Tetration

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in math prevents it from being rendered as HTML, which

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can't handle nested exponents -->

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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.

Related Topics:
Hyper operator - Portmanteau word - Reuben Louis Goodstein - Tetra- - Iteration - Notation of very large numbers

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Tetration follows exponentiation in the sequence:

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• addition
• :$a+b$
• multiplication
• :
• exponentiation
• :
• tetration
• :

where each operation is defined by iterating the previous one.

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

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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:

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:,!2^{2^{2^2}} = 2^{left(2^{left(2^2 ight)} ight)} = 2^{left(2^4 ight)} = 2^{16} = 65,!536

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:,!2^{2^{2^2}} is not equal to ,! left({left(2^2 ight)}^2 ight)^2 = 256

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

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:,! left(left(2^2 ight)^2 ight)^2 = 2^{2 cdot 2 cdot 2} = 2^{2^3}.

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Thus, its general form still uses ordinary exponentiation notation.

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The notations in which tetration can be written (some of which allow even higher levels of iteration) include:

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• Standard notation: $\{\}^ba$ — first used by 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 equivalently, an indexed arrow
• Conway chained arrow notation: $a\; ightarrow\; b\; ightarrow\; 2$ — allows extension by increasing the number 2 (equivalent with the extensions above), but also, even more powerfully, by extending the chain
• hyper4 notation: $a^\{(4)\}b$ = hyper4 (a, b) = 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 = A(4, b−3) + 3, i.e. A(4, n) =

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2 uparrowuparrow (n+3) − 3

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

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Tetration: Examples ►