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Complex plot of the holomorphic tetration

{}^{z}e

, with hue representing the function argument and brightness representing magnitude

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{}^{n}x

, for n = 1, 2, 3 ..., showing convergence to the infinitely iterated exponential between the two dots

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 ${^{n}a}$ means ${a^{a^{\cdot ^{\cdot ^{a}}}}}$ , the application of exponentiation $n-1$ times.

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

Addition

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

n copies of 1 added to a.
Multiplication

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

n copies of a combined by addition.
Exponentiation

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

n copies of a combined by multiplication.
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 not an elementary function.

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 ( $an$ ) is also a primary operation, though for natural numbers it can be thought of as a chained addition involving $n$ numbers $a$ ; and exponentiation ( $a^{n}$ ) can be thought of as a chained multiplication involving $n$ numbers $a$ . Analogously, tetration ( $^{n}a$ ) 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).

Definition[edit]

For any positive real $a>0$ and non-negative integer $n\geq 0$ , we define $\,\!{^{n}a}$ by:

{^{n}a}:={\begin{cases}1&{\text{if }}n=0\\a^{\left(^{(n-1)}a\right)}&{\text{if }}n>0\end{cases}}

Iterated powers vs. iterated exponentials[edit]

As we can see from the definition, when evaluating tetration expressed as an "exponentiation tower", the exponentiation is done at the deepest level first (in the notation, at the highest level). For example:

\,\!\ ^{4}2=2^{2^{2^{2}}}=2^{\left(2^{\left(2^{2}\right)}\right)}=2^{\left(2^{4}\right)}=2^{16}=65,\!536

Note that exponentiation is not associative, so evaluating the expression in the other order will lead to a different answer:

\,\!2^{2^{2^{2}}}\neq \left({\left(2^{2}\right)}^{2}\right)^{2}=4^{2\cdot 2}=256

Exponential towers must be evaluated from top to bottom (or right to left). Computer programmers refer to this choice as right-associative.

When a and 10 are coprime, we can compute the last m decimal digits of $\,\!\ ^{n}a$ using Euler's theorem, for any integer m.

Terminology[edit]

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^[1] (generalizing the recursive base-representation used in Goodstein's theorem to use higher operations), 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.^[2] It was used earlier by Ed Nelson in his book Predicative Arithmetic, Princeton University Press, 1986.
The term hyperpower^[3] 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^[4] is occasionally used, in the form "the power tower of order n" for ${\ \atop {\ }}{{\underbrace {a^{a^{\cdot ^{\cdot ^{a}}}}} } \atop n}$ . This is a misnomer, however, because tetration cannot be expressed with iterated power functions (see above), since it is an iterated exponential function.
The term snap is occasionally used in informal contexts, in the form "a snap n" for ${\ \atop {\ }}{{\underbrace {a^{a^{\cdot ^{\cdot ^{a}}}}} } \atop n}$ . This term is not yet widely accepted, although it is used within select communities. It is believed to be a reference to jounce, the fourth derivative of position in physics, since tetration is the fourth hyperoperation.

Owing in part to some shared terminology and similar notational symbolism, tetration is often confused with closely related functions and expressions. Here are a few related terms:

Form	Terminology
$a^{a^{\cdot ^{\cdot ^{a^{a}}}}}$	Tetration
$a^{a^{\cdot ^{\cdot ^{a^{x}}}}}$	Iterated exponentials
$a_{1}^{a_{2}^{\cdot ^{\cdot ^{a_{n}}}}}$	Nested exponentials (also towers)
$a_{1}^{a_{2}^{a_{3}^{\cdot ^{\cdot ^{\cdot }}}}}$	Infinite exponentials (also towers)

In the first two expressions a is the base, and the number of times a appears is the height (add one for x). In the third expression, n 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[edit]

There are many different notation styles that can be used to express tetration (also known as hyper-4; some of them can be used as well for hyper-5, hyper-6, and higher hyperoperations).

Name	Form	Description
Rudy Rucker notation	$\,{}^{n}a$	Used by Maurer [1901] and Goodstein [1947]; Rudy Rucker's book Infinity and the Mind popularized the notation.
Knuth's up-arrow notation	$a{\uparrow \uparrow }n$	Allows extension by putting more arrows, or, even more powerfully, an indexed arrow.
Conway chained arrow notation	$a\rightarrow n\rightarrow 2$	Allows extension by increasing the number 2 (equivalent with the extensions above), but also, even more powerfully, by extending the chain
Ackermann function	${}^{n}2=\operatorname {A} (4,n-3)+3$	Allows the special case $a=2$ to be written in terms of the Ackermann function.
Iterated exponential notation	${}^{n}a=\exp _{a}^{n}(1)$	Allows simple extension to iterated exponentials from initial values other than 1.
Hooshmand notations^[5]	$\operatorname {uxp} _{a}n$ $a^{\frac {n}{}}$
Hyperoperation notations	$a[4]n$ $H_{4}(a,n)$	Allows extension by increasing the number 4; this gives the family of hyperoperations
Text notation	`a^^n` `a[4]n`	Since the up-arrow is used identically to the caret (`^`), tetration may be written as (`^^`); convenient for ASCII.
Bowers's operators	{a,b,4,1} or a {4} b

One notation above uses iterated exponential notation; in general this is defined as follows:

\exp _{a}^{n}(x)=a^{a^{\cdot ^{\cdot ^{a^{x}}}}}

with n "a"s.

There are not as many notations for iterated exponentials, but here are a few:

Name	Form	Description
Standard notation	$\exp _{a}^{n}(x)$	Euler coined the notation $\exp _{a}(x)=a^{x}$ , and iteration notation $f^{n}(x)$ has been around about as long.
Knuth's up-arrow notation	$(a{\uparrow })^{n}(x)$	Allows for super-powers and super-exponential function by increasing the number of arrows; used in the article on large numbers.
Text notation	`exp_a^n(x)`	Based on standard notation; convenient for ASCII.
J Notation	`x^^:(n-1)x`	Repeats the exponentiation. See J (programming language)^[6]

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