Tetration Guide, Meaning , Facts, Information and Description

Tetration (also hyper4, power tower, super-exponentiation) is iterated exponentiation.

Tetration follows exponentiation in the sequence:

1. addition

2. multiplication

3. exponentiation

4. tetration

where each operation is defined by iterating the previous one.

We can think of multiplication () as B instances of A added together, and we can consequently think of exponentiation () as B instances of A multiplied together. So we can go a step further, and think of tetration () as B instances of A exponentiated together.

Note that when solving multiple-level exponentiation, the exponentiation is done at the deepest level first (in the notation, at the highest level). In other words:

not:

.

There is no standard notation for tetration. The notations in which it can be written (some of which allow further iteration) include:

• The Ackermann function: A(a, b, 3); this is the most generally accepted notation.
• Knuth's up-arrow notation: .
• Conway chained arrow notation: .
• Rudy Rucker's notation: .
• Hyper operator notation: .

The up-arrow is used identically to the caret (^), so that the tetration operator may be written as ^^ in ASCII: a^^b.

Table of contents

1 Examples 2 Properties 3 Complex Tetration

Examples

• = = 1
• = = 4
• = = 27
• = = 256
• = = 3,125
• = = 46,656
• = = 823,543
• = = 16,777,216
• = = 387,420,489
• = = 10,000,000,000

• = = 1
• = = 16
• = = 7,625,597,484,987
• = =
• = = = (over 2,000 digits long)
• = = = (over 35,000 digits long)

• = = 1
• = = 65,536
• = = (over three trillion digits long)

• = = 1
• = = = (nearly 20,000 digits long)

Properties

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 . Therefore, must remain an undefined quantity as well. (The figure can safely be defined as 1, however.)

Again, is an undefined quantity, so 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, but would conflict with the standard undefinedness of .

Complex Tetration

Since complex numbers can be raised to powers, tetration can be applied to numbers of the form , where i 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:

i^(a+bi) = e^{iπ/2 (a+bi)} = e^-bπ/2 (cos(aπ/2) + i sin(aπ/2)) .

This suggests a recursive definition for given any :

a' = e^-bπ/2 cos(aπ/2) and b' = e^-bπ/2 sin(aπ/2)

The following approximate values can be derived, where is ordinary exponentiation (ie. iⁿ).

• = i
• = = 0.2079
• = = 0.9472+ 0.3208i
• = = 0.0501+ 0.6021i
• = = 0.3872+ 0.0305i
• = = 0.7823+ 0.5446i
• = = 0.1426+ 0.4005i
• = = 0.5198+ 0.1184i
• = = 0.5686+ 0.6051i

Solving the relation yields the expected = 1 and = 0, with negative values of k giving infinite results on the imaginary axis. Plotted in the complex plane, the entire sequence spirals to the limit 0.4383+ 0.3606i, which could be interpreted as the value where k is infinite.

See http://home.earthlink.net/~mrob/pub/math/ln-notes1.html#real-hyper4 for attempts to extend tetration to real numbers.

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