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 in the sequence:
#
addition
#
#multiplication
#
#
exponentiation
#
#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.
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 in the sequence:
- addition
- multiplication
- exponentiation
- 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'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: a^^b.
Examples
| n = n??1 | n??2 | n??3 | n??4 |
|---|
| 1 | 1 | 1 | 1 |
| 2 | 4 | 16 | 65,536 |
| 3 | 27 | 7.63×1012 | |
| 4 | 256 | 1.34×10154 | |
| 5 | 3,125 | 1.91×102,184 | |
| 6 | 46,656 | 2,70×1036,305 | |
| 7 | 823,543 | 3.76×10695,974 | |
| 8 | 16,777,216 | 6.01×1015,151,335 | |
| 9 | 387,420,489 | 4.28×10369,693,099 | |
| 10 | 10,000,000,000 | 1010,000,000,000 | |
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 numbers can be raised to powers, tetration can be applied to numbers of the form , where is the
square root 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, 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; for example, if = 10:
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.
See also
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, .
