{{위키데이터 속성 추적}}
{{번역 확장 필요|en|Tetration}}
[[파일:TetrationComplexColor.png|섬네일|테두리|268px|[[정칙함수|정칙]] 테트레이션 <math>{}^{z}e</math>의 복소수 그래프, [[색상]]은 함수의 [[편각 (수학)|편각]]을 나타내고, [[명도]]는 크기를 나타낸다.]]
[[파일:TetrationConvergence2D.png|섬네일|''n'' = 1, 2, 3 ...일 때 <math>{}^{n}x</math>가 두 점 사이에서 무한히 반복되는 지수가 수렴하는 것과 수렴선을 나타내고 있다.]]
[[파일:Infinite power tower.svg|섬네일|무한으로 발산하는 <math>\textstyle \lim_{n\rightarrow \infty} {}^nx</math>가 <math>\textstyle (e^{-1})^e \le x \le e^{e^{-1}}) </math>를 기준으로 수렴하는 모습.]]
[[수학]]에서 '''테트레이션'''({{llang|en|tetration, hyper-4}})은 [[거듭제곱]]의 다음 차례에 오는 [[하이퍼 연산]]으로, 거듭제곱의 반복으로 정의된다. 이 단어는 [[루벤 루이스 굿스타인]]이 4를 뜻하는 tetra-와 [[반복함수]]를 의미하는 iteration을 합성한 것이다. 테트레이션은 보통 [[큰 수]]를 표기하는 데 이용한다. 표기법 <math>{^{n}a}</math>은 <math>{a^{a^{\cdot^{\cdot^{a}}}}}</math>에서 거듭제곱이 <math>n-1</math>번 적용된 것을 의미한다.

여기에 나타낸 것은 처음 네 [[하이퍼연산]]을 나타낸 것으로, 테트레이션이 네 번째이다(그리고 [[다음수 함수|증분]], 증분은 <math>a' = a + 1</math>로 정의된 단항 연산으로 <math>a</math>를 넣으면 <math>a</math>의 다음 수를 얻는 연산이 0번째다):
# [[덧셈]]
#: <math>a + n = a + \underbrace{1 + 1 + \cdots + 1}_n</math>
#:: ''n''번 ''a''에 1이 더해졌다.
# [[곱셈]]
#: <math>a \times n = \underbrace{a + a + \cdots + a}_n</math>
#:: ''n''번 ''a''가 덧셈으로 결합했다.
# [[거듭제곱]]
#: <math>a^n = \underbrace{a \times a \times \cdots \times a}_n</math>
#:: ''n''번 ''a''가 곱셈으로 결합했다.
# 테트레이션
#: <math>{^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_n</math>
#:: ''n''번 ''a''가 오른쪽에서 왼쪽으로 결합했다.

위의 예시들은 "''a''의 ''n번째'' 테트레이션"(the ''nth'' tetration of ''a'')이라고 읽는다. 각 연산은 이전의 연산을 반복하는 것으로 정의한다(이 수열의 다음 연산은 [[펜테이션]]이다). 테트레이션은 [[초등함수]]가 아니다.
<!--
증분 {{math|(''a''' {{=}} ''a'' + 1)}}은 가장 기본적인 연산이다. 덧셈 ({{math|''a + n''}})은 일차 연산이고  is a primary operation, though for natural numbers it can be thought of as a chained succession of {{mvar|n}} successors of {{mvar|a}}; multiplication ({{mvar|an}}) is also a primary operation, though for natural numbers it can be thought of as a chained addition involving {{mvar|n}} numbers {{mvar|a}}; and exponentiation (<math>a^n</math>) can be thought of as a chained multiplication involving {{mvar|n}} numbers {{mvar|a}}. Analogously, tetration (<math>^{n}a</math>) can be thought of as a chained power involving {{mvar|n}} numbers {{mvar|a}}. The parameter {{mvar|a}} may be called the base-parameter in the following, while the parameter {{mvar|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 number|real]] <math> a > 0 </math> and non-negative [[integer]] <math> n \ge 0 </math>, we define <math>\,\! {^{n}a} </math> by:
:<math>{^{n}a} := \begin{cases} 1 &\text{if }n=0 \\ a^{\left[^{(n-1)}a\right]} &\text{if }n>0 \end{cases} </math>

