NationMaster - Encyclopedia: Tetration

Tetration (also exponential map, hyperpower, power tower, super-exponentiation, and hyper4) is iterated exponentiation, the first hyper operator after exponentiation. The portmanteau word tetration was coined by Reuben Louis Goodstein from tetra- (four) and iteration. Tetration is used for the notation of very large numbers. Tetration follows exponentiation in this sequence: The hyper operators forming the hypern family are as follows: hypern (a, b) = (See Knuths up-arrow notation and Conway chained arrow notation. ... A portmanteau (IPA: ) is a word or morpheme that fuses two or more words or word parts to give a combined or loaded meaning. ... Reuben Louis Goodstein (born 15 December 1912 in London, died 8 March 1985 in Leicester) was an English mathematician with a strong interest in the philosophy and teaching of mathematics. ... Tetra is a Greek prefix meaning 4 (in constrast to quad- as in quadruped, which is Latin). ... The word iteration is sometimes used in everyday English with a meaning virtually identical to repetition. ... Big numbers redirects here. ... â€œExponentâ€ redirects here. ...

addition

${{a + b} atop ,} {= atop ,} {a , + atop , } {{underbrace{1 + 1 + cdots + 1}} atop b}$
multiplication

${{a times b = } atop { }} {{underbrace{a + a + cdots + a}} atop b}$
exponentiation

${{a^b = } atop { }} {{underbrace{a times a times cdots times a}} atop b}$
tetration

${ ^{b}a = atop { }} {{underbrace{a^{a^{cdot^{cdot^{a}}}}}} atop b}$

where each operation is defined by iterating the previous one. 3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ... In mathematics, multiplication is an elementary arithmetic operation. ... â€œExponentâ€ redirects here. ...

Addition (a+b) can be thought of as being b iterations of the "add one" function applied to a, multiplication (ab) can be thought of as a chained addition involving b numbers a, and exponentiation ( $a b$ ) can be thought of as a chained multiplication involving b numbers a. Analogously, tetration ( $b a$ ) can be thought of as a chained power involving b numbers a. The parameter a may be called the base-parameter in the following, while the parameter b in the following may be called the height-parameter (which is integral in the first approach but may be generalized to fractional, real and complex heights, see below)

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:

$,! ^{4}2 = 2^{2^{2^2}} = 2^{left(2^{left(2^2right)}right)} = 2^{left(2^4right)} = 2^{16} = 65,!536$

The convention for iterated exponentiation is to work from the right to the left. Thus,

$,!2^{2^{2^2}}$ is not equal to $,! left({left(2^2right)}^2right)^2 = 2^{2times2times2} = 256$ .

1 Notation
2 Examples
3 Extension to low and negative (integer) values of height (second operand)
4 Tetration with integer height (second operand) and complex base (first operand) "Complex tetration"
5 Extension to fractional or real height (second operand)
6 Approaches to a definition for the Inverse
7 Infinitely high power towers
8 See also
9 References
10 External links

Notation

To generalize the first case (tetration) above, a new notation is needed (see below); however, the second case can be written as

$,! left({left(2^2right)}^2right)^2 = 2^{2 cdot 2 cdot 2} = 2^{2^3}$

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: $b a$ — first used by Hans Maurer; Rudy Rucker's book Infinity and the Mind popularized the notation.
Knuth's up-arrow notation: $a uparrowuparrow b$ — allows extension by putting more arrows, or, even more powerfully, an indexed arrow
Conway chained arrow notation: $a rightarrow b rightarrow 2$ — allows extension by increasing the number 2 (equivalent with the extensions above), but also, even more powerfully, by extending the chain
hyper4 notation: $a^{(4)}b = operatorname{hyper4}(a, b) = operatorname{hyper}(a, 4, b)$ — allows extension by increasing the number 4; this gives the family of hyper operators

For the Ackermann function we have $2 uparrowuparrow b = operatorname{A}(4, b - 3) + 3$ , i.e. $operatorname{A}(4, n) = 2 uparrowuparrow (n+3) - 3$ Hans Maurer was a West German bobsledder who competed during the early 1960s. ... Rudy Rucker, Fall 2004, photo by Georgia Rucker. ... In mathematics, Knuths up-arrow notation is a notation for very large integers introduced by Donald Knuth in 1976. ... Conway chained arrow notation, created by mathematician John Conway, is a means of expressing certain extremely large numbers. ... The hyper operators forming the hypern family are as follows: hypern (a, b) = (See Knuths up-arrow notation and Conway chained arrow notation. ... In recursion theory, the Ackermann function or Ackermann-PÃ©ter function is a simple example of a general recursive function that is not primitive recursive. ...

