# Tetration

In mathematics, **tetration** (or **hyper-4**) is the next hyper operator 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. Shown here are examples of the first four hyper operators, with tetration as the fourth (and succession, the unary operation denoted taking and yielding the number after , as the 0th):

- Addition
- Multiplication
- Exponentiation
- Tetration

where each operation is defined by iterating the previous one (the next operation in the sequence is pentation). Tetration is not an elementary function.

Tetration is not an elementary recursive function.^{[1]}

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

For any positive real and non-negative integer , we define by:

## Iterated powers vs. iterated bases/exponentiation

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). In other words:

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

Thus, the 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 using Euler's theorem.

## 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*^{[2]}(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.^{[3]}It was used earlier by Ed Nelson in his book Predicative Arithmetic, Princeton University Press, 1986. - The term
*hyperpower*^{[4]}is a natural combination of*hyper*and*power*, which aptly describes tetration. The problem lies in the meaning of*hyper*with respect to the hyper operator hierarchy. When considering hyper operators, 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*^{[5]}is occasionally used, in the form "the power tower of order*n*" for

Tetration is often confused with closely related functions and expressions. This is because much of the terminology that is used with them can be used with tetration. Here are a few related terms:

Form Terminology Tetration Iterated exponentials Nested exponentials (also towers) 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

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

Name Form Description Standard notation Used by Maurer [1901] and Goodstein [1947]; Rudy Rucker's book *Infinity and the Mind*popularized the notation.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 (equivalent with the extensions above), 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 ^{[6]}

Hyper operator notations

Allows extension by increasing the number 4; this gives the family of hyper operations ASCII notation `a^^n`

Since the up-arrow is used identically to the caret ( `^`

), the tetration operator may be written as (`^^`

).Bowers' array notation {a,b,2}

One notation above uses iterated exponential notation; in general this is defined 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. Ioannis Galidakis' notation Allows for large expressions in the base. ^{[7]}ASCII (auxiliary) `a^^n@x`

Based on the view that an iterated exponential is *auxiliary tetration*.ASCII (standard) `exp_a^n(x)`

Based on standard notation. J Notation `x^^:(n-1)x`

Repeats the exponentiation. See J (programming language) ^{[8]}

## Examples

In the following table, most values are too large to write in scientific notation, so iterated exponential notation is employed to express them in base 10. The values containing a decimal point are approximate.

1 1 1 1 2 4 16 65,536 3 27 7,625,597,484,987 4 256 1.34078*10 ^{154}5 3,125 6 46,656 7 823,543 8 16,777,216 9 387,420,489 10 10,000,000,000

## Extensions

Tetration can be extended to define and other domains as well.

### Extension of domain for bases

#### Extension to base zero

The exponential is not consistently defined. Thus, the tetrations are not clearly defined by the formula given earlier. However, is well defined, and exists:

Thus we could consistently define . This is equivalent to defining .

Under this extension, , so the rule from the original definition still holds.

#### Extension to complex bases

Since complex numbers can be raised to powers, tetration can be applied to *bases* of the form , where . For example, where , tetration is achieved by using the principal branch of the natural logarithm, and using Euler's formula we get the relation:

This suggests a recursive definition for given any :

The following approximate values can be derived:

Solving the inverse relation as in the previous section, yields the expected and , with negative values of *n* giving infinite results on the imaginary axis. Plotted in the complex plane, the entire sequence spirals to the limit , which could be interpreted as the value where *n* is infinite.

Such tetration sequences have been studied since the time of Euler but are poorly understood due to their chaotic behavior. Most published research historically has focused on the convergence of the power tower function. Current research has greatly benefited by the advent of powerful computers with fractal and symbolic mathematics software. Much of what is known about tetration comes from general knowledge of complex dynamics and specific research of the exponential map.

### Extensions of the domain for (iteration) "heights"

#### Extension to infinite heights

Tetration can be extended to infinite heights (*n* in ). This is because for bases within a certain interval, tetration converges to a finite value as the height tends to infinity. For example, converges to 2, and can therefore be said to be equal to 2. The trend towards 2 can be seen by evaluating a small finite tower:

In general, the infinite power tower , defined as the limit of as *n* goes to infinity, converges for *e*^{−e} ≤ *x* ≤ *e*^{1/e}, roughly the interval from 0.066 to 1.44, a result shown by Leonhard Euler.^{[9]} The limit, should it exist, is a positive real solution of the equation *y* = *x*^{y}. Thus, *x* = *y*^{1/y}. The limit defining the infinite tetration of *x* fails to converge for *x* > *e*^{1/e} because the maximum of *y*^{1/y} is *e*^{1/e}.

This may be extended to complex numbers *z* with the definition:

where *W*(*z*) represents Lambert's W function.

As the limit *y* = ^{∞}*x* (if existent, i.e. for *e*^{−e} < *x* < *e*^{1/e}) must satisfy *x*^{y} = *y* we see that *x* ↦ *y* = ^{∞}*x* is (the lower branch of) the inverse function of *y* ↦ *x* = *y*^{1/y}.

