Octonion Hyper-Operations

[Catullus]

How does tetration work for quaternions? What about for octonions? How do pentation and higher hyper-operations work for those?

[tommy1729]

We try to walk before we can run

We do not even understand tetration well for octonions ( or even complex numbers lol )

And there are many tetrations and many hyper-operators.

So I assume it might be case specific.

I like the question , but it seems very very general.

Wish I could say more ... sorry.


regards

tommy1729

[tommy1729]

I think most (known) methods rely on properties such as commutative associative etc.

So that might imply they all fail for most non-complex numbers.

Note that it is also problematic to define derivatives for non-complex functions. ( zero-divisors , depending on the type of infinitesimal , ... ) 
so that makes formulas that use calculus concepts * uncomfortable *. 

Functional composition is associative ... but if we plug in non-associative numbers as parameters ... feels * delicate *.

Also we might get extra fixpoints in say the quaternions and such that harrass our convergeance domains and such.
Or as we learned before ; functions do no agree on the fixpoint. ( see the bummer thread )

There are also issues with combinatorial interpretations/methods but those are complicated.

Best I hope for is that the taylor series expansions of the solutions we have can be used ;

plug in a power-associative number x instead of the complex z and just compute the values.
... It should agree with the complex solution and by extension also the field/ring/whatever of x.

hopefully.



regards

tommy1729

[Catullus]

How do they work for matrices?