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	Dedekind cut - Wikipedia, the free encyclopedia
		For example it is shown that the typical Dedekind cut in the real numbers is either a pair with A the interval (−∞, a), in which case B must be [ a, +∞); or a pair with A the interval (−∞, a ], in which case B must be (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.
		A construction similar to Dedekind cuts is used for the construction of surreal numbers.
	en.wikipedia.org /wiki/Dedekind_cut (511 words)

	Dedekind cut (Site not responding. Last check: 2007-10-21)
		In mathematics, a Dedekind cut in a totally ordered set ''S is a partition of it, (A, B), such that A is closed downwards (meaning that whenever a is in A and x ≤ a, then x is in A as well) and B is closed upwards.
		The Dedekind cut is named after Richard Dedekind, who invented this construction in order to represent the real number s as Dedekind cuts of the rational number s.
		Dedekind, Richard (1831-1916) study of CONTINUITY and definition of the real numbers in terms of Dedekind "cuts", the nature of number and mathematical induction, definition of finite and infinite sets; algebraic number fields, concept of RINGS.
	www.serebella.com /encyclopedia/article-Dedekind_cut.html (992 words)

	Richard Dedekind - Wikipedia, the free encyclopedia
		Dedekind was born in Braunschweig (Brunswick) the youngest of four children of Julius Levin Ulrich Dedekind.
		Dedekind received his doctorate in 1852 and he was Gauss's last student.
		For example, the square root of 2 is a cut putting the negative numbers and the numbers with square smaller than 2 into the lower, and the positive numbers with square greater than 2 into the higher class.
	en.wikipedia.org /wiki/Richard_Dedekind (951 words)

	Dedekind cut - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-10-21)
		In mathematics, a Dedekind cut in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for any element x in S, if a is in A and x ≤ a, then x is in A as well) and B is closed upwards.
		For example it is shown that the typical Dedekind cut in the real numbers is either a pair with A the interval (−∞, a), in which case B must be
		is a Dedekind cut we could call (−∞, a); by identifying a with it, the linearly ordered set S is embedded in the set of all Dedekind cuts of S. If the linearly ordered set S does not enjoy the least-upper-bound property, then the set of Dedekind cuts will be strictly bigger than S.
	encyclopedia.worldsearch.com /dedekind_cut.htm (382 words)

	Dedekind domain (Site not responding. Last check: 2007-10-21)
		In other words, a Dedekind domain is a commutative ring which is not a field, doesn't have zero divisors, and in which every ideal is finitely generated, every nonzero prime ideal is a maximal ideal, and which is integrally closed in its fraction field.
		The most important examples of Dedekind domains, and historically the motivating ones, arise from algebraic number field s: start with a finite field extension F of the rational number s 'Q' and consider the set of all elements of F which are algebraic integers (in other words, the integral closure of 'Z' in F).
		The study of Dedekind domains began when Dedekind introduced the notion of ideal in a ring in the hopes of compensating for the failure of unique factorization into primes in rings of algebraic integers.
	www.serebella.com /encyclopedia/article-Dedekind_domain.html (843 words)

	PlanetMath: Dedekind cuts
		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.
	www.planetmath.org /encyclopedia/DedekindCuts.html (594 words)

	Dedekind (Site not responding. Last check: 2007-10-21)
		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.
		Dedekind's brilliance consisted not only of the theorems and concepts that he studied but, because of his ability to formulate and express his ideas so clearly, he introduced a new style of mathematics that been a major influence on mathematicians ever since.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Dedekind.html (1962 words)

	Dedekind cut - Term Explanation on IndexSuche.Com
		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.
		A typical Dedekind cut of the rational_numbers is given by :A = \{ a\in\textbf{Q} : a^2 :B = \{ b\in\textbf{Q} : b^2 \ge 2 \land b > 0 \}.
		A construction similar to Dedekind cuts is used for the construction of surreal_numbers.
	www.indexsuche.com /Dedekind_cut.html (976 words)

