Tetration

In mathematics, tetration (or hyper-4) is the next hyper operator after exponentiation, and is defined as iterated exponentiation. The word was coined by Reuben Louis Goodstein, from tetra- (four) and iteration. Tetration is used for the notation of very large numbers. Shown here are examples of the first four hyper operators, with tetration as the fourth (and succession, the unary operation denoted taking and yielding the number after, as the 0th):

1. Addition

   a succeeded n times.

2. Multiplication

   a added to itself, n times.

3. Exponentiation

   a multiplied by itself, n times.

4. Tetration

   a exponentiated by itself, n times.

where each operation is defined by iterating the previous one (the next operation in the sequence is pentation). The peculiarity of the tetration among these operations is that the first three (addition, multiplication and exponentiation) are generalized for complex values of n, while for tetration, no such regular generalization is yet established; and tetration is not considered an elementary function.

Addition is the most basic operation, multiplication is also a primary operation, though for natural numbers it can be thought of as a chained addition involving n numbers a, and exponentiation can be thought of as a chained multiplication involving n numbers a. Analogously, tetration can be thought of as a chained power involving n numbers a. The parameter a may be called the base-parameter in the following, while the parameter n in the following may be called the height-parameter (which is integral in the first approach but may be generalized to fractional, real and complex heights, see below).