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

allowing for the holomorphic extension of tetration to non-natural numbers such as real, complex, and ordinal numbers, which was proved in 2017.

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.

The first four hyperoperations are shown here, with tetration being considered the fourth in the series. The unary operation succession, defined as ${\displaystyle a'=a+1}$, is considered to be the zeroth operation.