Tetration

Not to be confused with Titration.

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 an operation based on iterated, or repeated, exponentiation. There is no universal notation for tetration, though Knuth's up arrow notation $a\uparrow \uparrow n$ and the left-exponent ${}^{n}a$ are common.

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

Tetration is the hyperoperation after exponentiation and before pentation. Along with the other hyperoperations, tetration is used for the notation of very large numbers. The name was coined by Reuben Goodstein from the prefix tetra- (meaning "four") and the word "iteration".

Tetration can also be defined recursively as

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

This form allows for the extension of tetration to more general domains for n than the natural numbers such as real, complex, or ordinal numbers.

The two inverses of tetration are called super-root and super-logarithm. They are respectively analogous to the operations of taking n-th roots and taking logarithms. None of the three functions are elementary.