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Jai Ganesh

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Registered: 2005-06-28

Posts: 48,422

Tetration

Tetration

In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though

  and the left-exponent

are common.

Under the definition as repeated exponentiation,

means

, where n copies of a are iterated via exponentiation, right-to-left, I.e. the application of exponentiation 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".

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 attempts to extend tetration to non-natural numbers such as real and complex numbers.

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.

Introduction

The first four hyperoperations are shown here, with tetration being considered the fourth in the series. The unary operation succession, defined as

, is considered to be the zeroth operation.

Addition

n copies of 1 added to a.

Multiplication

n copies of a combined by addition.

Exponentiation

n copies of a combined by multiplication.

Tetration

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

Succession, (a′ = a + 1), is the most basic operation; while addition (a + n) is a primary operation, for addition of 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 analogously be thought of as a chained addition involving n numbers of a. Exponentiation can be thought of as a chained multiplication involving n numbers of a and tetration

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.

The parameter a is referred to as the base, while the parameter n may be referred to as the height. In the original definition of tetration, the height parameter must be a natural number; for instance, it would be illogical to say "three raised to itself negative five times" or "four raised to itself one half of a time." However, just as addition, multiplication, and exponentiation can be defined in ways that allow for extensions to real and complex numbers, several attempts have been made to generalize tetration to negative numbers, real numbers, and complex numbers. One such way for doing so is using a recursive definition for tetration; for any positive real

and non-negative integer

, we can define

recursively as:

The recursive definition is equivalent to repeated exponentiation for natural heights; however, this definition allows for extensions to the other heights such as

,

, and

as well – many of these extensions are areas of active research.

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