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Topic: Richard Dedekind

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	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 (1066 words)

	Dedekind cut - Wikipedia, the free encyclopedia
		Dedekind used cuts to prove the completeness of the reals without using the axiom of choice (proving the existence of a complete ordered field to be independent of said axiom).
		For example it is shown that the typical Dedekind cut in the real numbers is a pair with A the interval (−∞, a), and B the interval [ a, +∞).
		The Dedekind cut is named after Richard Dedekind, who invented this construction in order to represent the real numbers as Dedekind cuts of the rational numbers.
	en.wikipedia.org /wiki/Dedekind_cut (558 words)

	PlanetMath: Dedekind cuts (Site not responding. Last check: 2007-10-20)
		The purpose of Dedekind cuts is to provide a sound logical foundation for the real number system.
		Dedekind defines a point to produce the division of the real line if this point is either the least or greatest element of either one of the classes mentioned above.
		This is version 23 of Dedekind cuts, born on 2002-05-16, modified 2004-02-15.
	planetmath.org /encyclopedia/DedekindCuts.html (601 words)

	Dedekind (Site not responding. Last check: 2007-10-20)
		Richard Dedekind's father was a professor at the Collegium Carolinum in Brunswick.
		Dedekind and Dirichlet soon became close friends and the relationship was in many ways the making of Dedekind, whose mathematical interests took a new lease of life with the discussions between the two.
		Dedekind's work was quickly accepted, partly because of the clarity with which he presented his ideas and partly since Heinrich Weber lectured to Hilbert on these topics at the University of Königsberg.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Dedekind.html (2040 words)

	Julius Wihelm Richard Dedekind (Site not responding. Last check: 2007-10-20)
		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)

	Dedekind's Real Numbers
		Richard Dedekind's characterization of the real numbers as the system of cuts of rational numbers is by now the standard in almost every mathematical book on analysis or number theory.
		Dedekind regards counting as the simplest arithmetic act, and describes this as ``the successive creation [Schöpfung] of the infinite series of positive integers in which each individual is defined by the one immediately preceeding'' (p.
		Dedekind's general aim is the reconstruction of the systems of numbers starting with the natural numbers, in such a way, that the building blocks of each system are only elements belonging to previously obtained systems.
	www.colorado.edu /StudentGroups/PhilosophyClub/reals.htm (1860 words)

	Essays on the Theory of Numbers Richard Dedekind Dover Publications Paperback (Site not responding. Last check: 2007-10-20)
		Richard Dedekind (1831-1916) is recognized as one of the great pioneers in the logical and philosophical analysis of the foundations of mathematics.
		Dedekind was not successful in imposing his terminology on later mathematicians.
		Richard Dedekind is one of the fathers of modern mathematical proofs.
	www.removal-spyware.com /software/viewproduct.php?country=us&asin=0486210103 (743 words)

	Dedekind, Julius Wilhelm Richard. The Columbia Encyclopedia, Sixth Edition. 2001-05
		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 /65/de/Dedekind.html (148 words)

	Dedekind, (Julius Wilhelm) Richard (Site not responding. Last check: 2007-10-20)
		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.
	www.cartage.org.lb /en/themes/Biographies/MainBiographies/D/Dedekind/1.html (152 words)

	Program Files\Netscape\Communicator\Program\dedekind
		Richard Dedekind was born into the life of research and experimental theory.
		However Richard's soon took a disliking to the fields of physics because of the imprecise logical structure and soon he returned to the field of Mathematics.
		Dedekind was then qualified as a university teacher and he began teaching at Göttingin giving courses on probability and geometry.
	www.andrews.edu /~calkins/math/biograph/199899/biodedek.htm (1430 words)

	MODERN PHILOSOPHY: Unclassified Philosophers - 4
		When Dedekind (picture) was seventy-three years old, he read in a mathematical annual, an obituary that stated he had died on September 4, 1899.
		Dedekind's principal works, Continuity and Irrational Numbers (1871) and The Nature and Meaning of Numbers (1888) are highly important contributions to the theory of numbers.
		Dedekind's "cut," in the first book, is considered to be the foundation of irrational numbers.
	radicalacademy.com /adiphilunclassified3.htm (2381 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 cuts (Site not responding. Last check: 2007-10-20)
		The first construction of the Real numbers from the Rationals is due to the German mathematician Richard Dedekind (1831 - 1916).
		Richard Dedekind, along with Bernhard Riemann was the last research student of Gauss.
		His arithmetisation of analysis was his most important contribution to mathematics, but was not enthusiastically received by leading mathematicians of his day, notably Kronecker and Weierstrass.
	turnbull.mcs.st-and.ac.uk /~john/analysis/Lectures/A3.html (364 words)

	Program Files\Netscape\Communicator\Program\dedexxx
		Dedekind still learned courses on mathematics throughout this time by attending courses on abelian functions and elliptic functions.
		Dedekind develoed the idea that both rational and irrational numbers could form a continuum(with no gaps) of real numbers, provided that the real numbers have a one-one relationship with points on a line.
		Dedekinds Cuts were developed as a way to prove a number rational or irrational.
	www.andrews.edu /~calkins/math/biograph/biodedek.htm (1253 words)

	References for Dedekind (Site not responding. Last check: 2007-10-20)
		I Dedekind, P Dugac, W-D Geyerand W Scharlau, Richard Dedekind, 1831-1981, in Eine Würdigung zu seinem 150.
		H Niederreiter, Richard Dedekind and the development of the theory of finite fields, Abh.
		H Stein, Eudoxos and Dedekind : on the ancient Greek theory of ratios and its relation to modern mathematics, Synthese 84 (2) (1990), 163-211.
	www-gap.dcs.st-and.ac.uk /~history/References/Dedekind.html (348 words)