The up-arrow is used identically to the caret (^), so that the tetration operator may be written as ^^ in ASCII: a^^b. Image:ASCII fullsvg There are 95 printable ASCII characters, numbered 32 to 126. ...

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	$10^{3.64 times 10^{12}}$
4	256	1.34×10¹⁵⁴	$10^{8.07 times 10^{153}}$
5	3,125	1.91×10^2,184	$10^{1.34 times 10^{2,184}}$
6	46,656	2.70×10^36,305	$10^{2.07 times 10^{36,305}}$
7	823,543	3.76×10^695,974	$10^{3.18 times 10^{695,974}}$
8	16,777,216	6.01×10^15,151,335	$10^{5.43 times 10^{15,151,335}}$
9	387,420,489	4.28×10^369,693,099	$10^{4.09 times 10^{369,693,009}}$
10	10,000,000,000	10^{10,000,000,000}	$10^{10^{10^{10}}}$

Extension to low and negative (integer) values of height (second operand)

Using the relation $n uparrowuparrow k = log_n left(n uparrowuparrow (k+1)right)$ (which follows from the definition of tetration), one can derive (or define) values for $n uparrowuparrow k$ where $k in {-1, 0, 1}$ .

$begin{matrix} n uparrowuparrow 1 & = & log_n left(n uparrowuparrow 2right) & = & log_{n} left(n^nright) & = & n log_{n} n & = & n n uparrowuparrow 0 & = & log_{n} left(n uparrowuparrow 1right) & = & log_{n} n & & & = & 1 n uparrowuparrow -1 & = & log_{n} left(n uparrowuparrow 0right) & = & log_{n} 1 & & & = & 0 end{matrix}$

This confirms the intuitive definition of $n uparrowuparrow 1$ as simply being $n$ . However, no further values can be derived by further iteration in this fashion, as $log n 0$ is undefined.

Similarly, since $log 11$ is also undefined ( $log_{1} 1 = begin{matrix}frac{log_n 1}{log_n 1} = frac{0}{0}end{matrix}$ ), the derivation above does not hold when $n$ = 1. Therefore, $1 uparrowuparrow {-1}$ must remain an undefined quantity as well. (The figure $1 uparrowuparrow {0}$ can safely be defined as 1, however.)

Sometimes, $00$ is taken to be an undefined quantity. In this case, values for $0uparrowuparrow{k}$ cannot be defined directly. However, $lim_{nrightarrow0} nuparrowuparrow{k}$ is well defined, and exists:

$lim_{nrightarrow0} nuparrowuparrow k = begin{cases} 1, & k mbox{ even} 0, & k mbox{ odd} end{cases}$

This limit holds for negative $n$ , as well. $0 uparrowuparrow {k}$ could be defined in terms of this limit and this would agree with a definition of $00 = 1$ (since 0 is even.) Zero objects, divided into two equal groups. ...

Tetration with integer height (second operand) and complex base (first operand) "Complex tetration"

Tetration by period

Tetration by escape

Since complex numbers can be raised to powers, tetration can be applied to bases z of the form $a + b i$ , where $i$ is the square root of −1. For example, $z uparrowuparrow k$ where $z = i$ , tetration is achieved by using the principal branch of the natural logarithm, and noting the relation: Download high resolution version (1024x768, 76 KB)Tetration by period. ... Download high resolution version (1024x768, 76 KB)Tetration by period. ... Download high resolution version (1024x768, 56 KB)Tetration by escape. ... Download high resolution version (1024x768, 56 KB)Tetration by escape. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ... In mathematics, a principal branch is a function which selects one branch, or slice, of a multi-valued function. ...

$i^{a+bi} = e^{{ipi over 2} (a+bi)} = e^{-{bpi over 2}} left(cos{api over 2} + i sin{api over 2}right)$

This suggests a recursive definition for $i uparrowuparrow (k+1) = a'+b'i$ given any $i uparrowuparrow k = a+bi$ :

$a' = e^{-{bpi over 2}} cos{api over 2}$

$b' = e^{-{bpi over 2}} sin{api over 2}$

The following approximate values can be derived, where $i uparrow n$ is ordinary exponentiation (i.e. $i n$ ).

