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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 normal number ： ウィキペディア英語版 normal number In mathematics, a normal number is a real number whose infinite sequence of digits in every base b〔The only bases considered here are natural numbers greater than 1〕 is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b, also all possible b2 pairs of digits are equally likely with density b−2, all b3 triplets of digits equally likely with density b−3, etc. Intuitively this means that no digit, or combination of digits, occurs more frequently than any other, and this is true whether the number is written in base 10, binary, or any other base. A normal number can be thought of as an infinite sequence of coin flips (binary) or rolls of a die (base 6). Even though there ''will'' be sequences such as 10, 100, or more consecutive tails (binary) or fives (base 6) or even 10, 100, or more repetitions of a sequence such as tail-head (two consecutive coin flips) or 6-1 (two consecutive rolls of a die), there will also be equally many of any other sequence of equal length. No digit or sequence is "favored". While a general proof can be given that almost all real numbers are normal (in the sense that the set of exceptions has Lebesgue measure zero), this proof is not constructive and only very few specific numbers have been shown to be normal. For example, it is widely believed that the numbers , π, and ''e'' are normal, but a proof remains elusive. == Definitions == Let Σ be a finite alphabet of ''b'' digits, and Σ∞ the set of all sequences that may be drawn from that alphabet. Let ''S'' ∈ Σ∞ be such a sequence. For each ''a'' in Σ let ''NS''(''a'', ''n'') denote the number of times the letter ''a'' appears in the first ''n'' digits of the sequence ''S''. We say that ''S'' is simply normal if the limit :$\backslash lim\_\; \backslash frac\; =\; \backslash frac$ for each ''a''. Now let ''w'' be any finite string in Σ∗ and let ''NS''(''w'', ''n'') to be the number of times the string ''w'' appears as a substring in the first ''n'' digits of the sequence ''S''. (For instance, if ''S'' = 01010101..., then ''NS''(010, 8) = 3.) ''S'' is normal if, for all finite strings ''w'' ∈ Σ∗, :$\backslash lim\_\; \backslash frac\; =\; \backslash frac),\; 0\; and\; 1\; each\; occur\; with\; frequency1⁄$2; 00, 01, 10, and 11 each occur with frequency 1⁄4; 000, 001, 010, 011, 100, 101, 110, and 111 each occur with frequency 1⁄8, etc. Roughly speaking, the probability of finding the string ''w'' in any given position in ''S'' is precisely that expected if the sequence had been produced at random. Suppose now that ''b'' is an integer greater than 1 and ''x'' is a real number. Consider the infinite digit sequence expansion ''Sx, b'' of ''x'' in the base ''b'' positional number system (we ignore the decimal point). We say that ''x'' is simply normal in base ''b'' if the sequence ''Sx, b'' is simply normal and that ''x'' is normal in base ''b'' if the sequence ''Sx, b'' is normal. The number ''x'' is called a normal number (or sometimes an absolutely normal number) if it is normal in base ''b'' for every integer ''b'' greater than 1. A given infinite sequence is either normal or not normal, whereas a real number, having a different base-''b'' expansion for each integer ''b'' ≥ 2, may be normal in one base but not in another. For bases ''r'' and ''s'' with log ''r'' / log ''s'' rational (so that ''r'' = ''b''''m'' and ''s'' = ''b''''n'') every number normal in base ''r'' is normal in base ''s''. For bases ''r'' and ''s'' with log ''r'' / log ''s'' irrational, there are uncountably many numbers normal in each base but not the other.〔 A disjunctive sequence is a sequence in which every finite string appears. A normal sequence is disjunctive, but a disjunctive sequence need not be normal. A ''rich number'' in base ''b'' is one whose expansion in base ''b'' is disjunctive: one that is disjunctive to every base is called ''absolutely disjunctive'' or is said to be a ''lexicon''. A number normal in base ''b'' is rich in base ''b'', but not necessarily conversely. The real number ''x'' is rich in base ''b'' if and only if the set is dense in the unit interval.〔〔''x bn'' mod 1 denotes the fractional part of ''x bn''.〕 We defined a number to be simply normal in base ''b'' if each individual digit appears with frequency 1/''b''. For a given base ''b'', a number can be simply normal (but not normal or ''b''-dense), ''b''-dense (but not simply normal or normal), normal (and thus simply normal and ''b''-dense), or none of these. A number is absolutely non-normal or absolutely abnormal if it is not simply normal in any base.〔 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「normal number」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.