Tetration


In mathematics, tetration is an operation based on iterated, or repeated, exponentiation. It is the next hyperoperation after exponentiation, but before pentation. The word was coined by Reuben Louis Goodstein from tetra- and iteration.
Under the definition as repeated exponentiation, the notation means, where ' copies of ' are iterated via exponentiation, right-to-left, I.e. the application of exponentiation times. ' is called the "height" of the function, while ' is called the "base," analogous to exponentiation. It would be read as "the th tetration of ".
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 the super-root and the 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.
  1. Addition
  2. :
  3. :: copies of 1 added to.
  4. Multiplication
  5. :
  6. ::' copies of ' combined by addition.
  7. Exponentiation
  8. :
  9. ::' copies of ' combined by multiplication.
  10. Tetration
  11. :
  12. ::' 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.
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:
TerminologyForm
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.
NameFormDescription
Rudy Rucker notationUsed by Maurer and Goodstein ; Rudy Rucker's book Infinity and the Mind popularized the notation.
Knuth's up-arrow notationAllows extension by putting more arrows, or, even more powerfully, an indexed arrow.
Conway chained arrow notationAllows extension by increasing the number 2, but also, even more powerfully, by extending the chain
Ackermann functionAllows the special case to be written in terms of the Ackermann function.
Iterated exponential notationAllows simple extension to iterated exponentials from initial values other than 1.
Hooshmand notationsUsed by M. H. Hooshmand .
Hyperoperation notationsAllows extension by increasing the number 4; this gives the family of hyperoperations.
Double caret notationSince 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:
NameFormDescription
Standard notationEuler coined the notation, and iteration notation has been around about as long.
Knuth's up-arrow notationAllows for super-powers and super-exponential function by increasing the number of arrows; used in the article on large numbers.
Text notationBased on standard notation; convenient for ASCII.
J NotationRepeats the exponentiation. See J

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.
11111
241665,5362 or
3277,625,597,484,987
42561.34078 × 10
53,1251.91101 × 10
646,6562.65912 × 10
7823,5433.75982 × 10
816,777,2166.01452 × 10
9387,420,4894.28125 × 10
1010,000,000,00010

Properties

Tetration has several properties that are similar to exponentiation, as well as properties that are specific to the operation and are lost or gained from exponentiation. Because exponentiation does not commute, the product and power rules do not have an analogue with tetration; the statements and are not necessarily true for all cases.
However, tetration does follow a different property, in which. This fact is most clearly shown using the recursive definition. From this property, a proof follows that, which allows for switching b and c in certain equations. The proof goes as follows:
When a number and 10 are coprime, it is possible to compute the last decimal digits of using Euler's theorem, for any integer.

Direction of evaluation

When evaluating tetration expressed as an "exponentiation tower", the serial exponentiation is done at the deepest level first. For example:
This order is important because exponentiation is not associative, and evaluating the expression in the opposite order will lead to a different answer:
Evaluating the expression the left to right is considered less interesting; evaluating left to right, any expression can be simplified to be. Because of this, the towers must be evaluated from right to left. Computer programmers refer to this choice as right-associative.

Extensions

Tetration can be extended in two different ways; in the equation, both the base and the height can be generalized using the definition and properties of tetration. Although the base and the height can be extended beyond the non-negative integers to different domains, including, complex functions such as, and heights of infinite, the more limited properties of tetration reduce the ability to extend tetration.

Extension of domain for bases

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

Complex bases

Since complex numbers can be raised to powers, tetration can be applied to bases of the form . For example, in with, tetration is achieved by using the principal branch of the natural logarithm; using Euler's formula we get the relation:
This suggests a recursive definition for given any :
The following approximate values can be derived:
Approximate value

Solving the inverse relation, as in the previous section, yields the expected and, with negative values of 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 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 infinitely iterated exponential 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 different heights

Infinite heights

Tetration can be extended to infinite heights; i.e., for certain and values in, there exists a well defined result for an infinite. 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 infinitely iterated exponential, defined as the limit of as goes to infinity, converges for, roughly the interval from 0.066 to 1.44, a result shown by Leonhard Euler. The limit, should it exist, is a positive real solution of the equation. Thus,. The limit defining the infinite tetration of fails to converge for because the maximum of is.
This may be extended to complex numbers with the definition:
where represents Lambert's W function.
As the limit must satisfy we see that is the inverse function of.