$i uparrowuparrow 1 = i$
$i uparrowuparrow 2 = i uparrowleft(i uparrowuparrow 1right) approx 0.2079$
$i uparrowuparrow 3 = i uparrowleft(i uparrowuparrow 2right) approx 0.9472 + 0.3208i$
$i uparrowuparrow 4 = i uparrowleft(i uparrowuparrow 3right) approx 0.0501 + 0.6021i$
$i uparrowuparrow 5 = i uparrowleft(i uparrowuparrow 4right) approx 0.3872 + 0.0305i$
$i uparrowuparrow 6 = i uparrowleft(i uparrowuparrow 5right) approx 0.7823 + 0.5446i$
$i uparrowuparrow 7 = i uparrowleft(i uparrowuparrow 6right) approx 0.1426 + 0.4005i$
$i uparrowuparrow 8 = i uparrowleft(i uparrowuparrow 7right) approx 0.5198 + 0.1184i$
$i uparrowuparrow 9 = i uparrowleft(i uparrowuparrow 8right) approx 0.5686 + 0.6051i$

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

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 fractional or real height (second operand)

x↑↑n, for n = 2, 3, 4, 5, 6 and 7.

Extending $x uparrowuparrow b$ to real numbers $x > 0$ is straightforward and gives, for each natural number $b$ , a super-power function $operatorname{f}(x) = x uparrowuparrow b$ . (The term super is sometimes replaced by hyper: hyper-power function). Image File history File links Download high resolution version (1276x784, 12 KB)Graph plotting x^^2, x^^3, x^^4, x^^5, x^^6 and x^^7 for positive values of x. ... Image File history File links Download high resolution version (1276x784, 12 KB)Graph plotting x^^2, x^^3, x^^4, x^^5, x^^6 and x^^7 for positive values of x. ...

As mentioned above, for positive integers $b$ the function tends to 1 for $x$ tending to 0 if $b$ is even, and to 0 if $b$ is odd, while for $b = 0$ and $b = - 1$ 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 values of $b$ , although it is an active area of research.

Consider the problem of finding a super-exponential function or hyper-exponential function $operatorname{f}(x) = a uparrowuparrow x$ which is an extension to real $x > - 2$ to what was defined above, satisfying (for $a > 1$ ):

$a uparrowuparrow(b+1) = a^{left(auparrow uparrow bright)}$
it is monotonically increasing
it is continuous

When $a uparrowuparrow x$ is defined for an interval of length one, the whole function easily follows for all $x > - 2$

A simple solution is given by $a uparrowuparrow x = x+1$ for $- 1 < x < 0$ , hence $a uparrowuparrow x = a^x$ for $0 < x < 1$ , $,!a uparrowuparrow x=a^{a^{(x-1)}}$ for $1 < x < 2$ , etc.

However, it is only piecewise differentiable; at integer values of x the derivative is multiplied by $log n a$ : $10 uparrowuparrow 0.99 = 9.77$ , $10 uparrowuparrow 1 = 10$ , $10 uparrowuparrow 1.01 = 10.55$ .

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 $a$ = 10: A double exponential function is a constant raised to the power of an exponential function. ...

$operatorname{f}(-1)=0$
$operatorname{f}(0)=1$
$operatorname{f}(1)=10$
$operatorname{f}(2)=10^{10}$
$operatorname{f}(2.3)=10^{100}$ (googol)
$,!operatorname{f}(3)=10^{10^{10}}$
$,!operatorname{f}(3.3)=10^{10^{100}}$ (googolplex)
It passes $,!10^{10^x}$ at $x = 2.376$ : $operatorname{f}(x) approx 4.83 times 10^{237}$

When defining $a uparrowuparrow x$ for every a, another possible requirement could be that $a uparrowuparrow x$ is monotonically increasing with a. Not to be confused with Google, the Internet company, and Nikolai Gogol, the author. ... This article is about a number. ...

Approaches to a definition for the Inverse

The inverse functions are called super-root or hyper-root, and super-logarithm or hyper-logarithm $slog a$ defined for all real numbers, also negative numbers. In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...

