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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: 1,605 Threads: 363 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: 739 Threads: 104 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: 1,605 Threads: 363 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: 1,605 Threads: 363 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 MphLee Long Time Fellow Posts: 262 Threads: 23 Joined: May 2013 06/18/2022, 08:56 AM Tommy, to be honest I just think that given like that, it an seems arbitrary set of conditions. I don't quite get the red line that is connecting all those properties together and connecting the various number systems nor what you are looking for. I believe this is part of some quest you are undertaking, but I believe you have not made the goal explicit. What you are going for Tommy? Sure it's a field in its own interesting and it "deserves more attention" but in giving these definition your going random or following some kind of map/logic? I ask because there are infinite kinds of algebraic structures, and since it is abstract algebra, if you have not a chart is easy to get lost into meaningless abstraction (category theory was born for this reason, in order to do not get lost into Hilbert/Bourbaki's kind of structural abstractness, in that it gives you conceptual tool for mapping and exploring the territory). MathStackExchange account:MphLee Fundamental Law $(\sigma+1)0=\sigma (\sigma+1)$ Catullus Fellow Posts: 128 Threads: 27 Joined: Jun 2022   06/18/2022, 09:06 AM (06/18/2022, 08:56 AM)MphLee Wrote: Tommy, to be honest I just think that given like that, it an seems arbitrary set of conditions. I don't quite get the red line that is connecting all those properties together and connecting the various number systems nor what you are looking for. I believe this is part of some quest you are undertaking, but I believe you have not made the goal explicit. What you are going for Tommy? Sure it's a field in its own interesting and it "deserves more attention" but in giving these definition your going random or following some kind of map/logic? I ask because there are infinite kinds of algebraic structures, and since it is abstract algebra, if you have not a chart is easy to get lost into meaningless abstraction (category theory was born for this reason, in order to do not get lost into Hilbert/Bourbaki's kind of structural abstractness, in that it gives you conceptual tool for mapping and exploring the territory).Happy 250th post!  You are a Long Time Fellow now!  Sincerely: Catullus MphLee Long Time Fellow Posts: 262 Threads: 23 Joined: May 2013 06/18/2022, 09:26 AM wow, cool ! MathStackExchange account:MphLee Fundamental Law $(\sigma+1)0=\sigma (\sigma+1)$ tommy1729 Ultimate Fellow Posts: 1,605 Threads: 363 Joined: Feb 2009 06/18/2022, 11:36 PM (06/18/2022, 08:56 AM)MphLee Wrote: Tommy, to be honest I just think that given like that, it an seems arbitrary set of conditions. I don't quite get the red line that is connecting all those properties together and connecting the various number systems nor what you are looking for. I believe this is part of some quest you are undertaking, but I believe you have not made the goal explicit. What you are going for Tommy? Sure it's a field in its own interesting and it "deserves more attention" but in giving these definition your going random or following some kind of map/logic? I ask because there are infinite kinds of algebraic structures, and since it is abstract algebra, if you have not a chart is easy to get lost into meaningless abstraction (category theory was born for this reason, in order to do not get lost into Hilbert/Bourbaki's kind of structural abstractness, in that it gives you conceptual tool for mapping and exploring the territory). Im still researching it. I did not want to flood this thread with mini ideas and mini results. My apologies for being vague , but I want to define things formally without being inconsistant. The basic ideas are 1) unital and commutative but nonassociative numbers. 2) power-associative numbers so we can use taylor theorems. 3) no nilpotent elements 4) every element has at least 1 square root. 5) the smallest ones 6) no subnumbers only real coefficients. and not iso to an extension of 2 type of numbers ( like complex coefficients or other extensions of smaller dimensions ) then there are 2 cases left the units sum to 0. the units are linear independant. assuming solutions exist ofcourse.  I conjecture yes. On the other hand I conjecture only a finite amount of them ... probably between 0 and 3. And all solutions having dimension below 28. The 8 dimensional number given here has nilpotent elements. So it violates one of the conditions. They always have a square root though. I will post a candidate soon. I was not able to find this relatively simple idea in the books. I see applications in physics and math as I believe they are the " next quaternion ". regards tommy1729 tommy1729 Ultimate Fellow Posts: 1,605 Threads: 363 Joined: Feb 2009 06/19/2022, 12:17 AM ok here is a 5 dim candidate the coefficients are " magnitudes " ; nonnegative reals. 1 + a + b + c + d + e = 0. so when " reduced " not all magnitudes can be nonzero. mod ( 1 + a + b + c + d + e ) if you want. ( so the dimension reduces from 6 ( 5 letters and real ) to 5 ) a^2 = b^2  = ... = 1. a b = c a c = d a d = e a e = b b c = e b d = a b e = d c d = b c e = a d e = c som and products are commutative. som and product behave distributive. regards tommy1729 « Next Oldest | Next Newest »

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