A fourth level

dyadic mathematic operation involving

exponentiation of

exponents.

There are two types; hyper

^{4} and hyper

_{4}.

hyper

^{4} is symbolized by either a^^b or a

^{(4)}b, and is defined as a

^{(4)}b = a^(a^(...^a)).

hyper

_{4} is symbolized by a

_{(4)} and is defined as a

_{(4)}b = ((a^a)^...)^a.

The former, hyper

^{4}, definition is used more often as an independent

operator because the other, hyper

_{4} operator can be

reduced to double

exponentiation in the form: a

_{(4)}b = a^(a^(b-1)), and thus is not really any different from

exponentiation. The hyper4 has not been extended to real numbers as

addition (1st level dyadic operation),

multiplication (2nd level) and

exponentiation (3rd level) have.

Alternate names for hyper4 can include:

tetration,

superpower,

superdegree, and

powerlog.

This concept can be extended to the fifth degree and beyond as well, however each definition becomes recursive upon the last one...

I.e. hyper5 = a^^^b = a

_{(5)}b = a

^{(4)
}a

^{(4)}...a

^{(4)}
or hyper6 = a^^^^b = a

_{(6)}b = a

^{(5)}a

^{(5)}...a

^{(5)}

For more

information on higher

dyadic and

triadic operators, see:

http://home.earthlink.net/~mrob/pub/math/largenum-2.html

P.S. If someone more knowledgeable wishes to rewrite this, feel free to do so.

This writeup is under the GNU Free Documentation License