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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 tetration ： ウィキペディア英語版 tetration In mathematics, tetration (or hyper-4) is the next hyperoperator after exponentiation, and is defined as iterated exponentiation. The word was coined by Reuben Louis Goodstein, from tetra- (four) and iteration. Tetration is used for the notation of very large numbers. Shown here are the first four hyperoperators, with tetration as the fourth (and succession, the unary operation denoted $a\text{'}\; =\; a\; +\; 1$ taking $a$ and yielding the number after $a$, as the 0th): #Addition #:$a\; +\; n\; =\; a\; +\; \backslash underbrace\_n$ #::''n'' copies of 1 added to ''a''. #Multiplication #:$a\; \backslash times\; n\; =\; \backslash underbrace\_n$ #::''n'' copies of ''a'' combined by addition. #Exponentiation #:$a^n\; =\; \backslash underbrace\_n$ #::''n'' copies of ''a'' combined by multiplication. #Tetration #:$=\; \backslash underbrace\}\}\_n$ #::''n'' copies of ''a'' combined by exponentiation, right-to-left. Each operation is defined by iterating the previous one (the next operation in the sequence is pentation). Tetration is neither an elementary function nor an elementary recursive function.〔It is easy to prove that for every elementary function ''f'', there is a constant ''c'' s.t. $f(x)\; \backslash leq\; \backslash underbrace\}\}\_c$. We can show (by using diagonal argument and above fact) that $(c,\; x)\; \backslash mapsto\; \backslash underbrace\}\}\_c$ is non-elementary, and so is tetration.〕 Here, succession ($a\text{'}\; =\; a\; +\; 1$) is the most basic operation; addition ($a\; +\; n$) is a primary operation, though for natural numbers it can be thought of as a chained succession of ''n'' successors of ''a''; multiplication ($an$) is also a primary operation, though for natural numbers it can be thought of as a chained addition involving ''n'' numbers ''a''; and exponentiation ($a^n$) can be thought of as a chained multiplication involving ''n'' numbers ''a''. Analogously, tetration ($^a$) can be thought of as a chained power involving ''n'' numbers ''a''. The parameter ''a'' may be called the base-parameter in the following, while the parameter ''n'' in the following may be called the ''height''-parameter (which is integral in the first approach but may be generalized to fractional, real and complex ''heights'', see below). == Definition == For any positive real $a\; >\; 0$ and non-negative integer $n\; \backslash ge\; 0$, we define $\backslash ,\backslash !$ by: : 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「tetration」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.