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- Dedekind_cut abstract "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.If B has a smallest element among the rationals, the cut corresponds to that rational. Otherwise, that cut defines a unique irrational number which, loosely speaking, fills the "gap" 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. An irrational cut is equated to an irrational number which is in neither set. Every real number, rational or not, is equated to one and only one cut of rationals.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 cut there corresponds a definite rational or irrational number ....More generally, a Dedekind cut is a partition of a totally ordered set into two non-empty parts (A and 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).In particular, as discussed below, Dedekind cuts among the real numbers may be considered defined as cuts among the rationals. It turns out that 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.Dedekind used the German word Schnitt (cut) in a visual sense rooted in Euclidean geometry. When two straight lines cross, one is said to cut the other. Dedekind's construction of the number line ensures that two crossing lines always have one point in common because each of them defines a Dedekind cut on the other.".
- Dedekind_cut thumbnail Dedekind_cut_sqrt_2.svg?width=300.
- Dedekind_cut wikiPageExternalLink 21016.
- Dedekind_cut wikiPageID "45569".
- Dedekind_cut wikiPageRevisionID "562682847".
- Dedekind_cut hasPhotoCollection Dedekind_cut.
- Dedekind_cut id "p/d030530".
- Dedekind_cut title "Dedekind cut".
- Dedekind_cut subject Category:Order_theory.
- Dedekind_cut subject Category:Rational_numbers.
- Dedekind_cut type Abstraction100002137.
- Dedekind_cut type ComplexNumber113729428.
- Dedekind_cut type DefiniteQuantity113576101.
- Dedekind_cut type Measure100033615.
- Dedekind_cut type Number113582013.
- Dedekind_cut type RationalNumber113730469.
- Dedekind_cut type RationalNumbers.
- Dedekind_cut type RealNumber113729902.
- Dedekind_cut comment "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.If B has a smallest element among the rationals, the cut corresponds to that rational. Otherwise, that cut defines a unique irrational number which, loosely speaking, fills the "gap" between A and B.".
- Dedekind_cut label "Cortes de Dedekind".
- Dedekind_cut label "Cortes de Dedekind".
- Dedekind_cut label "Coupure de Dedekind".
- Dedekind_cut label "Dedekind cut".
- Dedekind_cut label "Dedekindscher Schnitt".
- Dedekind_cut label "Dedekindsnede".
- Dedekind_cut label "Przekrój Dedekinda".
- Dedekind_cut label "Sezione di Dedekind".
- Dedekind_cut label "Дедекиндово сечение".
- Dedekind_cut label "حد ديديكايند".
- Dedekind_cut label "デデキント切断".
- Dedekind_cut label "分划".
- Dedekind_cut sameAs Dedekindův_řez.
- Dedekind_cut sameAs Dedekindscher_Schnitt.
- Dedekind_cut sameAs Τομή_Ντέντεκιντ.
- Dedekind_cut sameAs Cortes_de_Dedekind.
- Dedekind_cut sameAs Coupure_de_Dedekind.
- Dedekind_cut sameAs Sezione_di_Dedekind.
- Dedekind_cut sameAs デデキント切断.
- Dedekind_cut sameAs Dedekindsnede.
- Dedekind_cut sameAs Przekrój_Dedekinda.
- Dedekind_cut sameAs Cortes_de_Dedekind.
- Dedekind_cut sameAs m.0cct6.
- Dedekind_cut sameAs Q851333.
- Dedekind_cut sameAs Q851333.
- Dedekind_cut sameAs Dedekind_cut.
- Dedekind_cut wasDerivedFrom Dedekind_cut?oldid=562682847.
- Dedekind_cut depiction Dedekind_cut_sqrt_2.svg.
- Dedekind_cut isPrimaryTopicOf Dedekind_cut.