Tetration
In mathematics, tetration is an operation based on iterated, or repeated, exponentiation. There is no universal notation for tetration, though Knuth's up arrow notation and the left-exponent are common.
Under the definition as repeated exponentiation, means, where copies of are iterated via exponentiation, right-to-left, i.e. the application of exponentiation times. The number is called the height of the function, while is called the base, analogous to exponentiation. It would be read as "the th tetration of ". For example, 2 tetrated to 4 is.
Tetration is the next hyperoperation after exponentiation, but before pentation. Along with the other hyperoperations, tetration is used for the notation of very large numbers. The name was coined by Reuben Goodstein from the prefix tetra- and the word "iteration".
Tetration can also be defined recursively as
This form allows for the extension of tetration to more general domains than the natural numbers such as real, complex, or ordinal numbers.
The two inverses of tetration are called super-root and super-logarithm. They are respectively analogous to the operations of taking th roots and taking logarithms. None of the three functions are elementary.
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 copies of 1 added to combined by succession.
- Multiplication copies of combined by addition.
- Exponentiation copies of combined by multiplication.
- Tetration copies of combined by exponentiation, right-to-left.
Succession,, is the most basic operation; while addition is a primary operation, for addition of natural numbers it can be thought of as a chained succession of successors of ; multiplication is also a primary operation, though for natural numbers it can analogously be thought of as a chained addition involving numbers of. Exponentiation can be thought of as a chained multiplication involving numbers of and tetration as a chained power involving numbers. 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 is referred to as the base, while the parameter 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.
Terminology
There are many terms for tetration, each of which has some logic behind it, but some have not become commonly used for one reason or another. Here is a comparison of each term with its rationale and counter-rationale.- The term tetration, introduced by Goodstein in his 1947 paper Transfinite Ordinals in Recursive Number Theory, has gained dominance. It was also popularized in Rudy Rucker's Infinity and the Mind.
- The term superexponentiation was published by Bromer in his paper Superexponentiation in 1987. It was used earlier by Ed Nelson in his book Predicative Arithmetic, Princeton University Press, 1986.
- The term hyperpower is a natural combination of hyper and power, which aptly describes tetration. The problem lies in the meaning of hyper with respect to the hyperoperation sequence. When considering hyperoperations, the term hyper refers to all ranks, and the term super refers to rank 4, or tetration. So under these considerations hyperpower is misleading, since it is only referring to tetration.
- The term power tower is occasionally used, in the form "the power tower of order " for. Exponentiation is easily misconstrued: note that the operation of raising to a power is right-associative. Tetration is iterated exponentiation, starting from the top right side of the expression with an instance a^a. Exponentiating the next leftward a, is to work leftward after obtaining the new value b^c. Working to the left, use the next a to the left, as the base b, and evaluate the new b^c. 'Descend down the tower' in turn, with the new value for c on the next downward step.
| Terminology | Form |
| Tetration | |
| Iterated exponentials | |
| Nested exponentials | |
| Infinite exponentials |
In the first two expressions is the base, and the number of times appears is the height. In the third expression, is the height, but each of the bases is different.
Care must be taken when referring to iterated exponentials, as it is common to call expressions of this form iterated exponentiation, which is ambiguous, as this can either mean iterated powers or iterated exponentials.
Notation
There are many different notation styles that can be used to express tetration. Some notations can also be used to describe other hyperoperations, while some are limited to tetration and have no immediate extension.| Name | Form | Description |
| Knuth's up-arrow notation | Allows extension by putting more arrows, or, even more powerfully, an indexed arrow. | |
| Conway chained arrow notation | Allows extension by increasing the number 2, but also, even more powerfully, by extending the chain. | |
| Ackermann function | Allows the special case to be written in terms of the Ackermann function. | |
| Iterated exponential notation | Allows simple extension to iterated exponentials from initial values other than 1. | |
| Hooshmand notations | Used by M. H. Hooshmand . | |
| Hyperoperation notations | Allows extension by increasing the number 4; this gives the family of hyperoperations. | |
| Double caret notation | Since the up-arrow is used identically to the caret, tetration may be written as ; convenient for ASCII. |
One notation above uses iterated exponential notation; this is defined in general as follows:
There are not as many notations for iterated exponentials, but here are a few:
| Name | Form | Description |
| Standard notation | Euler coined the notation, and iteration notation has been around about as long. | |
| Knuth's up-arrow notation | Allows for super-powers and super-exponential function by increasing the number of arrows; used in the article on large numbers. | |
| Text notation | Based on standard notation; convenient for ASCII. | |
| J notation | Repeats the exponentiation. See J. | |
| Infinity barrier notation | Jonathan Bowers coined this, and it can be extended to higher hyper-operations. |
Examples
Because of the extremely fast growth of tetration, most values in the following table are too large to write in scientific notation. In these cases, iterated exponential notation is used to express them in base 10. The values containing a decimal point are approximate. Usually, the limit that can be calculated in a numerical calculation program such as Wolfram Alpha is 3↑↑4, and the number of digits up to 3↑↑5 can be expressed.| 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 4 | 16 | 65,536 | 2.00353 × 10 | ||
| 3 | 27 | 7,625,597,484,987 | 1.25801 × 10 | |||
| 4 | 256 | 1.34078 × 10 | ||||
| 5 | 3,125 | 1.91101 × 10 | ||||
| 6 | 46,656 | 2.65912 × 10 | ||||
| 7 | 823,543 | 3.75982 × 10 | ||||
| 8 | 16,777,216 | 6.01452 × 10 | ||||
| 9 | 387,420,489 | 4.28125 × 10 | ||||
| 10 | 10,000,000,000 | 10 |
Remark: If does not differ from 10 by orders of magnitude, then for all. For example, in the above table, and the difference is even smaller for the following rows.