# Tetration

Domain coloring of the holomorphic tetration ${\displaystyle {}^{z}e}$, with hue representing the function argument and brightness representing magnitude
${\displaystyle {}^{n}x}$, for n = 1, 2, 3 ..., showing convergence to the infinitely iterated exponential between the two dots

In mathematics, tetration (or hyper-4) is the next hyperoperation after exponentiation, but before pentation and is as such 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. The notation ${\displaystyle {^{n}a}}$ means ${\displaystyle {a^{a^{\cdot ^{\cdot ^{a}}}}}}$, which is the application of exponentiation ${\displaystyle n-1}$ times.

The first four hyperoperations are shown here, with tetration being the fourth of these (in this case, the unary operation succession, ${\displaystyle a'=a+1}$, is considered to be the zeroth operation).

${\displaystyle a+n=a+\underbrace {1+1+\cdots +1} _{n}}$
n copies of 1 added to a.
2. Multiplication
${\displaystyle a\times n=\underbrace {a+a+\cdots +a} _{n}}$
n copies of a combined by addition.
3. Exponentiation
${\displaystyle a^{n}=\underbrace {a\times a\times \cdots \times a} _{n}}$
n copies of a combined by multiplication.
4. Tetration
${\displaystyle {^{n}a}=\underbrace {a^{a^{\cdot ^{\cdot ^{a}}}}} _{n}}$
n copies of a combined by exponentiation, right-to-left.

Here, succession (a' = 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 (${\displaystyle a\times n}$) is also a primary operation, though for natural numbers it can be thought of as a chained addition involving n numbers a. Exponentiation (${\displaystyle a^{n}}$) can be thought of as a chained multiplication involving n numbers a, and analogously, tetration (${\displaystyle ^{n}a}$) can be thought of as a chained power involving n numbers a. Each of the operations above are defined by iterating the previous one; however, unlike the operations before it, tetration is not an elementary function.

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). Tetration is read as "the nth tetration of a"

## Formal definition

For any positive real ${\displaystyle a>0}$ and non-negative integer ${\displaystyle n\geq 0}$, we can define ${\displaystyle \,\!{^{n}a}}$ recursively as:

${\displaystyle {^{n}a}:={\begin{cases}1&{\text{if }}n=0\\a^{\left(^{(n-1)}a\right)}&{\text{if }}n>0\end{cases}}}$

This formal definition is equivalent to repeated exponentiation for natural heights; however, this definition allows for extensions to other heights such as ${\displaystyle ^{0}a}$, ${\displaystyle ^{-1}a}$, and ${\displaystyle ^{i}a}$.

Other Languages
čeština: Tetrace
dansk: Tetration
Deutsch: Potenzturm
español: Tetración
Esperanto: Supereksponento
français: Tétration
한국어: 테트레이션
italiano: Tetrazione
עברית: טטרציה
magyar: Tetráció
Nederlands: Tetratie
polski: Tetracja
português: Tetração
română: Tetrație
русский: Тетрация
Simple English: Tetration
српски / srpski: Тетрација
suomi: Tetraatio
svenska: Tetraering
Türkçe: Tetrasyon
українська: Тетрація
Tiếng Việt: Túc thừa