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+--- Thread: " tommy quaternion " (/showthread.php?tid=1288)
RE: " tommy quaternion " - marraco - 01/24/2021
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.
RE: " tommy quaternion " - tommy1729 - 02/12/2021
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
RE: " tommy quaternion " - JmsNxn - 02/14/2021
Off topic but I like that you use a day planner/agenda to write your math in, lol.
RE: " tommy quaternion " - tommy1729 - 03/23/2021
My friend posted the question at Mathoverflow :
https://mathoverflow.net/questions/387113/nonassociative-algebras-closed-under-sqrt
So it is more formal now.
Some of you are probably on MO.
regards
tommy1729
RE: " tommy quaternion " - tommy1729 - 09/16/2021
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
RE: " tommy quaternion " - MphLee - 06/18/2022
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).
RE: " tommy quaternion " - Catullus - 06/18/2022
(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.Happy 250th post!
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).
You are a Long Time Fellow now!
RE: " tommy quaternion " - MphLee - 06/18/2022
wow, cool !
RE: " tommy quaternion " - tommy1729 - 06/18/2022
(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
RE: " tommy quaternion " - tommy1729 - 06/19/2022
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