The function $slog a$ satisfies:

slog a a b = 1 + slog a b

slog a b = 1 + slog a log a b

slog a b > - 2

Examples:

$slog 10 - 3 = - 1 + slog 10 0.001 = - 1 + - 0.999 = - 1.999$
$slog 10 3 = log 10 3 = .477$
$mathrm{slog}_{10} 10^{6times 10^{23}} = 1 + mathrm{slog}_{10} 6times 10^{23} = 2 + mathrm{slog}_{10} 23.778 = 3 + mathrm{slog}_{10} 1.376 = 3 + log_{10} 1.376 = 3.139$

Infinitely high power towers

$sqrt{2}^{sqrt{2}^{sqrt{2}^{sqrt{2}^{sqrt{2}^{sqrt{2}^{..}}}}}}$ 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:

$sqrt{2}^{sqrt{2}^{sqrt{2}^{sqrt{2}^{sqrt{2}^{1.41}}}}} = sqrt{2}^{sqrt{2}^{sqrt{2}^{sqrt{2}^{1.63}}}} = sqrt{2}^{sqrt{2}^{sqrt{2}^{1.76}}} = sqrt{2}^{sqrt{2}^{1.84}} = sqrt{2}^{1.89} = 1.93$

In general, the infinite power tower $x^{x^{x^{..}}}$ , defined as the limit of $x uparrowuparrow n$ as n goes to infinity, converges for $e - e < x < e 1 / e$ . For arbitrary real $r$ with $0 < r < e$ , let $x = r 1 / r$ , then the limit is $r$ . There is no convergence for $x > e 1 / e$ (max of $r 1 / r$ is $e 1 / e$ ).

This may be extended to complex numbers $z$ with the definition:

$z^{z^{z^{.^{.^{.}}}}} = -frac{mathrm{W}(-ln{z})}{ln{z}}$

where $W(z)$ represents Lambert's W function. In mathematics, Lamberts W function, named after Johann Heinrich Lambert, also called the Omega function or product log, is the inverse function of where ew is the exponential function and w is any complex number. ...

References

Daniel Geisler, tetration.org
I.N. Galidakis, On extending hyper4 to nonintegers (undated, 2006 or earlier) (A simpler, easier to read review of the next reference)
I.N. Galidakis, On Extending hyper4 and Knuth's Up-arrow Notation to the Reals (undated, 2006 or earlier).
Robert Munafo, Extension of the hyper4 function to reals (An informal discussion about extending tetration to the real numbers.)
Lode Vandevenne, Tetration of the Square Root of Two, (2004). (Attempt to extend tetration to real numbers.)
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
Joseph MacDonell, Some Critical Points of the Hyperpower Function.
Dave L. Renfro, Web pages for infinitely iterated exponentials (Compilation of entries from questions about tetration on sci.math.)
Andrew Robbins, Home of Tetration (An infinitely differentiable extension of tetration to real numbers.)
R. Knobel. "Exponentials Reiterated." American Mathematical Monthly 88, (1981), p. 235-252.
Hans Maurer. "Über die Funktion $y=x^{[x^{[x(cdots)]}]}$ für ganzzahliges Argument (Abundanzen)." Mittheilungen der Mathematische Gesellschaft in Hamburg 4, (1901), p. 33-50. (Reference to usage of $^ba$ from Knobel's paper.)
Reuben Louis Goodstein. "Transfinite ordinals in recursive number theory." Journal of Symbolic Logic 12, (1947).

Ioannis N. Galidakis (born 1964) is a Greek mathematician, programmer, lab spectroscopist and amateur baroque composer. ... Ioannis N. Galidakis (born 1964) is a Greek mathematician, programmer, lab spectroscopist and amateur baroque composer. ... Ioannis N. Galidakis (born 1964) is a Greek mathematician, programmer, lab spectroscopist and amateur baroque composer. ... The American Mathematical Monthly is a mathematical journal published 10 times each year by the Mathematical Association of America since 1894. ... Hans Maurer was a West German bobsledder who competed during the early 1960s. ... Reuben Louis Goodstein (born 15 December 1912 in London, died 8 March 1985 in Leicester) was an English mathematician with a strong interest in the philosophy and teaching of mathematics. ... The Association for Symbolic Logic (ASL) is an international organization of specialists in mathematical logic and philosophical logicâ€”the largest such organization in the world. ...

External links

Categories: Arithmetic | Binary operations | Large numbers

Results from FactBites:

Tetration - Wikipedia, the free encyclopedia (1055 words)
	Tetration (also exponential map, hyperpower, power tower, super-exponentiation, and hyper4) is iterated exponentiation, the first hyper operator after exponentiation.
	The portmanteau word tetration was coined by Reuben Louis Goodstein from tetra- (four) and iteration.
	Tetration is used for the notation of very large numbers.

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