Tetration

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 ${\displaystyle \uparrow \uparrow }$ 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. The number 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 ${\displaystyle {^{4}2}=2^{2^{2^{2}}}=2^{2^{4}}=2^{16}=65536}$.

Tetration is the next hyperoperation after exponentiation, but 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

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

This form allows for the extension of tetration to more general domains 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 nth roots and taking logarithms. None of the three functions are elementary.