Tetration

Tetration is the hyperoperation which comes after exponentiation. [math]\displaystyle{ ^{x}{y} }[/math] means y exponentiated by itself, (x-1) times. List of first 4 natural number hyperoperations, the inverse of tetration is the super root shown in the example

1. Addition

   [math]\displaystyle{ a + n = a + \underbrace{1 + 1 + \cdots + 1}_n }[/math]

   n copies of 1 added to a.

2. Multiplication

   [math]\displaystyle{ a \times n = \underbrace{a + a + \cdots + a}_n }[/math]

   n copies of a combined by addition.

3. Exponentiation

   [math]\displaystyle{ a^n = \underbrace{a \times a \times \cdots \times a}_n }[/math]

   n copies of a combined by multiplication.

4. Tetration

   [math]\displaystyle{ {^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_n }[/math]

n copies of a combined by exponentiation, right-to-left.

The above example is read as "the nth tetration of a".

Examples

• [math]\displaystyle{ ^{2}3 = 3^3 = 27 }[/math]
• [math]\displaystyle{ ^{3}3 = 3^{({3^3})} = 3^{27} = 7,625,597,484,987 }[/math]

[math]\displaystyle{ x }[/math]	[math]\displaystyle{ {}^{2}x }[/math]	[math]\displaystyle{ {}^{3}x }[/math]	[math]\displaystyle{ {}^{4}x }[/math]
1	1 (11)	1 (11)	1 (11)
2	4 (22)	16 (24)	65,536 (216)
3	27 (33)	7,625,597,484,987 (327)	1.258015 × 103,638,334,640,024
4	256 (44)	1.34078 ×10154 (4256)	[math]\displaystyle{ \exp_{10}^3(2.18726) }[/math] (8.1 × 10153 digits)
5	3,125 (55)	1.91101 × 102,184 (53,125)	[math]\displaystyle{ \exp_{10}^3(3.33928) }[/math] (1.3 × 102,184 digits)
6	46,656 (66)	2.65912 × 1036,305 (646,656)	[math]\displaystyle{ \exp_{10}^3(4.55997) }[/math] (2.1 × 1036,305 digits)
7	823,543 (77)	3.75982 × 10695,974 (7823,543)	[math]\displaystyle{ \exp_{10}^3(5.84259) }[/math] (3.2 × 10695,974 digits)
8	16,777,216 (88)	6.01452 × 1015,151,335	[math]\displaystyle{ \exp_{10}^3(7.18045) }[/math] (5.4 × 1015,151,335 digits)
9	387,420,489 (99)	4.28125 × 10369,693,099	[math]\displaystyle{ \exp_{10}^3(8.56784) }[/math] (4.1 × 10369,693,099 digits)
10	10,000,000,000 (1010)	1010,000,000,000	[math]\displaystyle{ \exp_{10}^4(1) }[/math] (1010,000,000,000 digits)

Tetration Media

• 

  Domain coloring of the holomorphic tetration {}^{z}e, with hue representing the function argument and brightness representing magnitude

• 

  {}^{n}x, for n = 2, 3, 4, ..., showing convergence to the infinitely iterated exponential between the two dots

• 

  Tetration by escape.*This is a fractal of tetration showing the points in the complex plane that escape to infinity under iteration. This fractal is similar to the Mandelbrot fractal except it is an exponential map instead of a quadratic map. The fractal was created using Fractint and shows from -4.5 to 3.5 of the real axis and from –3i to 3i on the imaginary axis.

• 

  \textstyle \lim_{n\rightarrow \infty} {}^nx of the infinitely iterated exponential converges for the bases \textstyle \left(e^{-1}\right)^e \le x \le e^{\left(e^{-1}\right)}

• 

  on the complex plane, showing the real-valued infinitely iterated exponential function (black curve)

• 

  \,{}^{x}e using linear approximation

• 

  A comparison of the linear and quadratic approximations (in red and blue respectively) of the function ^{x}0.5, from x = −2 to x = 2

• 

  = 1, e^{\pm 1}, e^{\pm 2}, \ldots and levels \arg(f) = 0, \pm 1, \pm 2, \ldots are shown with thick curves.

• 

  The graph y = xs{\displaystyle {\sqrt {x}}_{s}}