1. Dedekind Cuts Class Notes/problem Set File Format PDF/Adobe Acrobat Quick View http://www.math.uiuc.edu/~dzaharo2/teaching/mathcampsets/dedekind_cuts.pdf

2. Dedekind Cuts A selection of articles related to Dedekind Cuts Dedekind cuts Encyclopedia Mathematical analysis. Analysis is the generic name given to any branch of mathematics which depends upon http://www.experiencefestival.com/dedekind_cuts

3. Dedekind_cuts | Define Dedekind_cuts At Dictionary.com –noun Mathematics . two nonempty subsets of an ordered field, as the rational numbers , such that one subset is the collection of upper bounds of the second and the second http://dictionary.reference.com/browse/Dedekind_cuts

4. Dedekind_cuts Synonyms, Dedekind_cuts Antonyms | Thesaurus.com No results found for dedekind_cuts Please try spelling the word differently, searching another resource, or typing a new word. Search another word or see dedekind_cuts on http://thesaurus.com/browse/Dedekind_cuts

5. Dedekind Cut - Handling Dedekind Cuts A selection of articles related to dedekind cut handling dedekind cuts http://www.experiencefestival.com/dedekind_cut_-_handling_dedekind_cuts

6. Dedekind Cuts Dedekind Cuts from WN Network. WorldNews delivers latest Breaking news including World News, US, politics, business, entertainment, science, http://wn.com/Dedekind_cuts

7. Dedekind Cut - Wikipedia, The Free Encyclopedia In mathematics, a Dedekind cut, named after Richard Dedekind, is a partition of the rational numbers into two nonempty parts A and B, such that all http://en.wikipedia.org/wiki/Dedekind_cut

Dedekind cut From Wikipedia, the free encyclopedia Jump to: navigation search Dedekind used his cut to construct the irrational real numbers In mathematics , a Dedekind cut , named after Richard Dedekind , is a partition of the rational numbers into two non-empty parts A and B , such that all elements of A are less than all elements of B and A contains no greatest element. The cut itself is, conceptually, the "gap" defined between A and B . In other words, A contains every rational number less than the cut, and B contains every rational number greater than the cut. The cut itself is in neither set. More generally, a Dedekind cut is a partition of a totally ordered set into two non-empty parts, ( A B ), such that A is closed downwards (meaning that for all a in A x a implies that x is in A as well) and B is closed upwards, and A contains no greatest element. See also completeness (order theory) The Dedekind cut resolves the contradiction between the continuous nature of the number line continuum and the discrete nature of the numbers themselves. Wherever a cut occurs and it is not on a real rational number , an irrational number (which is also a real number ) is created by the mathematician. Through the use of this device, there is considered to be a real number, either rational or irrational, at every point on the number line continuum, with no discontinuity.

8. PlanetMath: Dedekind Cuts May 16, 2002 The purpose of Dedekind cuts is to provide a sound logical foundation for the real number system. Dedekind s motivation behind this project http://planetmath.org/encyclopedia/DedekindCuts.html

(more info) Math for the people, by the people. donor list find out how Encyclopedia Requests ... Advanced search Login create new user name: pass: forget your password? Main Menu sections Encyclopædia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About Dedekind cuts (Definition) The purpose of Dedekind cuts is to provide a sound logical foundation for the real number system. Dedekind's motivation behind this project is to notice that a real number , intuitively, is completely determined by the rationals strictly smaller than and those strictly larger than . Concerning the completeness or continuity of the real line , Dedekind notes in [ ] that If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions. 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. He further notes that the completeness

9. Dedekind Cuts A dedekind cut comprises two nonempty sets of rationals, L and R, such that each rational appears in exactly one of the two sets, and all the rationals in L http://www.mathreference.com/top-ms,dcuts.html

