Non-integer base of numeration (original) (raw)

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A non-integer representation uses non-integer numbers as the radix, or base, of a positional numeral system. For a non-integer radix β > 1, the value of is The numbers di are non-negative integers less than β. This is also known as a β-expansion, a notion introduced by and first studied in detail by . Every real number has at least one (possibly infinite) β-expansion. The set of all β-expansions that have a finite representation is a subset of the ring Z[β, β−1]. There are applications of β-expansions in coding theory and models of quasicrystals .

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dbo:abstract A non-integer representation uses non-integer numbers as the radix, or base, of a positional numeral system. For a non-integer radix β > 1, the value of is The numbers di are non-negative integers less than β. This is also known as a β-expansion, a notion introduced by and first studied in detail by . Every real number has at least one (possibly infinite) β-expansion. The set of all β-expansions that have a finite representation is a subset of the ring Z[β, β−1]. There are applications of β-expansions in coding theory and models of quasicrystals . (en) Une numération en base non entière ou représentation non entière d'un nombre utilise, comme base de la notation positionnelle, un nombre qui n'est pas un entier. Si la base est notée , l'écriture dénote, comme dans les autres notations positionnelles, le nombre . Les nombres sont des entiers positifs ou nuls plus petits que . L'expression est aussi connue sous le terme β-développement (en anglais β-expansion). Tout nombre réel possède au moins un, et éventuellement une infinité de β-développements. La notion a été introduite par le mathématicien hongrois Alfréd Rényi en 1957 et étudiée en détail ensuite par William Parry en 1960. Depuis, de nombreux développements ultérieurs ont été apportés, dans le cadre de la théorie des nombres et de l’informatique théorique. Il y a des applications en théorie des codes et dans la modélisation de quasi-cristaux. (fr) 非整数进位制是指底数不是正整數的进位制。對於一個非正整數的底數β > 1，以下的數值:為 而數字di為小於β的非負整數。此進位制可以配合所使用β，稱為β进制或β展開，後者的名稱是數學家Rényi在1957年開始使用，而數學家Parry在1960年第一個進行相關的研究。每一個實數至少有一個β进位制的表示方式（也可能是無限多個）。 β进制可以應用在编码理论及準晶體模型的描述。 (zh)
dbo:wikiPageExternalLink http://intlpress.com/site/pub/pages/journals/items/mrl/content/vols/0008/0004/00019835/index.html http://www.americanscientist.org/issues/pub/third-base/2%7Cyear=2001%7Cvolume=89%7Cissue=6%7Cpages=490%E2%80%93494%7Cdoi=10.1511/2001.40.3268 https://books.google.com/books%3Fid=I3fC6batwokC&pg=PA154 https://web.archive.org/web/20160324100419/http:/www.americanscientist.org/issues/pub/2001/11/third-base/2
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dbp:title Base (en)
dbp:urlname Base (en)
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rdfs:comment A non-integer representation uses non-integer numbers as the radix, or base, of a positional numeral system. For a non-integer radix β > 1, the value of is The numbers di are non-negative integers less than β. This is also known as a β-expansion, a notion introduced by and first studied in detail by . Every real number has at least one (possibly infinite) β-expansion. The set of all β-expansions that have a finite representation is a subset of the ring Z[β, β−1]. There are applications of β-expansions in coding theory and models of quasicrystals . (en) 非整数进位制是指底数不是正整數的进位制。對於一個非正整數的底數β > 1，以下的數值:為 而數字di為小於β的非負整數。此進位制可以配合所使用β，稱為β进制或β展開，後者的名稱是數學家Rényi在1957年開始使用，而數學家Parry在1960年第一個進行相關的研究。每一個實數至少有一個β进位制的表示方式（也可能是無限多個）。 β进制可以應用在编码理论及準晶體模型的描述。 (zh) Une numération en base non entière ou représentation non entière d'un nombre utilise, comme base de la notation positionnelle, un nombre qui n'est pas un entier. Si la base est notée , l'écriture dénote, comme dans les autres notations positionnelles, le nombre . (fr)
rdfs:label Numération en base non entière (fr) Non-integer base of numeration (en) 非整数进位制 (zh)
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