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 " tommy quaternion " marraco Fellow Posts: 100 Threads: 12 Joined: Apr 2011 01/24/2021, 06:53 AM I have the intuition that tetration requires the introduction of numbers with real dimension (non integer, like complex or quaternion sets). A problem is that those numbers have to include hypercomplex as a subset, but, for example a set with the dimension of a serpinsky triangle, would be a subset of the complex, and it cannot be closed under addition/multiplication, without including all of the complex numbers, but then it would just be equal to the complex. So, closure has to be abandoned, or the set has to be a set of sets with all the possible dimensions. Also, sometimes tetration leads to step functions or dirac delta functions. A diract delta is similar to a segment of line with unit length. It is one dimensional, meanwhile an hypercomplex number is a point, which is zero-dimensional. It is analogous to surreal numbers, where a segment of line would be like a number infinitely larger than a point. So maybe the serpinsky triangle should be taken as something in between a point and a surface, instead of a subset of the complex. Instead of thinking of numbers as points, we should think of then as fractal sets. An exponentiation to a real exponent, in general, has infinite solutions (because a root has infinite solutions in the complex field). Those infinite solutions are a set. Probably we should think of tetration as operations between sets instead of between numbers. Probably fractal sets. Possibly singularities in functions are just numbers with a different dimension. A pole in a complex function may be a line number with length equal to the pole's residue. Possibly Taylor series require numbers with a different dimension outside of his radius of convergence, to be able to calculate pass singularities. I have the result, but I do not yet know how to get it. tommy1729 Ultimate Fellow Posts: 1,507 Threads: 358 Joined: Feb 2009 02/12/2021, 11:17 PM For the " tommy octonion " we have the following Jacobi matrix. By that I mean the Jacobi matrix for taking f(X) = X^2 in that number system. see pictures. The inverse of that matrix corresponds to taking the square root whenever the matrix is invertible (the determinant of the original Jacobi is not zero). ( see : https://en.wikipedia.org/wiki/Inverse_function_theorem ) regards tommy1729 Tom Marcel Raes Attached Files Image(s)         JmsNxn Long Time Fellow Posts: 628 Threads: 102 Joined: Dec 2010 02/14/2021, 02:24 AM Off topic but I like that you use a day planner/agenda to write your math in, lol. tommy1729 Ultimate Fellow Posts: 1,507 Threads: 358 Joined: Feb 2009 03/23/2021, 01:21 PM My friend posted the question at Mathoverflow : https://mathoverflow.net/questions/38711...under-sqrt So it is more formal now. Some of you are probably on MO. regards tommy1729 tommy1729 Ultimate Fellow Posts: 1,507 Threads: 358 Joined: Feb 2009 09/16/2021, 11:34 PM another idea are my " xyz numbers ". They are also a commutative 4d type of number. x*x = y*y = -1 x*y = y*x = 1. x*z= z*x = y y*z = z*y = - x z*z = - 1 + x + y notice many properties are not present ; not associative , nilpotent , no unique inverses etc For instance  (x+y)^2 = 0 but x+y is not 0. -x*x = y*x = 1. but -x is not y. still investigating. what do you think ? regards tommy1729 « Next Oldest | Next Newest »

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