	Dedekind cut -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		The original and most important cases are Dedekind cuts for (An integer or a fraction) rational numbers and (Any rational or irrational number) real numbers.
		For example it is shown that the typical Dedekind cut in the real numbers is either a pair with A the (The difference in pitch between two notes) interval (−∞, a), in which case B must be
		The Dedekind cut is named after (Click link for more info and facts about Richard Dedekind) Richard Dedekind, who invented this construction in order to represent the (Any rational or irrational number) real numbers as Dedekind cuts of the (An integer or a fraction) rational numbers.
	www.absoluteastronomy.com /encyclopedia/d/de/dedekind_cut.htm (435 words)

	dedekind completion (Site not responding. Last check: 2007-10-21)
		A Dedekind cut in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that whenever a is in A and x ≤ a, then x is in A as well), B is closed upwards.
		A typical Dedekind cut of the rational numbers is given by A = { a in Q : a
		A construction very similar to Dedekind cuts is used for the construction of surreal numbers.
	www.yourencyclopedia.net /Dedekind_completion.html (356 words)

	Talk:Dedekind cut - Wikipedia, the free encyclopedia
		The main result is that a Dedekind cut of the reals must be at some real x: so, (-∞,x) and [x,+∞) or the other one with x in the left interval.
		Arguably the business of Dedekind cuts of the rationals giving one the reals should be at construction of real numbers, which currently has no detail to speak of.
		Referencing between construction of real numbers and Dedekind cut should have to be correspondingly precise, and edits on either page (ideally) synchronized.
	en.wikipedia.org /wiki/Talk:Dedekind_cut (1732 words)

	Dedekind cut (Site not responding. Last check: 2007-10-21)
		In mathematics, a Dedekind cut in a totally ordered set S is a partition of it, (A, B), such that A isclosed downwards (meaning that for any element x in S, if a is in A and x ≤a, then x is in A as well) and B is closed upwards.
		Regard one Dedekind cut { A, B } as less than another Dedekind cut {C, D } if A is a proper subset of C, or, equivalently D is a proper subset ofB.
		The Dedekind cut is named after Richard Dedekind, who inventedthis construction in order to represent the real numbers as Dedekind cuts ofthe rational numbers.
	www.therfcc.org /dedekind-cut-32817.html (485 words)

	Encyclopedia: Dedekind cut (Site not responding. Last check: 2007-10-21)
		Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 â€“ February 12, 1916) was a German mathematician and Ernst Eduard Kummers closest follower in arithmetic.
		In mathematics, the supremum of an ordered set S is the least element (not necessarily in S) which is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound.
		A typical Dedekind cut of the rational numbers is given by Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 â€“ February 12, 1916) was a German mathematician and Ernst Eduard Kummers closest follower in arithmetic.
	www.nationmaster.com /encyclopedia/Dedekind-cut (1079 words)

	Math Forum - Ask Dr. Math
		The idea is that if you have the set of rational numbers (all numbers of the form p/q, where p and q are integers and q is not zero; the integers can be positive or negative), it ought to be possible to somehow create numbers like e and pi from this raw material.
		For example, Dedekind said that it would be possible to think of the number sqrt(2), which is not a rational number, as two sets of rational numbers.
		Dedekind devised a way of constructing the real numbers from the rationals by utilizing the "Dedekind Cut".
	mathforum.org /library/drmath/view/52511.html (945 words)

	Richard Julius Wilhelm Dedekind
		Dedekind's accomplishment was to define irrational numbers in terms of rationals.
		This cut is defined as a subdivision of the rational numbers into two nonempty sets satisfying the condition that any member of the first set is less than any member of the second and the first set has no largest members.
		note] Although Dedekind's work in defining irrational numbers seems to be of relatively little consequence in the discipline of Discrete Mathematics, the notion of the Dedekind cut is an early example of a formal procedure that can be used to partition a set (with the understanding that certain "cuts", i.e.
	www.engr.iupui.edu /~orr/webpages/cpt120/mathbios/rdedek.htm (843 words)

