Copyright © Philip M. Parker, INSEAD. Terms of Use.
(From Wikipedia, the free Encyclopedia)
The hypern family and hyper are very closely related to
Knuth's up-arrow notation.
The family has not been extended to real numbers for n>3, due to nonassociativity in the "obvious" ways of doing it.
Known aliases for hyper4 include tetration, superpower, superdegree, and powerlog; other notation,
hyper4(a,b)=ba.
But this suffers a kind of collapse,
failing to form the "power tower" traditionally expected of hyper4:
How can a(n)b and a(n)b suddenly diverge for n>3? This 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.
Derivation of the notation
It can be seen as an answer to the question "what's next in this sequence:
summation (+),
multiplication (×),
exponentiation (^),…?"
Noting that
recursively define an
infix triadic operator
a(n+1)b = a(n)(a(n+1)(b-1))
with
a(1)b = a+b
then define hypern(a,b)=a(n)b
and
hyper(a,n,b)=a(n)bExtensions to the notation
These operators can be generalised in another way: careful readers will ask "what about expansions to the opposite side?" Since
define
a(n+1)b = (a(n+1)(b-1))(n)a
with
a(1)b = a+b
a(4)b = a^(a^(b-1))See Also
External Links
Source: the above text is adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Hyper4."
Copyright © Philip M. Parker, INSEAD. Terms of Use.