Negative heights

We can use the recursive rule for tetration,
to prove :
Substituting −1 for gives
Smaller negative values cannot be well defined in this way. Substituting −2 for in the same equation gives
which is not well defined. They can, however, sometimes be considered sets.
For, any definition of is consistent with the rule because

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. There have, however, been multiple approaches towards the issue, and different approaches are outlined below.
In general, the problem is finding — for any real — a super-exponential function over real that satisfies
To find a more natural extension, one or more extra requirements are usually required. This is usually some collection of the following:
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.
Linear approximation for real heights
A linear approximation is given by:
hence:
ApproximationDomain
for
for
for

and so on. However, it is only piecewise differentiable; at integer values of the derivative is multiplied by. It is continuously differentiable for if and only if. For example, using these methods and
A main theorem in Hooshmand's paper states: Let. If is continuous and satisfies the conditions:
then is uniquely determined through the equation
where denotes the fractional part of and is the -iterated function of the function.
The proof is that the second through fourth conditions trivially imply that is a linear function on.
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.
The proof is much the same as before; the recursion equation ensures that and then the convexity condition implies that is linear on.
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.
Higher order approximations for real heights
Beyond linear approximations, a quadratic approximation is given by:
which is differentiable for all, but not twice differentiable. For example, If this is the same as the linear approximation.
Because of the way it is calculated, this function does not "cancel out", contrary to exponents, where. Namely,
Just as there is a quadratic approximation, cubic approximations and methods for generalizing to approximations of degree also exist, although they are much more unwieldy.

Complex heights

It has now been proven that there exists a unique function which is a solution of the equation and satisfies the additional conditions that and approaches the fixed points of the logarithm as approaches and that is holomorphic in the whole complex -plane, except the part of the real axis at. This proof confirms a previous conjecture. The construction of such a function was originally demonstrated by Kneser in 1950. The complex map of this function is shown in the figure at right. The proof also works for other bases besides e, as long as the base is bigger than. Subsequent work extended the construction to all complex bases. The complex double precision approximation of this function is available online.
The requirement of the tetration being holomorphic is important for its 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 satisfies the tetration equations,, and if and approach 0 fast enough it will be analytic on a neighborhood of the positive real axis. However, if some elements of or are not zero, then function 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 and 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.

Non-elementary recursiveness

Tetration is not an elementary recursive function. One can prove by induction that for every elementary recursive function, there is a constant such that
We denote the right hand side by. Suppose on the contrary that tetration is elementary recursive. is also elementary recursive. By the above inequality, there is a constant such that. By letting, we have that, a contradiction.

Inverse operations

has two inverse operations; roots and logarithms. Analogously, the inverses of tetration are often called the super-root, and the super-logarithm ; e.g., in the function, the two inverses are the cube super-root of and the super logarithm base of.

Super-root

The super-root is the inverse operation of tetration with respect to the base: if, then is an th super root of .
For example,
so 2 is the 4th super-root of 65,536.

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:
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 may not have a single solution. Unlike square roots, determining the number of square super-roots of may be difficult. In general, if, then has two positive square super-roots between 0 and 1; and if, then has one positive square super-root greater than 1. If 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 not equal to 1. The function has been used to determine the size of data clusters.
At :

Other super-roots

For each integer, the function is defined and increasing for, and, so that the th super-root of,, exists for.
However, if the linear approximation above is used, then if, so cannot exist.
In the same way as the square super-root, terminology for other super roots can be based on the normal roots: "cube super-roots" can be expressed as ; the "4th super-root" can be expressed as ; and the "th super-root" is. Note that may not be uniquely defined, because there may be more than one root. For example, has a single super-root if is odd, and up to two if is even.
Just as with the extension of tetration to infinite heights, the super-root can be extended to, being well-defined if. Note that and thus that. Therefore, when it is well defined, and, unlike normal tetration, is an elementary function. For example,.
It follows from the Gelfond–Schneider theorem that super-root for any positive integer is either integer or transcendental, and is either integer or irrational. It is still an open question whether irrational super-roots are transcendental in the latter case.

Super-logarithm

Once a continuous increasing definition of tetration,, is selected, the corresponding super-logarithm or is defined for all real numbers, and.
The function satisfies:

Open questions

Other than the problems with the extensions of tetration, there are several open questions concerning tetration, particularly when concerning the relations between number systems such as integers and irrational numbers: