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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 Hyperoperation ： ウィキペディア英語版 Hyperoperation In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called ''hyperoperations'')〔〔〔 that starts with the unary operation of successor (''n'' = 0), then continues with the binary operations of addition (''n'' = 1), multiplication (''n'' = 2), and exponentiation (''n'' = 3), after which the sequence proceeds with further binary operations extending beyond exponentiation, using right-associativity. For the operations beyond exponentiation, the ''n''th member of this sequence is named by Reuben Goodstein after the Greek prefix of ''n'' suffixed with ''-ation'' (such as tetration (''n'' = 4), pentation (''n'' = 5), hexation (''n'' = 6), etc.)〔 and can be written as using ''n'' − 2 arrows in Knuth's up-arrow notation. Each hyperoperation may be understood recursively in terms of the previous one by: :$$ \begin a \uparrow^m b & = & \underbrace (a \uparrow^ (...(a \uparrow^ (a \uparrow^ a))...)))} \\ & & b\mboxa \end (''m'' ≥ 0) It may also be defined according to the recursion rule part of the definition, as in Knuth's up-arrow version of the Ackermann function: :$a\; \backslash uparrow^m\; b\; =\; a\; \backslash uparrow^\; (a\; \backslash uparrow^m\; (b-1))$ (''m'' ≥ -1) This can be used to easily show numbers much larger than those which scientific notation can, such as Skewes' number and googolplexplex, but there are some numbers which even they cannot easily show, such as Graham's number and TREE(3). This recursion rule is common to many variants of hyperoperations (see below). == Definition == The ''hyperoperation sequence'' is the sequence of binary operations $H\_n(a,b)\; \backslash ,:\backslash ,\; (\backslash mathbb\_0)^3\; \backslash rightarrow\; \backslash mathbb\_0\backslash ,\backslash !$, defined recursively as follows: :$$ H_n(a,b) = \begin b + 1 & \text n = 0 \\ a &\text n = 1, b = 0 \\ 0 &\text n = 2, b = 0 \\ 1 &\text n \ge 3, b = 0 \\ H_(a,H_n(a,b-1)) & \text \end\,\! (Note that for ''n'' = 0, the binary operation essentially reduces to a unary operation (successor function) by ignoring the first argument.) For ''n'' = 0, 1, 2, 3, this definition reproduces the basic arithmetic operations of successor (which is a unary operation), addition, multiplication, and exponentiation, respectively, as :$H\_0(a,b)\; =\; b\; +\; 1\backslash ,\backslash !,$ :$H\_1(a,b)\; =\; a\; +\; b\backslash ,\backslash !,$ :$H\_2(a,b)\; =\; a\; \backslash cdot\; b\backslash ,\backslash !,$ :$H\_3(a,b)\; =\; a^\backslash ,\backslash !,$ and for ''n'' ≥ 4 it extends these basic operations beyond exponentiation to what can be written in Knuth's up-arrow notation as :$H\_4(a,b)\; =\; a\backslash uparrow\backslash uparrow\backslash ,\backslash !,$ :$H\_5(a,b)\; =\; a\backslash uparrow\backslash uparrow\backslash uparrow\backslash ,\backslash !,$ :... :$H\_n(a,b)\; =\; a\backslash uparrow^b\; \backslash text\; n\; \backslash ge\; 3\backslash ,\backslash !,$ :... Knuth's notation could be extended to negative indices ≥ -2 in such a way as to agree with the entire hyperoperation sequence, except for the lag in the indexing: :$H\_n(a,b)\; =\; a\; \backslash uparrow^b\backslash text\; n\; \backslash ge\; 0.\backslash ,\backslash !$ The hyperoperations can thus be seen as an answer to the question "what's next" in the sequence: successor, addition, multiplication, exponentiation, and so on. Noting that * $a\; +\; b\; =\; (a\; +\; (b\; -\; 1))\; +\; 1$ * $a\; \backslash cdot\; b\; =\; a\; +\; (a\; \backslash cdot\; (b\; -\; 1))$ * $a\; ^\; b\; =\; a\; \backslash cdot\; (a\; ^\; )$ * $a\backslash uparrow\backslash uparrow\; b=\; a\; ^\{a\; \backslash uparrow\backslash uparrow\; \{(b\; -\; 1)\}\}$ the relationship between basic arithmetic operations is illustrated, allowing the higher operations to be defined naturally as above. The parameters of the hyperoperation hierarchy are sometimes referred to by their analogous exponentiation term;〔 〕 so ''a'' is the ''base'', ''b'' is the ''exponent'' (or ''hyperexponent''),〔 and ''n'' is the ''rank'' (or ''grade'').〔 In common terms, the hyperoperations are ways of compounding numbers that increase in growth based on the iteration of the previous hyperoperation. The concepts of successor, addition, multiplication and exponentiation are all hyperoperations; the successor operation (producing ''x''+1 from ''x'') is the most primitive, the addition operator specifies the number of times 1 is to be added to itself to produce a final value, multiplication specifies the number of times a number is to be added to itself, and exponentiation refers to the number of times a number is to be multiplied by itself. 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「Hyperoperation」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.