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+--- Forum: Hyperoperations and Related Studies (https://tetrationforum.org/forumdisplay.php?fid=11)
+--- Thread: Negative, Fractional, and Complex Hyperoperations (/showthread.php?tid=876)
Negative, Fractional, and Complex Hyperoperations - KingDevyn - 05/30/2014
Is there a way to continue the patterns we see within the natural numbers of current hyper-operations (Hyper-1, Hyper-2, Hyper-3, Hyper-4, ect...) or at least prove that we cannot extend the value of the operation to fractional numbers? E.g. Hyper-1/2. Negative numbers? E.g. Hyper-(-2) Or even imaginary numbers? E.g. Hyper-3i.
They need not be defined, but are these operations technically there, just without practical use? Or are our names for the hyper-operations strictly for listing and naming purposes, with no way to derive meaning from such a number?
Could a fractional, or negative hyper-operation represent an operator we have already defined? E.g. Hyper-(-2)= Division, or Hyper-1/2 = Division?
Comments on the controversy of Zeration are also encouraged.
Thanks!
RE: Negative, Fractional, and Complex Hyperoperations - MphLee - 05/30/2014
\( s \)-rank hyperoperations have meaning as long as we can iterate \( s \) times a function \( \Sigma \) defined in the set of the binary functions over the naturals numbers (or defined over a set of binary functions.)
let me explain why.
There are many differente Hyperoperations sequences, end they are all defined in a different way:
we start with an operation \( * \) and we obtain its successor operation \( *' \) applying a procedure \( \Sigma \) (usually a recursive one).
\( \Sigma(*)=*' \)
So every Hyperoperation sequence is obtained applying that recursive procedure \( \Sigma \) to a base operation \( * \) (aka the first step of the sequence)
\( *_0:=* \)
\( *_1:=\Sigma(*_0) \)
\( *_2:=\Sigma(\Sigma(*_0)) \) and so on
or in a formal way
\( *_0:=* \)
\( *_{n+1}:=\Sigma(*_n) \)
That is the same as
\( *_{n}:=\Sigma^{\circ n}(*_{0}) \)
so if we can extend the iteration of \( \Sigma^{\circ n} \) from \( n \in \mathbb{N} \) to the real-complex numbers the work is done.
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RE: Negative, Fractional, and Complex Hyperoperations - MphLee - 05/30/2014
I'm not sure but I think that bo198214(Henrik Trappmann) had this idea in 2008 http://math.eretrandre.org/tetrationforum/showthread.php?tid=157&highlight=Abel+function
With his idea we can reduce the problem of real-rank hyperoperations to an iteration problem
Later this idea was better developed by JmsNxn (2011) with the concept of "meta-superfunctions"
http://math.eretrandre.org/tetrationforum/showthread.php?tid=708
I'm still working on his point of view but there is a lot of work to do...