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