In mathematics, tetration (also known as hyper-4) is an iterated exponential, the first hyper operator after exponentiation. The word tetration was coined by English mathematician Reuben Louis Goodstein from tetra- (four) and iteration. Tetration is used for the notation of very large numbers but has few practical applications, so it has only been studied in pure mathematics. Shown here are examples of the first four hyper operators, with tetration as the fourth:
- addition The primary and simplest operation.
- multiplication generally also one of the primary operations, but in special case (for natural numbers) can be seen as a added to itself, n times.
- exponentiation a multiplied by itself, n times.
- tetration a exponentiated by itself, n times.
In mathematics, the hyperoperation sequence[nb 1] is a sequence of binary operations that starts with addition, multiplication and exponentiation, called hyperoperations in general. The nth member of this sequence is named by Reuben Goodstein after the Greek prefix of n suffixed with -ation (such as tetration, pentation) and can be written using (n − 2) arrows in Knuth's up-arrow notation. Each hyperoperation is defined recursively in terms of the previous one, as is the case with arrow notation. The part of the definition that does this is the recursion rule of the Ackermann function:
which is common to many variants of hyperoperations (see below).