Tetration

In mathematics, tetration (or hyper-4) is iterated, or repeated, exponentiation. 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 used for the notation of very large numbers. The notation n a {displaystyle {^{n}a}} means a a ⋅ ⋅ a {displaystyle {a^{a^{cdot ^{cdot ^{a}}}}}} , which is the application of exponentiation n − 1 {displaystyle n-1} times. In mathematics, tetration (or hyper-4) is iterated, or repeated, exponentiation. 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 used for the notation of very large numbers. The notation n a {displaystyle {^{n}a}} means a a ⋅ ⋅ a {displaystyle {a^{a^{cdot ^{cdot ^{a}}}}}} , which is the application of exponentiation n − 1 {displaystyle n-1} times. The first four hyperoperations are shown here, with tetration being the fourth of these (in this case, the unary operation succession, a ′ = a + 1 {displaystyle a'=a+1} , is considered to be the zeroth operation). Here, succession (a′ = 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 (a × n) is also a primary operation, though for natural numbers it can be thought of as a chained addition involving n numbers of a. Exponentiation ( a n {displaystyle a^{n}!} ) can be thought of as a chained multiplication involving n numbers of a, and analogously, tetration ( n a {displaystyle ^{n}a!} ) can be thought of as a chained power involving n numbers a. Each of the operations above are defined by iterating the previous one; however, unlike the operations before it, tetration is not an elementary function. In what follows, the parameter a may be referred to as the base, while the parameter n may be referred to as the height (which is integral in the first approach, but may be generalized to fractional, real and complex numbers — see the section Extensions for more). Tetration is read as 'the nth tetration of a'. For any positive real a > 0 {displaystyle a>0} and non-negative integer n ≥ 0 {displaystyle ngeq 0} , we can define n a {displaystyle ,!{^{n}a}} recursively as: This formal definition is equivalent to repeated exponentiation for natural heights; however, this definition allows for extensions to other heights such as 0 a {displaystyle ^{0}a} , − 1 a {displaystyle ^{-1}a} , and i a {displaystyle ^{i}a} as well.