"Une num\u00E9ration en base non enti\u00E8re ou repr\u00E9sentation non enti\u00E8re d'un nombre utilise, comme base de la notation positionnelle, un nombre qui n'est pas un entier. Si la base est not\u00E9e , l'\u00E9criture d\u00E9note, 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 \u03B2-d\u00E9veloppement (en anglais \u03B2-expansion). Tout nombre r\u00E9el poss\u00E8de au moins un, et \u00E9ventuellement une infinit\u00E9 de \u03B2-d\u00E9veloppements. La notion a \u00E9t\u00E9 introduite par le math\u00E9maticien hongrois Alfr\u00E9d R\u00E9nyi en 1957 et \u00E9tudi\u00E9e en d\u00E9tail ensuite par William Parry en 1960. Depuis, de nombreux d\u00E9veloppements ult\u00E9rieurs ont \u00E9t\u00E9 apport\u00E9s, dans le cadre de la th\u00E9orie des nombres et de l\u2019informatique th\u00E9orique. Il y a des applications en th\u00E9orie des codes et dans la mod\u00E9lisation de quasi-cristaux."@fr . "Une num\u00E9ration en base non enti\u00E8re ou repr\u00E9sentation non enti\u00E8re d'un nombre utilise, comme base de la notation positionnelle, un nombre qui n'est pas un entier. Si la base est not\u00E9e , l'\u00E9criture d\u00E9note, comme dans les autres notations positionnelles, le nombre ."@fr . . . "\u975E\u6574\u6570\u8FDB\u4F4D\u5236"@zh . . . . . . . . "\u975E\u6574\u6570\u8FDB\u4F4D\u5236\u662F\u6307\u5E95\u6570\u4E0D\u662F\u6B63\u6574\u6578\u7684\u8FDB\u4F4D\u5236\u3002\u5C0D\u65BC\u4E00\u500B\u975E\u6B63\u6574\u6578\u7684\u5E95\u6578\u03B2 > 1\uFF0C\u4EE5\u4E0B\u7684\u6578\u503C:\u70BA \u800C\u6578\u5B57di\u70BA\u5C0F\u65BC\u03B2\u7684\u975E\u8CA0\u6574\u6578\u3002\u6B64\u9032\u4F4D\u5236\u53EF\u4EE5\u914D\u5408\u6240\u4F7F\u7528\u03B2\uFF0C\u7A31\u70BA\u03B2\u8FDB\u5236\u6216\u03B2\u5C55\u958B\uFF0C\u5F8C\u8005\u7684\u540D\u7A31\u662F\u6578\u5B78\u5BB6R\u00E9nyi\u57281957\u5E74\u958B\u59CB\u4F7F\u7528\uFF0C\u800C\u6578\u5B78\u5BB6Parry\u57281960\u5E74\u7B2C\u4E00\u500B\u9032\u884C\u76F8\u95DC\u7684\u7814\u7A76\u3002\u6BCF\u4E00\u500B\u5BE6\u6578\u81F3\u5C11\u6709\u4E00\u500B\u03B2\u8FDB\u4F4D\u5236\u7684\u8868\u793A\u65B9\u5F0F\uFF08\u4E5F\u53EF\u80FD\u662F\u7121\u9650\u591A\u500B\uFF09\u3002 \u03B2\u8FDB\u5236\u53EF\u4EE5\u61C9\u7528\u5728\u7F16\u7801\u7406\u8BBA\u53CA\u6E96\u6676\u9AD4\u6A21\u578B\u7684\u63CF\u8FF0\u3002"@zh . . . . . . . . . . . . "A non-integer representation uses non-integer numbers as the radix, or base, of a positional numeral system. For a non-integer radix \u03B2 > 1, the value of is The numbers di are non-negative integers less than \u03B2. This is also known as a \u03B2-expansion, a notion introduced by and first studied in detail by . Every real number has at least one (possibly infinite) \u03B2-expansion. The set of all \u03B2-expansions that have a finite representation is a subset of the ring Z[\u03B2,\u2009\u03B2\u22121]. There are applications of \u03B2-expansions in coding theory and models of quasicrystals ."@en . . . . . . . . "14849"^^ . . . . . . . . . . . "1123661732"^^ . . . . . . "\u975E\u6574\u6570\u8FDB\u4F4D\u5236\u662F\u6307\u5E95\u6570\u4E0D\u662F\u6B63\u6574\u6578\u7684\u8FDB\u4F4D\u5236\u3002\u5C0D\u65BC\u4E00\u500B\u975E\u6B63\u6574\u6578\u7684\u5E95\u6578\u03B2 > 1\uFF0C\u4EE5\u4E0B\u7684\u6578\u503C:\u70BA \u800C\u6578\u5B57di\u70BA\u5C0F\u65BC\u03B2\u7684\u975E\u8CA0\u6574\u6578\u3002\u6B64\u9032\u4F4D\u5236\u53EF\u4EE5\u914D\u5408\u6240\u4F7F\u7528\u03B2\uFF0C\u7A31\u70BA\u03B2\u8FDB\u5236\u6216\u03B2\u5C55\u958B\uFF0C\u5F8C\u8005\u7684\u540D\u7A31\u662F\u6578\u5B78\u5BB6R\u00E9nyi\u57281957\u5E74\u958B\u59CB\u4F7F\u7528\uFF0C\u800C\u6578\u5B78\u5BB6Parry\u57281960\u5E74\u7B2C\u4E00\u500B\u9032\u884C\u76F8\u95DC\u7684\u7814\u7A76\u3002\u6BCF\u4E00\u500B\u5BE6\u6578\u81F3\u5C11\u6709\u4E00\u500B\u03B2\u8FDB\u4F4D\u5236\u7684\u8868\u793A\u65B9\u5F0F\uFF08\u4E5F\u53EF\u80FD\u662F\u7121\u9650\u591A\u500B\uFF09\u3002 \u03B2\u8FDB\u5236\u53EF\u4EE5\u61C9\u7528\u5728\u7F16\u7801\u7406\u8BBA\u53CA\u6E96\u6676\u9AD4\u6A21\u578B\u7684\u63CF\u8FF0\u3002"@zh . . . . . . . . . . . . "6832938"^^ . . . "Num\u00E9ration en base non enti\u00E8re"@fr . . . . . . . . "Non-integer base of numeration"@en . . . . "A non-integer representation uses non-integer numbers as the radix, or base, of a positional numeral system. For a non-integer radix \u03B2 > 1, the value of is The numbers di are non-negative integers less than \u03B2. This is also known as a \u03B2-expansion, a notion introduced by and first studied in detail by . Every real number has at least one (possibly infinite) \u03B2-expansion. The set of all \u03B2-expansions that have a finite representation is a subset of the ring Z[\u03B2,\u2009\u03B2\u22121]. There are applications of \u03B2-expansions in coding theory and models of quasicrystals ."@en . "Base"@en . . . . . "Base"@en . . . . . . . . . . . .