Home > Hyper operator

hypern (a, b) =

(See Knuth's up-arrow notation and Conway chained arrow notation.)

For n = 4 we have hyper4 or tetration, super-exponentiation or power towers in terms of an extension of standard operators:

See also Tables of values .

1 Derivation of the notation

It can be seen as an answer to the question "what's next in this sequence:

Noting that

recursively define an

with , , and

for

then define

and

This gives:

as further explained in the separate article tetration

Known aliases for hyper4 include superpower, superdegree, and powerlog; other notation, .

The family has not been extended from natural numbers to real numbers in general for n>3, due to non associativity in the "obvious" ways of doing it.

2 Evaluation from left to right

An alternative for these operators is obtained by evaluation from left to right. Since

define (with subscripts instead of superscripts)

with , , and

for

But this suffers a kind of collapse, failing to form the "power tower" traditionally expected of hyper4:

How can be so different from for n>3? This is because of a symmetry called associativity that's defined into + and × (see field) but which ^ lacks. It is more apt to say the two (n)s were decreed to be the same for n<4. (On the other hand, one can object that the field operations were defined to mimic what had been "observed in nature" and ask why "nature" suddenly objects to that symmetry…)

The other degrees do not collapse in this way, and so this family has some interest of its own as lower (perhaps lesser or inferior) hyper operators.

For example:

moser = (..(2^^2)^^..2)^^2 (258 numbers 2)

3 See also

• Ackermann function
• Tetration
• Tables of values .

4 External links

• The Dictionary of Large Numbers (Dictionary's author Robert Munafo claims the (n) notation as his own—it's alright to use it, but it's not a standard.)
• Lynz and the Clarkkkkson
• On extending hyper4 to nonintegers