#### (Limited) extension to negative heights

In order to preserve the original rule:

for negative values of we must use the recursive relation:

Thus:

However smaller negative values cannot be well defined in this way because

which is not well defined.

Note further that for any definition of is consistent with the rule because

#### Extension to real heights

At this time there is no commonly accepted solution to the general problem of extending tetration to the real or complex values of . Various approaches are mentioned below.

In general the problem is finding, for any real *a* > 0, a *super-exponential function* over real *x* > −2 that satisfies

The fourth requirement differs from author to author, and between approaches. There are two main approaches to extending tetration to real heights, one is based on the *regularity* requirement, and one is based on the *differentiability* requirement. These two approaches seem to be so different that they may not be reconciled, as they produce results inconsistent with each other.

When is defined for an interval of length one, the whole function easily follows for all *x* > −2.

##### Linear approximation for the extension to real heights

A *linear approximation* (solution to the continuity requirement, approximation to the differentiability requirement) is given by:

hence:

and so on. However, it is only piecewise differentiable; at integer values of x the derivative is multiplied by .

###### Examples

A main theorem in Hooshmand's paper^{[6]} states: Let . If is continuous and satisfies the conditions:

then is uniquely determined through the equation

where denotes the fractional part of x and is the -iterated function of the function .

The proof is that the second through fourth conditions trivially imply that f is a linear function on [−1, 0].

The linear approximation to natural tetration function is continuously differentiable, but its second derivative does not exist at integer values of its argument. Hooshmand derived another uniqueness theorem for it which states:

If is a continuous function that satisfies:

then . [Here is Hooshmand's name for the linear approximation to the natural tetration function.]

The proof is much the same as before; the recursion equation ensures that and then the convexity condition implies that is linear on (−1, 0).

Therefore the linear approximation to natural tetration is the only solution of the equation and which is convex on . All other sufficiently-differentiable solutions must have an inflection point on the interval (−1, 0).

##### Higher order approximations for the extension to real heights

A *quadratic approximation* (to the differentiability requirement) is given by:

which is differentiable for all , but not twice differentiable. If this is the same as the linear approximation.

Note that this function does not satisfy condition that tetration "cancels out" (for example as in raising to power: ), because it is calculated top-down (as explained in section Iterated powers above) namely:

A cubic approximation and a method for generalizing to approximations of degree *n* are given at.^{[10]}

#### Extension to complex heights

There is a conjecture^{[11]} that there exists a unique function *F* which is a solution of the equation *F*(*z*+1)=exp(*F*(*z*)) and satisfies the additional conditions that *F*(0)=1 and *F*(*z*) approaches the fixed points of the logarithm (roughly 0.31813150520476413531 ± 1.33723570143068940890*i*Template:Undue precision)
as *z* approaches ±*i*∞ and that *F* is holomorphic in the whole complex *z*-plane, except the part of the real axis at *z*≤−2.
This function is shown in the figure at right.
The complex double precision approximation of this function is available online.^{[12]}

The requirement of holomorphism of tetration is important for the uniqueness. Many functions can be constructed as

where and are real sequences which decay fast enough to provide the convergence of the series, at least at moderate values of .

The function *S* satisfies the tetration equations *S*(*z*+1)=exp(*S*(*z*)), *S*(0)=1, and if *α _{n}* and

*β*approach 0 fast enough it will be analytic on a neighborhood of the positive real axis. However, if some elements of {

_{n}*α*} or {

*β*} are not zero, then function

*S*has multitudes of additional singularities and cutlines in the complex plane, due to the exponential growth of sin and cos along the imaginary axis; the smaller the coefficients {

*α*} and {

*β*} are, the further away these singularities are from the real axis.

The extension of tetration into the complex plane is thus essential for the uniqueness; the real-analytic tetration is not unique.

## Open questions

- It is not known if
^{n}π or^{n}*e*is an integer for any positive integer*n*. Particularly, it is not known if^{4}π is an integer. - It is not known if
^{n}*q*is an integer for any positive integer*n*and positive non-integer rational*q*.^{[13]}Particularly, it is not known if the positive root of the equation^{2}*x*= 2 is a rational number.

## Inverse relations

Exponentiation has two inverse relations; roots and logarithms. Analogously, the inverse relations of tetration are often called the *super-root*, and the *super-logarithm*.

### Super-root

The super-root is the inverse relation of tetration with respect to the base: if , then *y* is an *n*th super root of *x*.
For example,

so 2 is the 4th super-root of 65,536 and

so 3 is the 3rd super-root (or super cube root) of 7,625,597,484,987.

#### Square super-root

The *2nd-order super-root*, *square super-root*, or *super square root* has two equivalent notations, and . It is the inverse of and can be represented with the Lambert W function:^{[14]}

The function also illustrates the reflective nature of the root and logarithm functions as the equation below only holds true when :

Like square roots, the square super-root of *x* may not have a single solution. Unlike square roots, determining the number of square super-roots of *x* may be difficult. In general, if , then *x* has two positive square super-roots between 0 and 1; and if , then *x* has one positive square super-root greater than 1. If *x* is positive and less than it doesn't have any real square super-roots, but the formula given above yields countably infinitely many complex ones for any finite *x* not equal to 1.^{[14]} The function has been used to determine the size of data clusters.^{[15]}

#### Other super-roots

For each integer *n* > 2, the function * ^{n}x* is defined and increasing for

*x*≥ 1, and

^{n}1 = 1, so that the

*n*th super-root of

*x*, , exists for

*x*≥ 1.

