Editing Tetration (section)

{{Short description|Arithmetic operation}}
{{Distinguish|Titration (disambiguation){{!}}Titration}}
{{for|repeated tetration|Pentation}}
{{Use dmy dates|date=August 2020|cs1-dates=y}}
[[File:TetrationComplexColor.png|thumb|alt=A colorful graphic with brightly colored loops that grow in intensity as the eye goes to the right|[[Domain coloring]] of the [[holomorphic function|holomorphic]] tetration <math>{}^{z}e</math>, with [[hue]] representing the function [[Argument (complex analysis)|argument]] and [[brightness]] representing magnitude]]
[[File:TetrationConvergence2D.svg|thumb|alt=A line graph with curves that bend upward dramatically as the values on the x-axis get larger|<math>{}^{n}x</math>, for {{math|1=''n'' = 2, 3, 4, ...}}, showing convergence to the infinitely iterated exponential between the two dots]]

In [[mathematics]], '''tetration''' (or '''hyper-4''') is an [[operation (mathematics)|operation]] based on [[iterated]], or repeated, [[exponentiation]]. There is no standard [[mathematical notation|notation]] for tetration, though [[Knuth's up arrow notation]] <math>\uparrow \uparrow</math> and the left-exponent <math>{}^{x}b</math> are common.

Under the definition as repeated exponentiation, <math>{^{n}a}</math> means <math>{a^{a^{\cdot^{\cdot^{a}}}}}</math>, where ''{{mvar|n}}'' copies of ''{{mvar|a}}'' are iterated via exponentiation, right-to-left, i.e. the application of exponentiation <math>n-1</math> times. ''{{mvar|n}}'' is called the "height" of the function, while ''{{mvar|a}}'' is called the "base," analogous to exponentiation. It would be read as "the {{mvar|n}}th tetration of {{mvar|a}}". For example, 2 tetrated to 4 (or the fourth tetration of 2) is <math>{^{4}2}=2^{2^{2^{2}}}=2^{2^{4}}=2^{16}=65536</math>.

It is the next [[hyperoperation]] after [[exponentiation]], but before [[pentation]]. The word was coined by [[Reuben Louis Goodstein]] from [[tetra-]] (four) and [[iterated function|iteration]].

Tetration is also defined recursively as
: <math>{a \uparrow \uparrow n} := \begin{cases} 1 &\text{if }n=0, \\ a^{a \uparrow\uparrow (n-1)} &\text{if }n>0, \end{cases}</math>
allowing for the [[holomorphic function|holomorphic]] extension of tetration to [[Natural numbers|non-natural numbers]] such as [[Real number|real]], [[Complex number|complex]], and [[ordinal number]]s, which was proved in 2017.

The two inverses of tetration are called '''[[#Super-root|super-root]]''' and '''[[#Super-logarithm|super-logarithm]]''', analogous to the [[nth root]] and the logarithmic functions. None of the three functions are [[elementary function|elementary]].

Tetration is used for the [[Large numbers#Standardized system of writing|notation of very large numbers]].