1916) wurde in Braunschweig als jüngstes von vier Kindern eines Jura-Professors geboren.
		Zunächst interessierte sich Dedekind hauptsächlich für Physik und Chemie, aber er wandte sich dann der Mathematik zu, weil ihm deren Art der exakteren Argumentation mehr zusagte.
		Dedekind war auch musikalisch begabt und komponierte sogar eine kleine Oper.
	www.mathe.tu-freiberg.de /~hebisch/cafe/dedekind.html (248 words)

	AllRefer.com - Julius Wilhelm Richard Dedekind (Mathematics, Biography) - Encyclopedia
		Julius Wilhelm Richard Dedekind[yOOl´yoos vil´helm rikh´Art dA´dukint] Pronunciation Key, 1831–1916, German mathematician.
		Dedekind studied at GOttingen under the German mathematician Carl Gauss and in 1852 received his doctorate there for a thesis on Eulerian integrals.
		In 1858 he went to ZUrich as a professor; in 1862 he returned to his home town Brunswick to become a professor there.
	reference.allrefer.com /encyclopedia/D/Dedekind.html (222 words)

	Program Files\Netscape\Communicator\Program\dedekind
		Dedekind was then qualified as a university teacher and he began teaching at Göttingin giving courses on probability and geometry.
		In the spring of 1858 the swiss councillor who made appointments came to Göttingin and Dedekind was quickly chosen for the post at the Polytecknikum in Zurich.
		While there, Dedekind met Weierstrass, Kummer, Borchardt and Kronecker, all great researchers in the fields of math and science.
	www.andrews.edu /~calkins/math/biograph/199899/biodedek.htm (1430 words)

	DEDEKIND
		Dedekind presenta el número real como una cortadura en el conjunto de los números racionales, dando al conjunto de los números reales una interpretación geométrica en forma de línea recta.
		La propiedad de continuidad de la recta, según Dedekind, consiste en que las cortaduras se encontrarán o en el punto más derecho o en el más izquierdo de una clase.
		El conjunto de los racionales no tiene la propiedad de la continuidad, introduciendo el número irracional, como tal cortadura en el conjunto de los racionales, en cuyas clases no hay ni puntos más derechos ni más izquierdos.
	almez.pntic.mec.es /~agos0000/Dedekind.html (140 words)

	Richard Dedekind: an Algebraic Foundation for Calculus free essays
		Richard Dedekind: An Algebraic Foundation for Calculus In 1858, while giving lectures on differential calculus, mathematician Richard Dedekind noted the lack of a truly scientific foundation of the arithmetic with which he taught his class.
		Julius Wilhelm Richard Dedekind was born October 6, 1831 in Brunswick, in what is today the country of Germany.
		Dedekind attended school in Brunswick at the Gymnasium Martino-Catharineum from the age of seven.
	www.needfreeessays.com /viewpaper/39802.html (281 words)

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		Cantor continued to correspond with Dedekind, sharing his ideas and seeking Dedekind's opinions, and he wrote to Dedekind in 1877 proving that there was a 1-1 correspondence of points on the interval [0, 1] and points in p-dimensional space.
		Dedekind, (Julius Wilhelm) Richard — (1831 — 1916) German mathematician who developed a major redefinition of irrational numbers in terms of arithmetic concept.
		Liouville, Joseph — (1809 - 1882) French mathematician known for his work in analysis, the theory of numbers, and differential geometry, and particularly for his discovery of transcendental number —i.e., numbers that are not the roots of algebraic equations having rational coefficients.
	debian.fmi.uni-sofia.bg /~emerald/fmi/semestar1/Aleksandra.doc (1353 words)

	Dedekind (Site not responding. Last check: 2007-10-26)
		Richard Dedekind completed his doctorate in 1852 at Göttingen under the supervision of Gauss; he was to be Gauss' last student.
		Gauss is, of course, one of the most admired and influential figures in mathematics, but he was also known as a very unpopular teacher.
		Dedekind was perhaps the first mathematician to put the real numbers on a firm foundation, constructing them from the rationals via his famous Dedekind cuts.
	www.math.fau.edu /schonbek/Modern_Analysis/calcmath21.html (100 words)

	Dedekind summary
		Dedekind's major contribution was a redefinition of irrational numbers in terms of Dedekind cuts.
		He introduced the notion of an ideal which is fundamental to ring theory.
		Introduction to Richard Dedekind - the man and the numbers
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Dedekind.html (45 words)

	Math Bihind the Project
		In the nineteenth century, there were many mathematicians researching problems in what would become the theory of ideals.
		However, the theory of ideals, in the form we know it today, is the result of work of one mathematician - Julius Wihelm Richard Dedekind.
		Also, Dedekind's ideas are considered a birthplace of the modern set-theoretic approach to the foundations of mathematics.
	www.ma.iup.edu /courses/ls499/semiring/math-b-p.html (675 words)

	Open Directory - Science: Math: Mathematicians: D: Dedekind, Richard (Site not responding. Last check: 2007-10-26)
		Julius Wihelm Richard Dedekind - Biography of the mathematician with links to relevant terms and related links.
		Julius Wihelm Richard Dedekind - Biography of the mathematician.
		Science and Modernism - Richard Dedekind - Biography of the mathematician.
	dmoz.org /Science/Math/Mathematicians/D/Dedekind,_Richard (91 words)

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