Tetration

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 ${\displaystyle \uparrow \uparrow }$ and the left-exponent ${\displaystyle {}^{x}b}$ are common.

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

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

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

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.