Dedekind cut

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Autor(es): dc.creatorSchreiber, Michael-
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Descrição: dc.descriptionEducação Superior::Ciências Exatas e da Terra::Matemática-
Descrição: dc.descriptionDedekind invented cuts to construct the real numbers from the rationals. Another method is to use Cauchy sequences. Split the rationals in two disjoint sets A and B, such that all the elements of A are smaller than all the element of B. This is called a cut. There are four cases: A has a largest element or not, and B has a smallest element or not. The case where A has a largest element x and B has a smallest element y is impossible. On the one hand, the average of x and y, being a rational, must belong to one of A or B. On the other hand, their average cannot belong to A (because x < (x+y)/2) nor to B (because (x+y)/2 < y). If there is a largest element of A or a smallest element of B, then the cut is rational. In the fourth case, the most interesting one, A does not have a largest element and B does not have a smallest element. In that case the cut is irrational. This visualization draws circles with rational radii smaller than 1. Examples of rational cuts are selected from these, with a red circle used to indicate that the rational is included in one of the two sets. Examples for irrational cuts are generated as multiples of √2/2-
Idioma: dc.languageen-
Publicador: dc.publisherWolfram demonstrations project-
Relação: dc.relationDedekindCut.nbp-
Direitos: dc.rightsDemonstration freeware using MathematicaPlayer-
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Palavras-chave: dc.subjectEducação Superior::Ciências Exatas e da Terra::Matemática::Teoria dos Números-
Palavras-chave: dc.subjectRepresentações de números-
Título: dc.titleDedekind cut-
???dc.description2???: dc.description2De acordo com dois conjuntos criados por Dedekind, a partir dos racionais, construir o conjunto dos números reais-
???dc.description3???: dc.description3This demonstration needs the "MathematicaPlayer.exe" to run. Find it in
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