where each operation is defined by iterating the previous one.
Addition (a+b) can be thought of as adding b iterations of the "add one" function applied to a, multiplication (ab) can be thought of as b iterations of the "add a" function applied to a, and exponentiation () can be thought of as b iterations of the "multiply by a" function to a. Analogously tetration () can be thought of as b iterations of the "raise to the power a" function applied to a.
Note that when evaluating multiple-level exponentiation, the exponentiation is done at the deepest level first (in the notation, at the highest level). In other words:
is not equal to .
(This is the general rule for the order of operations involving repeated exponentiation.)
Notation
To generalize the first case (tetration) above, a new notation is needed (see below); 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 (some of which allow even higher levels of iteration) include:
Standard notation: — first used by Hans 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 (equivalent with the extensions above), 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
The up-arrow is used identically to the caret (^), so that the tetration operator may be written as ^^ in ASCII: a^^b.
Examples
(The values containing a decimal point are approximate.)
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 (which follows from the definition of tetration), one can derive (or define) 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. (The figure can safely be defined as 1, however.)
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 (since 0 is even.)
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 (i.e. ).
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. (The term super is sometimes replaced by hyper: hyper-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 (for ):
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.
I.N. 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
Hans Maurer. "Über die Funktion für ganzzahliges Argument (Abundanzen)." Mittheilungen der Mathematische Gesellschaft in Hamburg4, (1901), p. 33-50. (Reference to usage of from Knobel's paper.)