What is "tetration"

n. (context arithmetic English) The arithmetic operator consisting of repeated exponentiation, by analogy with exponentiation being repeated multiplication and multiplication being repeated addition, ^''b''''a'' denoting ''a'' to the power of ''a'' to the power of ... to the power of ''a'', in which ''a'' appears ''b'' times. Notation: ^''b''''a'', ''$auparrowuparrow\; b$ ''or'' $arightarrow\; brightarrow\; 2$

In mathematics, tetration (or hyper-4) is the next hyperoperation 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. The notation a means a, the application of exponentiation n times.

Shown here are the first four hyperoperations, with tetration as the fourth (and succession, the unary operation denoted aʹ = a + 1 taking a and yielding the number after a, as the 0th):

1. Addition

   
   $$a + n = a + \underbrace{1 + 1 + \cdots + 1}_n$$
   

   n copies of 1 added to a.
2. Multiplication

   
   $$a \times n = \underbrace{a + a + \cdots + a}_n$$
   

   n copies of a combined by addition.
3. Exponentiation

   
   $$a^n = \underbrace{a \times a \times \cdots \times a}_n$$
   

   n copies of a combined by multiplication.
4. Tetration

   
   $${^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_n$$
   

   n copies of a combined by exponentiation, right-to-left.

The above example is read as "the nth tetration of a". Each operation is defined by iterating the previous one (the next operation in the sequence is pentation). Tetration is neither an elementary function nor an elementary recursive function.

Here, succession (aʹ = a + 1) is the most basic operation; addition (a + n) is a primary operation, though for natural numbers it can be thought of as a chained succession of n successors of a; multiplication (an) is also a primary operation, though for natural numbers it can be thought of as a chained addition involving n numbers a; and exponentiation (a) can be thought of as a chained multiplication involving n numbers a. Analogously, tetration (a) 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).