Metric Spaces, Dedekind Cuts Dedekind Cuts Like a cauchy sequence, a dedekind cut defines a real number. We will show that these definitions are equivalent. However, the cauchy sequence is more general, because it can be applied to arbitrary mettric spaces. Also, most people find cauchy sequences more intuitive, so dedekind cuts are primarily of historical interest. You can skip this page if you like. A dedekind cut comprises two nonempty sets of rationals, L and R, such that each rational appears in exactly one of the two sets, and all the rationals in L (left) are less than all the rationals in R (right). We have cut the line in two, and the cut point becomes the real number. If b is an upper bound for L and c is a lower bound for R, c cannot be less than b, and there can't be any gap between, hence b = c. Since each point is suppose to be in just one set, decide arbitrarily that b belongs to L. In this case the real number is the rational b. One cut is less than another if R contains points not in R . Show this is a partial ordering; in fact it is a linear ordering.

10. Dedekind Cut -- From Wolfram MathWorld Oct 11, 2010 Dedekind Cuts. §2.2.6 in What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England Oxford University http://mathworld.wolfram.com/DedekindCut.html

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Interactive Demonstrations Dedekind Cut A set partition of the rational numbers into two nonempty subsets and such that all members of are less than those of and such that has no greatest member. Real numbers can be defined using either Dedekind cuts or Cauchy sequences SEE ALSO: Cantor-Dedekind Axiom Cauchy Sequence REFERENCES: Courant, R. and Robbins, H. "Alternative Methods of Defining Irrational Numbers. Dedekind Cuts." §2.2.6 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 71-72, 1996. Jeffreys, H. and Jeffreys, B. S. "Nests of Intervals: Dedekind Section." §1.031 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 6-8, 1988. CITE THIS AS: Weisstein, Eric W. "Dedekind Cut." From MathWorld A Wolfram Web Resource. http://mathworld.wolfram.com/DedekindCut.html Contact the MathWorld Team Wolfram Research, Inc.

11. Dedekind Cuts The purpose of Dedekind cuts is to provide a sound logical foundation for the real number system. Dedekind s motivation behind this project is to notice http://myyn.org/m/article/dedekind-cuts/

12. Dedekind Cuts | Homework Help A detailed tutorial on how to determine Dedekind cuts. Step by step tutorial including several examples of Dedekind cuts for reference. http://homeworkhowto.com/dedekind-cuts/

Search Home Math Algebra Arithmetic ... Writing Dedekind Cuts Tags: algebra between cut Dedekind ... than January 5, 2010 No Comments How to Determine Dedekind Cuts Description A detailed tutorial on how to determine Dedekind cuts. Step by step tutorial including several examples of Dedekind cuts for reference. Overview A Dedekind cut is a partition of rational numbers into two non-empty sets A and B, such that all elements of A are less than elements of B, and A has no greatest element. The cut itself is a gap that is located between A and B, which is normally found by creating a new, irrational number, and setting it in the gap. What irrational number you use depends on what numbers you have partitioned into the two sets. It is like the number line of advanced algebra, that has both rational and irrational numbers on it instead of just integers. The Dedekind cut was named after Richard Dedekind. Bookmark It Hide Sites VN:F [1.7.5_995] Rating: (from votes) Comments Leave a Reply Click here to cancel reply. You must be logged in to post a comment.

13. Math Forum - Ask Dr. Math Sep 30, 2002 Date 10/23/96 at 193015 From mat stern Subject Dedekind cut I have already figured out Dedekind s theory of the rings and number notation. http://mathforum.org/library/drmath/view/52511.html

Associated Topics Dr. Math Home Search Dr. Math Dedekind Cuts Date: 10/23/96 at 19:30:15 From: mat stern Subject: Dedekind cut I have already figured out Dedekind's theory of the rings and number notation. I still cannot figure out what his theory of the Dedekind cut is. Could you please send me a couple of sentences of what his cut is all aboutso that a 7th grader can understand? I have contacted several mathamatic instructors and they know but cannot tell me in language that I can understand. http://mathforum.org/dr.math/ http://mathforum.org/dr.math/ Associated Topics High School History/Biography High School Number Theory Search the Dr. Math Library: Find items containing (put spaces between keywords): Click only once for faster results: [ Choose "whole words" when searching for a word like age. all keywords, in any order at least one, that exact phrase parts of words whole words Submit your own question to Dr. Math Math Forum Home Math Library Quick Reference ... Math Forum Search Ask Dr. Math TM http://mathforum.org/dr.math/

14. Dedekind Cuts Of Rational Numbers -- Math Fun Facts One way to do this was proposed by Dedekind in 1872, who suggested looking at cuts . A cut C is a proper subset of rational numbers that is nonempty, http://www.math.hmc.edu/funfacts/ffiles/30005.3.shtml

hosted by the Harvey Mudd College Math Department Francis Su The Math Fun Facts App is now in the App Store Subscribe to our RSS feed or Any Easy Medium Advanced The Math Fun Facts App! List All List Recent List Popular ... Francis Edward Su From the Fun Fact files, here is a Fun Fact at the Advanced level: Dedekind Cuts of Rational Numbers Given a number line with equally spaced tick marks one unit apart, we know how to measure rational lengths: the length m/n can be obtained by dividing a length m line segment into n equal parts (if you like, this can be done by straightedge and compass). A very natural question you might ask is whether all lengths on the line are rational length? The Greeks knew that this was not the case; the square root of two is in fact irrational and can be obtained as the hypotenuse of a right triangle with side lengths 1 and 1. And there are other lengths (like Pi) which are irrational, but cannot be constructed by straightedge and compass? These numbers (representing lengths) have an ordering, thus can be associated with points along a line. What the above remarks show is that the set rational numbers in this line has "gaps". How does one "fill in the gaps" between the rational numbers One way to do this was proposed by Dedekind in 1872, who suggested looking at "cuts". A

15. Dedekind Cut: Definition From Answers.com Dedekind cut ( d d kint k t ) ( mathematics ) A set of rational numbers satisfying certain properties, with which a unique real number may be. http://www.answers.com/topic/dedekind-cut

16. Dedekind Cut In NLab Idea. Dedekind cuts are a way to make precise the idea that a real number is that which can be approximated (in the absolute value? metric) by rational numbers. http://ncatlab.org/nlab/show/Dedekind cut

nLab Dedekind cut Skip the Navigation Links Home Page All Pages Recently Revised ... Export Dedekind cuts Idea Definitions Motivation Dedekind completion ... References Idea Dedekind cuts are a way to make precise the idea that a real number is that which can be approximated (in the absolute value metric) by rational numbers In 1872, Richard Dedekind published Stetigkeit und irrationale Zahlen (Continuity and irrational numbers), in which he pointed out that a real number may be uniquely determined its order relationships with rational numbers. That is, the real number x is determined by its lower set L x and its upper set U x L x a a x U x b x b Dedekind had found great success understanding the ‘ideal numbers’ of ring theory as certain sets, which are what we now understand as ideals . So he defined a ‘real number’ as a pair of sets of rational numbers, the lower and upper sets shown above (actually a slight variation). Such a pair of sets of rational numbers he called a ‘Schnitt’ (English ‘cut’), nowadays called a ‘Dedekind cut’. Definitions Exactly which pairs of sets of rational numbers can appear this way? We will define a

17. Dedekind Cut - Wikivisual In this way, the set of all Dedekind cuts is itself a linearly ordered set, and, moreover, it has the leastupper-bound property, i.e., its every nonempty subset that has an upper http://en.wikivisual.com/index.php/Dedekind_cut

