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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 Lochs' theorem ： ウィキペディア英語版 Lochs' theorem In number theory, Lochs' theorem is a theorem concerning the rate of convergence of the continued fraction expansion of a typical real number. A proof of the theorem was published by Gustav Lochs in 1964.〔.〕 The theorem states that for almost all real numbers in the interval (0,1), the number of terms ''m'' of the number's continued fraction expansion that are required to determine the first ''n'' places of the number's decimal expansion behaves asymptotically as follows: :$\backslash lim\_\; \backslash frac\; =\; \backslash frac\; \backslash approx\; 0.97027014$ . As this limit is only slightly smaller than 1, this can be interpreted as saying that each additional term in the continued fraction representation of a "typical" real number increases the accuracy of the representation by approximately one decimal place. The decimal system is the last positional system for which each digit carries less information than one continued fraction quotient; going to base-11 (changing $\backslash ln\; 10$ to $\backslash ln\; 11$ in the equation) makes the above value exceed 1. The reciprocal of this limit, :$\backslash frac\; \backslash approx\; 1.03064083$ , is twice the base-10 logarithm of Lévy's constant. ==References== 〔 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「Lochs' theorem」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.