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Negative, Fractional, and Complex Hyperoperations
#1
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!Tongue
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#2
-rank hyperoperations have meaning as long as we can iterate times a function 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 (usually a recursive one).



So every Hyperoperation sequence is obtained applying that recursive procedure to a base operation (aka the first step of the sequence)



and so on

or in a formal way




That is the same as



so if we can extend the iteration of from to the real-complex numbers the work is done.

----------------------

MathStackExchange account:MphLee
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#3
I'm not sure but I think that bo198214(Henrik Trappmann) had this idea in 2008 http://math.eretrandre.org/tetrationforu...l+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/tetrationforu...hp?tid=708

I'm still working on his point of view but there is a lot of work to do...
MathStackExchange account:MphLee
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