See Also

Tetration

Tetration is iterated exponentiation, the first hyper operator after exponentiation. The portmanteau word tetration was coined by Reuben Louis Goodstein from tetra-  and iteration. Tetration is used for the notation of very large numbers. Tetration follows exponentiation Exponentiation

Exponentiation is a mathematical [i] operation [i], written a'n, invo ... 

 in the sequence: #addition Addition

Addition is the mathematical operation [i] of increasing one amount by another. ... 

# #multiplication # #exponentiation Exponentiation

Exponentiation is a mathematical [i] operation [i], written a'n, invo ... 

# #tetration # where each operation is defined by iterating the previous one. Multiplication can be thought of as B instances of A added together, and consequently exponentiation can be thought of as B instances of A multiplied together.

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Encyclopedia

Tetration is iterated exponentiation, the first hyper operator after exponentiation. The portmanteau word tetration was coined by Reuben Louis Goodstein from tetra-  and iteration. Tetration is used for the notation of very large numbers.
Tetration follows exponentiation Exponentiation

Exponentiation is a mathematical [i] operation [i], written a'n, invo... 

 in the sequence:
  1. addition Addition

    Addition is the mathematical operation [i] of increasing one amount by another. ... 

  2. multiplication
  3. exponentiation Exponentiation

    Exponentiation is a mathematical [i] operation [i], written a'n, invo... 

  4. tetration


where each operation is defined by iterating the previous one.

Multiplication can be thought of as B instances of A added together, and consequently exponentiation can be thought of as B instances of A multiplied together. So a step further can be taken, and tetration can be thought of as B instances of A exponentiated together.

Note that when evaluating multiple-level exponentiation, the exponentiation is done at the deepest level first . In other words:
is not equal to .

Notation


To generalize the first case above, a new notation is needed ; however, the second case can be written as
Thus, its general form still uses ordinary exponentiation notation.

The notations in which tetration can be written include:

  • Standard notation: — first used by Maurer; Rudy Rucker Rudy Rucker

    Rudolf von Bitter Rucker is an American computer scientist [i] and science fiction [i] author, and is on... 

    's book Infinity and the Mind popularized the notation.
  • Knuth's up-arrow notation: — allows extension by putting more arrows, or equivalently, an indexed arrow
  • Conway chained arrow notation: — allows extension by increasing the number 2 , but also, even more powerfully, by extending the chain
  • hyper4 notation: — allows extension by increasing the number 4; this gives the family of hyper operators


For the Ackermann function we have , i.e.

The up-arrow is used identically to the caret , so that the tetration operator may be written as ^^ in ASCII ASCII

ASCII , generally pronounced [i] , is a character encoding [i] based on the English alphabet [i] ... 

: a^^b.

Examples












n = n??1n??2n??3n??4
1111
241665,536
3277.63×1012
42561.34×10154
53,1251.91×102,184
646,6562,70×1036,305
7823,5433.76×10695,974
816,777,2166.01×1015,151,335
9387,420,4894.28×10369,693,099
1010,000,000,0001010,000,000,000

Extension to low values of the second operand


Using the relation , one can derive values for where .

This confirms the intuitive definition of as simply being . However, no further values can be derived by further iteration in this fashion, as is undefined.

Similarly, since is also undefined , the derivation above does not hold when = 1. Therefore, must remain an undefined quantity as well.

Sometimes, is taken to be an undefined quantity. In this case, values for cannot be defined directly. However, is well defined, and exists:
This limit holds for negative , as well. could be defined in terms of this limit and this would agree with a definition of .

Complex tetration






Since complex number Complex number

In mathematics [i], a complex number is a number [i] of the form
... 

s can be raised to powers, tetration can be applied to numbers of the form , where is the square root Square root

In mathematics [i], a square root of a number x is a number whose square [i] is x. ... 

 of −1. For example, where , tetration is achieved by using the principal branch of the natural logarithm, and noting the relation:

This suggests a recursive definition for given any :

The following approximate values can be derived, where is ordinary exponentiation .


Solving the relation yields the expected and , with negative values of giving infinite results on the imaginary axis. Plotted in the complex plane Complex plane

In mathematics [i], the complex plane is a geometric space of the complex numbers [i] as set up by the ' ... 

, 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 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.

Extension to real numbers




Extending to real numbers is straightforward and gives, for each natural number , a super-power function . .

As mentioned above, for positive integers the function tends to 1 for tending to 0 if is even, and to 0 if is odd, while for and the function is constant, with values 1 and 0, respectively.

At this time there is no commonly accepted solution to the general problem of extending tetration to the real or complex numbers, although it is an active area of research.

Consider the problem of finding a super-exponential function or hyper-exponential function which is an extension to real to what was defined above, satisfying :
  • it is monotonically increasing
  • it is continuous


When is defined for an interval of length one, the whole function easily follows for all

A simple solution is given by for , hence for , for , etc.

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

Other, more complicated solutions may be smoother and/or satisfy additional properties.

A super-exponential function grows even faster than a double-exponential function Exponential function

The exponential function is one of the most important function [i]s in mathematics [i]. ... 

; for example, if = 10:
  • It passes at :


When defining for every a, another possible requirement could be that is monotonically increasing with a.

The inverse functions are called super-root or hyper-root, and super-logarithm or hyper-logarithm defined for all real numbers, also negative numbers.

The function satisfies:

Examples:


Infinitely high power towers


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 converges for . For arbitrary real with , let , then the limit is . There is no convergence for .

This may be extended to complex numbers with the definition:




where represents Lambert's W function Lambert W function

In mathematics [i], The Lambert W function, named after Johann Heinrich Lambert [i], also called the ... 

.

See also


  • Ackermann function

References

  • Daniel Geisler,
  • Daniel Geisler,
  • I.N. Galidakis, '
  • I.N. Galidakis, .
  • Robert Munafo, '
  • Lode Vandevenne, , . '
  • I.N. Galidakis, , '
  • Galidakis, Ioannis and Weisstein, Eric W.
  • Joseph MacDonell, .
  • Dave L. Renfro, '
  • Andrew Robbins, '
  • R. Knobel. "Exponentials Reiterated." Amer. Math. Monthly 88, , p. 235-252.
  • Hans Maurer. "Über die Funktion für ganzzahliges Argument ." Mittheilungen der Mathematische Gesellschaft in Hamburg 4, , p. 33-50. '
  • Reuben Louis Goodstein. "Transfinite ordinals in recursive number theory." Journal of Symbolic Logic 12, .