Japan’s largest platform for academic e-journals: J-STAGE is a full text database for reviewed academic papers published by Japanese societies. de deux règles de verre accolées, déterminant trois lignes parallèles horizontales. qui lui apporte la théorie des coupures venue de Dedekind par Poincaré. des approximations de Théon de Smyrne Ainsi, m, · V2 coupures d’Eudoxe et de Dedekind ne.

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Every real number, rational or not, is equated to one and only one cut of rationals. March Learn how and when to remove this template message. This page was last edited on dedemind Novemberat Retrieved from ” https: From Wikipedia, the free encyclopedia. One completion of S is the set of its downwardly closed subsets, ordered by inclusion.

Order theory Rational numbers. By relaxing the first two requirements, we formally obtain the extended real number line. Similarly, every cut of reals is identical to the cut produced by a specific real number which can be identified as the smallest element of the B set. I, the copyright holder of this work, release this work into the public domain. The Dedekind-MacNeille completion is the smallest complete lattice with S embedded in it.

A related completion that preserves all existing sups and infs of S is obtained by the following construction: I grant anyone the right to use this work for any purposewithout any conditions, unless such conditions are required by law. Views Dedejind Edit History. This article needs additional citations for verification.

Contains information outside the scope of the article Please help improve this article if you can. See also completeness order theory. Unsourced material may be challenged and removed.

It is straightforward to show that a Dedekind cut among the real numbers is uniquely defined by the corresponding cut among the rational numbers.

## Dedekind cut

The set of all Dedekind cuts is itself a linearly ordered set of sets. Integer Dedekind cut Dyadic rational Half-integer Superparticular ratio. From now on, therefore, to every definite cut there corresponds a definite rational or irrational number This page was last edited on 28 Octoberat In other words, the number line where every real number is defined as a Dedekind cut of rationals is a complete continuum without any further gaps.

Moreover, the set of Dedekind cuts has the least-upper-bound propertyi. These operators form cpupures Galois connection. The notion of complete lattice generalizes the least-upper-bound property of the reals. Thus, constructing the set of Dedekind cuts serves the purpose of embedding the original ordered set Swhich might not have had the least-upper-bound property, within a usually larger linearly ordered set that does have this useful property.

More generally, if S is a partially ordered seta completion of S means a complete lattice L with an order-embedding fedekind S into L. It can be a simplification, in terms of notation if nothing more, to concentrate on one “half” — say, the lower one — and call any downward closed set A without greatest element a “Dedekind cut”. In this case, we say that b is represented by the cut AB. Description Dedekind cut- square root of two.

A similar construction to that used by Dedekind cuts was used in Euclid’s Elements book V, definition 5 to define proportional segments.

### KUNUGUI : Sur une Généralisation de la Coupure de Dedekind

The cut can represent a number beven though the numbers contained in the two sets A and B do not actually include the number b that their cut represents. Views Read Edit View history. Summary [ edit ] Description Dedekind cut- square root of two.

Articles needing additional references from March All articles needing additional references Articles needing cleanup from June All pages needing cleanup Cleanup tagged articles with a reason field from June Wikipedia pages needing cleanup from June In this way, set inclusion can be used to represent the ordering of numbers, and all other relations greater thanless than or equal tocoupurs toand so on can be similarly created from set relations.

### Sur une Généralisation de la Coupure de Dedekind

It is more symmetrical to use the AB notation for Dedekind cuts, but each of A and B does determine the other. An irrational cut is equated to an irrational number which is in neither set. This file contains additional information such as Exif metadata which may have been added by decekind digital camera, scanner, or software program used to create or digitize it.

By using this site, you agree to the Terms of Use and Privacy Policy. If the file has been modified from its original state, some details such as the timestamp may not fully reflect ciupures of the original file. The cut itself can represent a number not in the original collection of numbers most often rational numbers. To establish this truly, one must show that this really is a cut and that it is the square root of two.

From Wikimedia Commons, the free media repository. In some countries this may not be legally possible; if so: Dedekind cut sqrt 2. All those whose square is less than two redand those whose square is equal to or greater than two blue. A Dedekind cut is a partition of the rational numbers into two non-empty cuopures A and Bsuch that all elements of A are less than all elements of Coupkresand A contains no greatest element.