Tetration

A colorful graphic with brightly colored loops that grow in intensity as the eye goes to the right — Domain coloring of the holomorphic tetration ${}^{z}e$ , with hue representing the function argument and brightness representing magnitude

A line graph with curves that bend upward dramatically as the values on the x-axis get larger — ${}^{n}x$ , for $n = 2, 3, 4, ...$ , showing convergence to the infinitely iterated exponential between the two dots

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

Under the definition as repeated exponentiation, ${^{n}a}$ means ${a^{a^{\cdot ^{\cdot ^{a}}}}}$ , where $n$ copies of $a$ are iterated via exponentiation, right-to-left, i.e. the application of exponentiation $n-1$ times. $n$ is called the "height" of the function, while $a$ is called the "base," analogous to exponentiation. It would be read as "the $n$ th tetration of $a$ ". An example is 3 tetrated to 4 is $3^{3^{3^{3}}}$

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 also defined recursively as

{a\uparrow \uparrow n}:={\begin{cases}1&{\text{if }}n=0,\\a^{a\uparrow \uparrow (n-1)}&{\text{if }}n>0,\end{cases}}

allowing for attempts to extend tetration to non-natural numbers such as real, complex, and ordinal numbers.

The two inverses of tetration are called super-root and super-logarithm, analogous to the nth root and the logarithmic functions. None of the three functions are elementary.

Tetration is used for the notation of very large numbers.