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:

1. addition

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

2. multiplication

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

3. exponentiation

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

4. tetration

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

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 ($a^{b}$) can be thought of as b iterations of the "multiply by a" function to a. Analogously tetration ($^{b}a$) 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:

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

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

(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

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

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: $a\uparrow \uparrow b$ — allows extension by putting more arrows, or equivalently, 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 $2\uparrow \uparrow b=\operatorname {A} (4,b-3)+3$, i.e. $\operatorname {A} (4,n)=2\uparrow \uparrow (n+3)-3$

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	$10^{3.64\times 10^{12}}$
4	256	1.34×10154	$10^{8.07\times 10^{153}}$
5	3,125	1.91×102,184	$10^{1.34\times 10^{2,184}}$
6	46,656	2.70×1036,305	$10^{2.07\times 10^{36,305}}$
7	823,543	3.76×10695,974	$10^{3.18\times 10^{695,974}}$
8	16,777,216	6.01×1015,151,335	$10^{5.43\times 10^{15,151,335}}$
9	387,420,489	4.28×10369,693,099	$10^{4.09\times 10^{369,693,009}}$
10	10,000,000,000	1010,000,000,000	$10^{10^{10^{10}}}$

Extension to low values of the second operand

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

${\begin{matrix}n\uparrow \uparrow 1&=&\log _{n}\left(n\uparrow \uparrow 2\right)&=&\log _{n}\left(n^{n}\right)&=&n\log _{n}n&=&n\\n\uparrow \uparrow 0&=&\log _{n}\left(n\uparrow \uparrow 1\right)&=&\log _{n}n&&&=&1\\n\uparrow \uparrow -1&=&\log _{n}\left(n\uparrow \uparrow 0\right)&=&\log _{n}1&&&=&0\end{matrix}}$

This confirms the intuitive definition of $n\uparrow \uparrow 1$ 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 $\log _{1}1$ 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, $1\uparrow \uparrow {-1}$ must remain an undefined quantity as well. (The figure $1\uparrow \uparrow {0}$ can safely be defined as 1, however.)

Sometimes, $0^{0}$ is taken to be an undefined quantity. In this case, values for $0\uparrow \uparrow {k}$ cannot be defined directly. However, $\lim _{n\rightarrow 0}n\uparrow \uparrow {k}$ is well defined, and exists:

$\lim _{n\rightarrow 0}n\uparrow \uparrow k={\begin{cases}1,&k{\mbox{ even}}\\0,&k{\mbox{ odd}}\end{cases}}$

This limit holds for negative $n$, as well. $0\uparrow \uparrow {k}$ could be defined in terms of this limit and this would agree with a definition of $0^{0}=1$ (since 0 is even.)

Complex tetration

File:Tetration period.gif

Tetration by period

File:Tetration escape.gif

Tetration by escape

Since complex numbers can be raised to powers, tetration can be applied to numbers of the form $a+bi$, where $i$ is the square root of −1. For example, $n\uparrow \uparrow k$ where $n=i$, tetration is achieved by using the principal branch of the natural logarithm, and noting the relation:

$i^{a+bi}=e^{{i\pi \over 2}(a+bi)}=e^{-{b\pi \over 2}}\left(\cos {a\pi \over 2}+i\sin {a\pi \over 2}\right)$

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

$a'=e^{-{b\pi \over 2}}\cos {a\pi \over 2}$

$b'=e^{-{b\pi \over 2}}\sin {a\pi \over 2}$

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

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

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

x↑↑n, for n = 2, 3, 4, 5, 6 and 7.

Extending $x\uparrow \uparrow b$ to real numbers $x>0$ is straightforward and gives, for each natural number $b$, a super-power function $\operatorname {f} (x)=x\uparrow \uparrow b$. (The term super is sometimes replaced by hyper: hyper-power function).

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 numbers, although it is an active area of research.

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

• $a\uparrow \uparrow (b+1)=a^{\left(a\uparrow \uparrow b\right)}$
• it is monotonically increasing
• it is continuous

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

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

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

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:

• $\operatorname {f} (-1)=0$
• $\operatorname {f} (0)=1$
• $\operatorname {f} (1)=10$
• $\operatorname {f} (2)=10^{10}$
• $\operatorname {f} (2.3)=10^{100}$ (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 $a\uparrow \uparrow x$ for every a, another possible requirement could be that $a\uparrow \uparrow x$ is monotonically increasing with a.

The inverse functions are called super-root or hyper-root, and super-logarithm or hyper-logarithm $\mathrm {slog} _{a}$ defined for all real numbers, also negative numbers.

The function $\mathrm {slog} _{a}$ satisfies:

$\mathrm {slog} _{a}a^{b}=1+\mathrm {slog} _{a}b$

$\mathrm {slog} _{a}b=1+\mathrm {slog} _{a}\log _{a}b$

$\mathrm {slog} _{a}b>-2$

Examples:

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

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 $x^{x^{x^{..}}}$ 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 $\mathrm {W} (z)$ represents Lambert's W function.

See also

• Ackermann function

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 $\ ^{b}a$ from Knobel's paper.)
• Reuben Louis Goodstein. "Transfinite ordinals in recursive number theory." Journal of Symbolic Logic 12, (1947).

External links

• Site on tetration
• http://www.tetration.org/