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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 Knuth's up-arrow notation ： ウィキペディア英語版 Knuth's up-arrow notation In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976. It is closely related to the Ackermann function and especially to the hyperoperation sequence. The idea is based on the fact that multiplication can be viewed as iterated addition and exponentiation as iterated multiplication. Continuing in this manner leads to iterated exponentiation (tetration) and to the remainder of the hyperoperation sequence, which is commonly denoted using Knuth arrow notation. ==Introduction== The ordinary arithmetical operations of addition, multiplication and exponentiation are naturally extended into a sequence of hyperoperations as follows. Multiplication by a natural number is defined as iterated addition: :$$ \begin a\times b & = & \underbrace \\ & & b\mboxa \end For example, :$$ \begin 4\times 3 & = & \underbrace & = & 12\\ & & 3\mbox4 \end Exponentiation for a natural power $b$ is defined as iterated multiplication, which Knuth denoted by a single up-arrow: :$$ \begin a\uparrow b= a^b = & \underbrace\\ & b\mboxa \end For example, :$$ \begin 4\uparrow 3= 4^3 = & \underbrace & = & 64\\ & 3\mbox4 \end To extend the sequence of operations beyond exponentiation, Knuth defined a “double arrow” operator to denote iterated exponentiation (tetration): :$$ \begin a\uparrow\uparrow b & = = & \underbrace} & = & \underbrace \\ & & b\mboxa & & b\mboxa \end For example, :$$ \begin 4\uparrow\uparrow 3 & = = & \underbrace & = & 4^ & \approx & 1.340~780~79\times 10^& \\ & & 3\mbox4 & & 3\mbox4 \end Here and below evaluation is to take place from right to left, as Knuth's arrow operators (just like exponentiation) are defined to be right-associative. According to this definition, :$3\backslash uparrow\backslash uparrow\; 2=3^3=27$ :$3\backslash uparrow\backslash uparrow\; 3=3^=3^=7~625~597~484~987$ :$3\backslash uparrow\backslash uparrow\; 4=3^\}=3^\backslash approx\; 1.258~014~3\backslash times\; 10^$ :$3\backslash uparrow\backslash uparrow\; 5=3^=3^=3^$ a\uparrow\uparrow\uparrow b= & \underbrace\\ & b\mboxa \end followed by a “quadruple arrow“ operator for iterated pentation (hexation): :$$ \begin a\uparrow\uparrow\uparrow\uparrow b= & \underbrace\\ & b\mboxa \end and so on. The general rule is that an $n$-arrow operator expands into a right-associative series of ($n\; -\; 1$)-arrow operators. Symbolically, :$$ \begin a\ \underbrace_\ b= \underbrace \ (a\ \underbrace_ \ (\dots \ \underbrace_ \ a))}_ \end Examples: :$3\backslash uparrow\backslash uparrow\backslash uparrow2\; =\; 3\backslash uparrow\backslash uparrow3\; =\; 3^\; =\; 3^=7~625~597~484~987$ :$$ \begin 3\uparrow\uparrow\uparrow3 = 3\uparrow\uparrow(3\uparrow\uparrow3) = 3\uparrow\uparrow(3\uparrow3\uparrow3) = & \underbrace \\ & 3\uparrow3\uparrow3\mbox3 \end \begin = & \underbrace \\ & \mbox \end \begin = & \underbrace}}}}}} \\ & \mbox \end The notation $a\; \backslash uparrow^n\; b$ is commonly used to denote $a\; \backslash uparrow\backslash uparrow\; \backslash dots\; \backslash uparrow\; b$ with ''n'' arrows. In fact, $a\; \backslash uparrow^n\; b$ is ''a'' () ''b'' with hyperoperation. For example,$39\backslash uparrow\backslash uparrow14$ can also be written as 39 () 14, the "()" means tetration, but it does not equal to 39 () 14 = 39 × 14 = 546, similarly, $77\; \backslash uparrow^\; 77$ = 77 () 77 instead of 77 () 77. 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「Knuth's up-arrow notation」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.