          Tetration Domain coloring of the holomorphic tetration ${}^{z}e$ , with hue representing the function argument and brightness representing magnitude ${}^{n}x$ , for n = 2, 3, 4, …, showing convergence to the infinitely iterated exponential between the two dots

In mathematics, tetration (or hyper-4) is iterated, or repeated, exponentiation. It is the next hyperoperation after exponentiation, but before pentation. 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 ${^{n}a}$ means ${a^{a^{\cdot ^{\cdot ^{a}}}}}$ , which is the application of exponentiation $n-1$ times.

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

$a+n=a+\underbrace {1+1+\cdots +1} _{n}$ n copies of 1 added to a.
2. Multiplication
$a\times n=\underbrace {a+a+\cdots +a} _{n}$ n copies of a combined by addition.
3. Exponentiation
$a^{n}=\underbrace {a\times a\times \cdots \times a} _{n}$ n copies of a combined by multiplication.
4. Tetration
${^{n}a}=\underbrace {a^{a^{\cdot ^{\cdot ^{a}}}}} _{n}$ 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 ((a × n) is also a primary operation, though for natural numbers it can be thought of as a chained addition involving n numbers a. Exponentiation ($a^{n}$ ) can be thought of as a chained multiplication involving n numbers a, and analogously, tetration ($^{n}a$ ) 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 $a>0$ and non-negative integer $n\geq 0$ , we can define $\,\!{^{n}a}$ recursively as:

${^{n}a}:={\begin{cases}1&{\text{if }}n=0\\a^{\left(^{(n-1)}a\right)}&{\text{if }}n>0\end{cases}}$ This formal definition is equivalent to repeated exponentiation for natural heights; however, this definition allows for extensions to other heights such as $^{0}a$ , $^{-1}a$ , and $^{i}a$ .

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  