Francais English Dedekind cut From Wikipedia, the free encyclopedia Jump to: navigation search In mathematics , a Dedekind cut , named after Richard Dedekind , in a totally ordered set S is a partition of it, ( A B ), such that A is closed downwards (meaning that for all a in A x a implies that x is in A as well) and B is closed upwards, and A contains no greatest element. The cut itself is, conceptually, the "gap" defined between A and B . The original and most important cases are Dedekind cuts for rational numbers and real numbers . 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). The Dedekind cut resolves the contradiction between the continuous nature of the number line continuum and the discrete nature of the numbers themselves. Wherever a cut occurs and it is not on a real rational number , an irrational number (which is also a real number ) is created by the mathematician. Through the use of this device, there is considered to be a real number, either rational or irrational, at every point on the number line continuum, with no discontinuity. Contents Handling Dedekind cuts Ordering Dedekind cuts The cut construction of the real numbers Additional structure on the cuts ... edit Handling Dedekind cuts It is more symmetrical to use the ( A B ) notation for Dedekind cuts, but each of

18. Arithmetic Of Dedekind Cuts Of Ordered Abelian Groups - Microsoft Academic Searc Authors Antongiulio Fornasiero, Marcello Mamino. Citations 0 We study Dedekind cuts on ordered Abelian groups. We introduce a monoid structure on them, and we characterise, via a http://academic.research.microsoft.com/Paper/4753965.aspx

var SiteRoot = 'http://academic.research.microsoft.com'; SHARE Author Conference Journal Year Look for results that meet for the following criteria: since equal to before Publication Arithmetic of Dedekind cuts of ordered Abelian groups Edit Arithmetic of Dedekind cuts of ordered Abelian groups Antongiulio Fornasiero Marcello Mamino We study Dedekind cuts on ordered Abelian groups. We introduce a monoid structure on them, and we characterise, via a suitable representation theorem, the universal part of the theory of such structures. Journal: Annals of Pure and Applied Logic - APAL View or Download The following links allow you to view and download full papers. These links are maintained by other sources not affiliated with Microsoft Academic Search. www.dm.unipi.it dx.doi.org www.informatik.uni-trier.de Reference ... W. Baur Published in 1976. Partially Ordered Algebraic Systems Citations: 126 L. Fuchs

19. Construction Of The Real Numbers - Wikipedia, The Free Encyclopedia We form the set of real numbers as the set of all Dedekind cuts A of , and define a total ordering on the real numbers as follows We embed the rational numbers into the reals by http://en.wikipedia.org/wiki/Construction_of_the_real_numbers

Construction of the real numbers From Wikipedia, the free encyclopedia Jump to: navigation search In mathematics , there are several ways of defining the real number system as an ordered field . The synthetic approach gives a list of axioms for the real numbers as a complete ordered field . Under the usual axioms of set theory , one can show that these axioms are categorical, in the sense that there is a model for the axioms, and any two such models are isomorphic . Any one of these models must be explicitly constructed, and most of these models are built using the basic properties of the rational number system as an ordered field. Contents Synthetic approach Explicit constructions of models Construction from Cauchy sequences Construction by Dedekind cuts ... edit Synthetic approach The synthetic approach axiomatically defines the real number system as a complete ordered field. Precisely, this means the following. A model for the real number system consists of a set R , two distinct elements and 1 of R , two binary operations + and * on R (called addition and multiplication , resp.), a

20. Real Number In NLab Two successful approaches were developed in 1872, Richard Dedekind? 's definition of real numbers as certain sets of rational numbers (called Dedekind cuts) and Georg Cantor? 's http://ncatlab.org/nlab/show/real number

nLab real number Skip the Navigation Links Home Page All Pages Recently Revised ... Export Contents Idea History of formalisations Definitions and characterizations Dedekind cuts ... Generalisations Idea A real number is something that may be approximated by rational numbers . The real numbers form the real line , also known as the continuum , which is the completion of the field of rational numbers. To be precise, we may use the absolute value metric on rational numbers and forming a metric completion , or alternatively (and equivalently) use their unique ordering as an ordered field and forming the Dedekind completion Together with its cartesian powers – the cartesian spaces n – the continuum encodes one basic idea of of continuous space ; see cartesian space . The notion of (especially smooth) manifold is modeled on this notion. These provides some of the basic models of space , notably the standard model for physical space and time (see spacetime ), at least in classical physics History of formalisations The original idea of a real number came from geometry ; one thinks of a real number as specifying a point on a line , with

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