	Dedekind cut (Site not responding. Last check: 2007-10-21)
		In mathematics a Dedekind cut in a totally ordered set S is a partition of it (A B) such that A is closed downwards (meaning that for element x in S if a is in A and x ≤ a then x is in A as well) and B is closed upwards.
		For example it is shown that typical Dedekind cut in the real numbers either a pair with A the interval (-∞ a) in which case B must be [ a +∞); or a pair with A the interval (-∞ a ] in which case B must be (a +∞).
		A construction similar to Dedekind cuts is for the construction of surreal numbers.
	www.freeglossary.com /Dedekind_completion (769 words)

	AllRefer.com - Julius Wilhelm Richard Dedekind (Mathematics, Biography) - Encyclopedia
		Dedekind studied at GOttingen 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.allrefer.com /encyclopedia/D/Dedekind.html (222 words)

	Dedekind Cut History - Dedekind Cut Information
		His idea was to define a "cut" of the rational numbers corresponding to each irrational number.
		A set K of rational numbers is said to be a cut if (i) K contains at least one rational number, but not every rational number; (ii) if p is in K and q is a rational number less than p, then q is in K; and (iii) K contains no largest rational number.
		That is, the arithmetic of cuts obeys the same operational rules that govern the rational numbers.
	www.bookrags.com /sciences/mathematics/dedekind-cut-wom.html (652 words)

	Dedekind cut (Site not responding. Last check: 2007-10-21)
		If a is a member of S then the set :\, \ \ is a Dedekind cut we could call −∞, a ; by identifying a with it, the linearly ordered set S is embedded in the set of all Dedekind cuts of S.
		If the linearly ordered set S does not enjoy the least upper bound least-upper-bound property, then the set of Dedekind cuts will be strictly bigger than S.
		Regard one Dedekind cut as less than another Dedekind cut if A is a proper subset of C, or, equivalently D is a proper subset of B.
	www.uk.fraquisanto.net /Dedekind_cut (411 words)

	Dedekind, Richard -- Britannica Concise Encyclopedia - The online encyclopedia you can trust!
		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.
		Such a cut, which corresponds to a given value, defines an irrational number if no largest or no smallest is present in either part; whereas a rational is defined as a cut in which one part contains a smallest or a largest.
		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.britannica.com /ebc/article-9029718 (1546 words)

	Reals: Construction by Dedekind Cuts (Site not responding. Last check: 2007-10-21)
		Definition A Dedekind cut is a nonempty bounded set of rationals that is closed downward and has no greatest member.
		We may think of the Dedekind cuts as partitions of the rationals into two sets: a lower set `L` which is closed downward, and an upper set `U`, the complementary set, which is closed upwards.
		Notes: The advantage of the definition by Dedekind cuts is its simplicity and intuitive appeal.
	alpha.fdu.edu /~mayans/core/real_numbers_pages/real_construct1.html (489 words)

	Archimedes Plutonium (Site not responding. Last check: 2007-10-21)
		Although at first sight quite different from the method of nested intervals or of cuts, it is equivalent to either of them, in the sense that the number systems defined in these three ways have the same properties.
		That new idea of mine that AC = Dedekind cut lay slowly cooking and then when I asked on the Internet sci.math whether anyone had proved Axiom of Choice = Dedekind cut = Reals was the time when the idea came full bloom like a spring flower into my mind.
		But, since noone before me realized that AC was none other than a fancy Dedekind Cut to manufacture the Reals Irrationals from out of the Rationals or a subset of the Rationals, AC was misused and so many so-called valid proofs are fakeries.
	www.iw.net /~a_plutonium/File107.html (2837 words)

	Program Files\Netscape\Communicator\Program\dedexxx
		At this time Dedekind was thinking about how to teach differential and integral calculus when the thought of a 'Dedekind cut' came to him.
		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)

	Amazon.ca: Books: Essays on the Theory of Numbers (Site not responding. Last check: 2007-10-21)
		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.amazon.ca /exec/obidos/ASIN/0486210103 (893 words)

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