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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Non-integer representation ： ウィキペディア英語版 Non-integer representation A non-integer representation uses non-integer numbers as the radix, or bases, of a positional numbering system. For a non-integer radix β > 1, the value of :$x=d\_n\backslash dots\; d\_2d\_1d\_0.d\_d\_\backslash dots\; d\_$ is :$x=\backslash beta^nd\_n\; +\; \backslash cdots\; +\; \backslash beta^2d\_2\; +\; \backslash beta\; d\_1\; +\; d\_0\; +\; \backslash beta^d\_\; +\; \backslash beta^d\_\; +\; \backslash cdots\; +\; \backslash beta^d\_.$ The numbers ''d''''i'' 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. There are applications of β-expansions in coding theory and models of quasicrystals . ==Construction== β-expansions are a generalization of decimal expansions. While infinite decimal expansions are not unique (for example, 1.000... = 0.999...), all finite decimal expansions are unique. However, even finite β-expansions are not necessarily unique, for example φ + 1 = φ2 for β = φ, the golden ratio. A canonical choice for the β-expansion of a given real number can be determined by the following greedy algorithm, essentially due to and formulated as given here by . Let be the base and ''x'' a non-negative real number. Denote by the floor function of ''x'', that is, the greatest integer less than or equal to ''x'', and let = ''x'' − ⌊''x''⌋ be the fractional part of ''x''. There exists an integer ''k'' such that . Set :$d\_k\; =\; \backslash lfloor\; x/\backslash beta^k\backslash rfloor$ and :$r\_k\; =\; \backslash .\backslash ,$ For , put :$d\_j\; =\; \backslash lfloor\backslash beta\; r\_\backslash rfloor,\; \backslash quad\; r\_j\; =\; \backslash .$ In other words, the canonical β-expansion of ''x'' is defined by choosing the largest ''d''''k'' such that , then choosing the largest ''d''''k''−1 such that , etc. Thus it chooses the lexicographically largest string representing ''x''. With an integer base, this defines the usual radix expansion for the number ''x''. This construction extends the usual algorithm to possibly non-integer values of β. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Non-integer representation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.