	Amazon.ca: Theory of Algebraic Integers: Books (Site not responding. Last check: 2007-10-20)
		Dedekind's memoir offers a candid account of the development of an elegant theory and provides blow by blow comments regarding the many difficulties encountered en route.
		This is Dedekind's famous creation of the theory of (algebraic number) rings and modules, as an appendix to his edition of Dirichlet's LECTURES ON NUMBER THEORY.
		And she was right, in a very deep sense the whole modern approach to abstract algebra is in Dedekind, though it took her phenomenal genius to find it there.
	www.amazon.ca /exec/obidos/ASIN/0521565189 (488 words)

	(Julius Wilhelm) Richard Dedekind Biography / Biography of (Julius Wilhelm) Richard Dedekind World of Mathematics ...
		Richard Dedekind is best known for his work in number theory.
		Although he was among the most capable and original mathematicians of his day, Dedekind was a modest man, spending most of his professional life as a teacher at the technical high school in his hometown of Brunswick.
		Dedeking was born Julius Wilhelm Richard Dedekind on October 6, 1831, in Brunswick (Braunschweig), now Germany, the last of four ch.....
	www.bookrags.com /biography-julius-wilhelm-richard-dedekind-wom (262 words)

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		Dedekind came to the conclusion that the essence of the continuity of a line segment is not due to a vague hang-togetherness, but to an exactly opposite property--the nature of the division of the segment into two parts by a point on the segment (Boyer, 1968, p.
		Dedekind asked us to imagine a blade with an infinitely thin blade (i.e., it has no thickness at all) which can be used to cut the continuum into two segments.
		Richard Dedekind has traditionally been credited with this lovely definition of infinite sets: "A system S is said to be infinite when it is similar to a proper part of itself, in the contrary case S is said to be a finite system (Dedekind, 1888).
	www.angelfire.com /super/magicrobin/peirce.htm (7996 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)

	Untitled Document (Site not responding. Last check: 2007-10-20)
		One can reasonably question the wisdom of including such an abstract topic in the course, and admittedly, part of the original reason for their inclusion in the course is to maintain a rigorous definition of the real numbers in the math ed curriculum, and justify the capstone course as a senior level mathematics course.
		Dedekind's attack rested on the assumptions that we “know” the rational numbers, and as such can use them to define the real numbers.
		Dedekind's insight was that the fundamental building block of the calculus was an understanding of the number line.
	gallery.carnegiefoundation.org /cbennett/Dedint.htm (648 words)

	Theory of Algebraic Integers - Cambridge University Press
		The invention of ideals by Dedekind in the 1870s was well ahead of its time, and proved to be the genesis of what today we would call algebraic number theory.
		This is a translation of that work by John Stillwell, who also adds a detailed introduction that gives the historical background as well as outlining the mathematical obstructions that Dedekind was striving to overcome.
		Dedekind’s memoir gives a candid account of his development of an elegant theory as well as providing blow-by-blow comments as he wrestled with the many difficulties encountered en route.
	www.cambridge.org /uk/catalogue/catalogue.asp?isbn=0521565189 (187 words)

	Richard Dedekind -- Britannica Student Encyclopedia
		In work originating from discussions on the foundations of the infinitesimal and derivative calculus by Baron Augustin-Louis Cauchy and Karl Weierstrauss, Cantor and Richard Dedekind developed methods of dealing with the large, and in...
		Richard Burbage was known as the first performer to play Shakespeare's Richard III, Othello, Romeo, Hamlet, Henry V, and Lear.
		The English writer and librarian Richard Garnett was the head of the Garnett family, which exerted a formative influence on the development of modern British writing.
	www.britannica.com /ebi/article-9029718 (777 words)

	Additional Reading (from Richard Dedekind) -- Encyclopædia Britannica
		Laid out in 1748 by Nicholas Scull and William Parsons on land owned by Thomas and Richard Penn (sons of William Penn, Pennsylvania's founder), it was built around Penn Common, a large open square, and named for the hometown of the Penn...
		The American author Richard Wright pictured with brutal realism what it meant to be fl in a white society.
		It was inventor Richard Trevithick who realized that a much smaller, lighter, and more powerful engine could be made by using high-pressure steam and allowing it to expand within the cylinder.
	www.britannica.com /eb/article-1769?tocId=1769 (757 words)

	Amazon.com: Theory of Algebraic Integers (Cambridge Mathematical Library): Books: Richard Dedekind,John Stillwell (Site not responding. Last check: 2007-10-20)
		Dedekind's invention of ideals in the 1870s was a major turning point in the development of algebra.
		This is Dedekind's famous creation of the theory of (algebraic number) rings and modules, which he presented as an appendix to his edition of Dirichlet's LECTURES ON NUMBER THEORY.
		Dedekind (most of the time) explicitly limits himself to modules of algebraic numbers, but Noether correctly saw that Dedekind already knew (many of) his theorems held for the whole abstract range she would explicate and develop.
	www.amazon.com /exec/obidos/tg/detail/-/0521565189?v=glance (1329 words)

	Richard Dedekind at opensource encyclopedia (Site not responding. Last check: 2007-10-20)
		es:Julius Wilhelm Richard Dedekind ja:リヒャルト・デーデキント nl:Julius Wilhelm Richard Dedekind sl:Julius Wilhelm Richard Dedekind
		In 1863, he published Dirichlet's lectures on number theory in Vorlesungen über Zahlentheorie (Essays on the Theory of numbers).
		As a first part of this work he published his cognitions on his major rigorous redefinition of irrational numbers in terms of Dedekind cut named Stetigkeit und irrationale Zahlen (Continuity and irrational numbers) in 1872.
	www.wiki.tatet.com /Richard_Dedekind.html (898 words)

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