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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 Midy's theorem ： ウィキペディア英語版 Midy's theorem In mathematics, Midy's theorem, named after French mathematician E. Midy, is a statement about the decimal expansion of fractions ''a''/''p'' where ''p'' is a prime and ''a''/''p'' has a repeating decimal expansion with an even period . If the period of the decimal representation of ''a''/''p'' is 2''n'', so that :$\backslash frac=0.\backslash overline\}$ then the digits in the second half of the repeating decimal period are the 9s complement of the corresponding digits in its first half. In other words, :$a\_i+a\_=9\; \backslash ,$ :$a\_1\backslash dots\; a\_n+a\_\backslash dots\; a\_=10^n-1.\; \backslash ,$ For example, :$\backslash frac=0.\backslash overline\backslash text076+923=999.\; \backslash ,$ :$\backslash frac=0.\backslash overline\backslash text05882352+94117647=99999999.\; \backslash ,$ ==Extended Midy's theorem== If ''k'' is any divisor of the period of the decimal expansion of ''a''/''p'' (where ''p'' is again a prime), then Midy's theorem can be generalised as follows. The extended Midy's theorem〔Bassam Abdul-Baki, (''Extended Midy's Theorem'' ), 2005.〕 states that if the repeating portion of the decimal expansion of ''a''/''p'' is divided into ''k''-digit numbers, then their sum is a multiple of 10''k'' − 1. For example, :$\backslash frac=0.\backslash overline\; \backslash ,$ has a period of 18. Dividing the repeating portion into 6-digit numbers and summing them gives :$052631+578947+368421=999999.$ Similarly, dividing the repeating portion into 3-digit numbers and summing them gives :$052+631+578+947+368+421=2997=3\backslash times999.$ ==Midy's theorem in other bases== Midy's theorem and its extension do not depend on special properties of the decimal expansion, but work equally well in any base ''b'', provided we replace 10''k'' − 1 with ''b''''k'' − 1 and carry out addition in base ''b''. For example, in octal :$$ \begin & \frac=0.\overline_8 \\() & 032_8+745_8=777_8 \\() & 03_8+27_8+45_8=77_8. \end In duodecimal (using inverted two and three for ten and eleven, respectively) : $$ \begin & \frac=0.\overline_ \\() & 076_+\mathcal45_=\mathcal_ \\() & 07_+6\mathcal_+45_=\mathcal_ \end 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「Midy's 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.