== Iterated powers vs. iterated exponentials ==
As we can see from the definition, when evaluating tetration expressed as an "exponentiation tower", the exponentiation is done at the deepest level first (in the notation, at the highest level). For example:
:<math>\,\!\ ^{4}2 = 2^{2^{2^2}} = 2^{\left[2^{\left(2^2\right)}\right]} = 2^{\left(2^4\right)} = 2^{16} = 65,\!536</math>

Note that exponentiation is not [[associative]], so evaluating the expression in the other order will lead to a different answer:
:<math>\,\! 2^{2^{2^2}} \ne \left[{\left(2^2\right)}^2\right]^2 = 2^{2 \cdot 2 \cdot2} = 256</math>

Exponential towers must be evaluated from top to bottom (or right to left). Computer programmers refer to this choice as [[right-associative]].

When ''a'' and 10 are [[coprime]], we can compute the last ''m'' decimal digits of <math>\,\!\ ^{n}a</math> using [[Euler's theorem]], for any integer ''m''.
-->
== 같이 보기 ==
* [[아커만 함수]]
* [[이중 지수 함수]]
* [[하이퍼 연산]]
* [[반복 로그]]
* [[대칭 레벨-인덱스 연산]]
* [[펜테이션]]

== 참고 문헌 ==
* Daniel Geisler, ''[http://www.tetration.org/ tetration.org]''
* Ioannis Galidakis, ''[https://web.archive.org/web/20090520164620/http://ioannis.virtualcomposer2000.com/math/exponents4.html On extending hyper4 to nonintegers]'' (undated, 2006 or earlier) ''(A simpler, easier to read review of the next reference)''
* Ioannis Galidakis, ''[https://web.archive.org/web/20131214131548/http://ioannis.virtualcomposer2000.com/math/papers/Extensions.pdf On Extending hyper4 and Knuth's Up-arrow Notation to the Reals]'' (undated, 2006 or earlier).
* Robert Munafo, ''[http://mrob.com/pub/math/hyper4.html#real-hyper4 Extension of the hyper4 function to reals]'' ''(An informal discussion about extending tetration to the real numbers.)''
* Lode Vandevenne, ''[http://groups.google.com/group/sci.math/browse_frm/thread/39a7019f9051c5d7/8c1c4facb7e4bd6d#8c1c4facb7e4bd6d Tetration of the Square Root of Two]'', (2004). ''(Attempt to extend tetration to real numbers.)''
* Ioannis Galidakis, ''[https://web.archive.org/web/20090420132250/http://ioannis.virtualcomposer2000.com/math/index.html Mathematics]'', ''(Definitive list of references to tetration research. Lots of information on the Lambert W function, Riemann surfaces, and analytic continuation.)''
* Galidakis, Ioannis and Weisstein, Eric W. [http://mathworld.wolfram.com/PowerTower.html Power Tower]
* Joseph MacDonell, ''[http://www.faculty.fairfield.edu/jmac/ther/tower.htm Some Critical Points of the Hyperpower Function] {{웨이백|url=http://www.faculty.fairfield.edu/jmac/ther/tower.htm |date=20100117161604 }}''.
* Dave L. Renfro, ''[http://mathforum.org/discuss/sci.math/t/350321 Web pages for infinitely iterated exponentials]'' ''(Compilation of entries from questions about tetration on sci.math.)''
* R. Knobel. "Exponentials Reiterated." ''[[American Mathematical Monthly]]'' '''88''', (1981), p.&nbsp;235–252.
* Hans Maurer. "Über die Funktion <math>y=x^{[x^{[x(\cdots)]}]}</math> für ganzzahliges Argument (Abundanzen)." ''Mittheilungen der Mathematische Gesellschaft in Hamburg'' '''4''', (1901), p.&nbsp;33–50. ''(Reference to usage of <math>\ {^{n} a}</math> from Knobel's paper.)''

== 외부 링크 ==
* [http://www.tetration.org/ Daniel Geisler's site on tetration]
* [http://math.eretrandre.org/tetrationforum/index.php Tetration Forum]
* [https://web.archive.org/web/20150203140427/http://mizugadro.mydns.jp/t/index.php/Tetration Tetration - TORI - Mizugadro, the research site by Dmitrii Kouznetsov]
* [http://go.helms-net.de/math/tetdocs/ Gottfried Helms' site on tetration]

{{큰 수}}

[[분류:이항연산]]
[[분류:큰 수]]
[[분류:거듭제곱]]