However, if the linear approximation above is used, then if −1 < *y* ≤ 0, so cannot exist.

Other super-roots are expressible under the same basisTemplate:Clarify used with normal roots: super cube roots, the function{{ safesubst:#invoke:Unsubst||$N=Dubious |date=__DATE__ |$B=
{{#invoke:Category handler|main}}^{[dubious – discuss]}
}} that produces *y* when , can be expressed as ; the 4th super-root can be expressed as ; and it can therefore be said that the *n*^{th} super-root is . Note that may not be uniquely defined, because there may be more than one *n*^{th} root. For example, *x* has a single (real) super-root if *n* is *odd*, and up to two if *n* is *even*.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
{{#invoke:Category handler|main}}{{#invoke:Category handler|main}}^{[citation needed]}
}}

The super-root can be extended to , and this shows a link to the mathematical constant *e* as it is only well-defined if 1/*e* ≤ *x* ≤ *e* (see extension of tetration to infinite heights). Note that implies that Template:Why and thus that . Therefore, when it is well defined, and thus it is an elementary function. For example, .

It follows from the Gelfond–Schneider theorem that super-root for any positive integer *n* is either integer or transcendental, and is either integer or irrational.^{[16]} But it is still an open question whether irrational super-roots are transcendental in the latter case.

### Super-logarithm

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Once a continuous increasing (in *x*) definition of tetration, ^{x}*a*, is selected, the corresponding super-logarithm slog_{a} *x* is defined for all real numbers *x*, and *a* > 1.

## See also

- Ackermann function
- Double exponential function
- Hyperoperation
- Iterated logarithm
- Symmetric level-index arithmetic

## References

- ↑ It is easy to prove that for every elementary function
*f*, there is a constant*c*s.t. . We can show (by using diagonal argument and above fact) that is non-elementary, and so is tetration. - ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ Weisstein, Eric W., "Power Tower",
*MathWorld*. - ↑
^{6.0}^{6.1}{{#invoke:Citation/CS1|citation |CitationClass=journal }} - ↑ Ioannis Galidakis. On Extending hyper4 and Knuth’s Up-arrow Notation to the Reals.
- ↑ Template:Cite web
- ↑ Euler, L. "De serie Lambertina Plurimisque eius insignibus proprietatibus."
*Acta Acad. Scient. Petropol. 2*, 29–51, 1783. Reprinted in Euler, L.*Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae*. Leipzig, Germany: Teubner, pp. 350–369, 1921. (facsimile) - ↑ Andrew Robbins. Solving for the Analytic Piecewise Extension of Tetration and the Super-logarithm.
- ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ Mathematica code for evaluation and plotting of the tetration and its derivatives.
- ↑ Marshall, Ash J., and Tan, Yiren, "A rational number of the form
*a*^{a}with*a*irrational", Mathematical Gazette 96, March 2012, pp. 106-109. - ↑
^{14.0}^{14.1}{{#invoke:Citation/CS1|citation |CitationClass=journal }} - ↑ BOSTON UNIVERSITY COLLEGE OF ENGINEERING – EFFICIENT SELF-ORGANIZATION OF LARGE WIRELESS SENSOR NETWORKS
- ↑ Marshall, Ash J., and Tan, Yiren, "A rational number of the form
*a*^{a}with*a*irrational", Mathematical Gazette 96, March 2012, pp. 106-109.

- Daniel Geisler,
*tetration.org* - Ioannis Galidakis,
*On extending hyper4 to nonintegers*(undated, 2006 or earlier)*(A simpler, easier to read review of the next reference)* - Ioannis Galidakis,
*On Extending hyper4 and Knuth's Up-arrow Notation to the Reals*(undated, 2006 or earlier). - Robert Munafo,
*Extension of the hyper4 function to reals**(An informal discussion about extending tetration to the real numbers.)* - Lode Vandevenne,
*Tetration of the Square Root of Two*, (2004).*(Attempt to extend tetration to real numbers.)* - Ioannis Galidakis,
*Mathematics*,*(Definitive list of references to tetration research. Lots of information on the Lambert W function, Riemann surfaces, and analytic continuation.)* - Galidakis, Ioannis and Weisstein, Eric W. Power Tower
- Joseph MacDonell,
*Some Critical Points of the Hyperpower Function*. - Dave L. Renfro,
*Web pages for infinitely iterated exponentials**(Compilation of entries from questions about tetration on sci.math.)* - R. Knobel. "Exponentials Reiterated."
*American Mathematical Monthly***88**, (1981), p. 235–252. - Hans Maurer. "Über die Funktion für ganzzahliges Argument (Abundanzen)."
*Mittheilungen der Mathematische Gesellschaft in Hamburg***4**, (1901), p. 33–50.*(Reference to usage of from Knobel's paper.)*