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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A Dedekind cut is a partition of the rational numbers into two non-empty sets A and Bsuch that all elements of A are less than all elements of Band A contains no greatest element. The set B may or may not have a smallest element among the rationals. If B dedeknd a smallest element among the rationals, the cut corresponds to that rational.
An irrational cut is equated to an irrational number dedeekind is in neither set. Every real number, rational or not, is equated to one and only one cut of rationals. See also completeness order theory. It is straightforward to show that a Dedekind cut among the real numbers is uniquely defined by the corresponding cut among the rational numbers.
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. 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. A similar construction to that used by Dedekind cuts was used in Euclid’s Elements book V, definition 5 to define proportional segments.
By relaxing the first two requirements, we formally obtain the extended real number line. It is more symmetrical to use the AB notation for Dedekind cuts, but each of A and B does determine the other. It can be a simplification, in terms of notation if nothing more, to concentrate on one “half” — say, the lower one — and call dr downward closed set A without greatest element a “Dedekind cut”.
In this case, we say that b is represented by the cut AB.
The important purpose of the Dedekind cut is to work with number sets that are not complete. The cut itself can represent a number not in the original collection of numbers most often rational numbers. 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.
In this way, set inclusion can be used to represent the ordering of numbers, and all other relations greater thanless than or equal toequal toand so on can be similarly created from set relations. The set of all Dedekind cuts is itself a linearly ordered set of sets. Moreover, the set of Dedekind cuts has the least-upper-bound propertyi.
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.
To establish this truly, one must show that this really is a cut and that it is the square root of two.
Dedekind cut – Wikipedia
However, neither claim is immediate. A construction similar to Dedekind cuts is used for the construction of surreal numbers.
More generally, if S is a partially ordered seta completion of S means a complete lattice L with an order-embedding of S into L. The notion of complete lattice generalizes the least-upper-bound property of the reals. One completion of S is the set of its downwardly closed subsets, ordered by inclusion. A related completion that preserves all existing sups and infs of S is obtained by the following construction: For each subset A of Slet A u denote the set of upper bounds of Aand let A l denote the set of lower bounds of A.
These operators form a Galois connection. The Dedekind-MacNeille completion is the smallest complete lattice with S embedded in it. From Wikipedia, the free encyclopedia. This article needs additional citations for verification.
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The specific problem is: Contains information outside the dedeind of the article Please help improve this article if you can. June Learn how and when to remove this template message. Whenever, then, we have to do with a cut produced by no rational number, we create a new irrational number, which we regard as completely defined by this cut From now on, therefore, to every definite ddekind there corresponds a definite rational or irrational number Integer Dedekind cut Dyadic rational Half-integer Superparticular ratio.
Retrieved from ” https: Order theory Rational numbers.
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