What do I need to create a coherent new set of numbers?

Im doing a research and im trying to create a new set of numbers to explain a new dimension, what is fundamental for a new set of numbers? What properties need to stay the same? etc.

[DGM]

Why is a new set of numbers required to explain a dimension?

How do any set of numbers explain any dimension?

Have you ever been so far as to do more like?

[Carlos Javier]

1. A new set of numbers is required to explain a dimension when the existing systems cannot fully capture or represent the properties, im trying to create a new set of numbers to explain "negative dimensions"
2. Different sets of numbers describe dimensions by providing coordinates, magnitudes, or relationships within a space.

Answer 1

[Pavl M.]

According to phylosophy, the novel set of numbers should be closed under:

• Distinction property (i.m. how to distinguish between one or another number). And more advance of this property is comparison property (comparator operator). If you make it closed also under comparison, then the set of numbers is just some transformation of the set of Regnal Numbers.

%New_number for new dimension evaluation class:
classdef new_n 
properties
      x
      y
      z
end
methods
    function obj = new_n(x1,y1,z1)
            obj.x = x1;
            obj.y = y1;
            obj.z = z1;
    end
    function r = X_(obj)
        r = obj.x;
    end
    function r = Y_(obj)
        r = obj.y;
    end
    function r = Z_(obj)
        r = obj.z;
    end
      function r = ThirdDimension(obj)
         r = obj.z;
      end
      function r = R(obj)
         r = sqrt(obj.x^2 + obj.y^2 + obj.z^2);;
      end
      function ang = angle(obj)
          u = [obj.x obj.y obj.z];
          v = [1 0 0]; % or v = [0 1 0];
          ang = atan2(norm(cross(u,v)), dot(u,v));
      end
end
end



%new_symbol = 0^0*k_s; % Matlab so far only has i,j symbols for Complex Numbers (max)
%new_number = [x y z];
%new_number = Re(new_number) + i*Im(new_number) + new_symbol*ThirdDimension(new_number);

%where Re and Im are the regular Real and Imaginary parts of Complex Number
%and ThirdDimension is the your new dimension. 
%Graphically it will be 3rd axis Z. So 3D plot instead for 2D for z = Real + j*Imaginary;
%It will be polar plot in 3D, just the radius will be R radius of sphere 

%R = sqrt(Re(new_number)^2 + Im(new_number)^2 + ThirdDimension(new_number)^2);

%The angle will be 

%u = [Re(new_number) Im(new_number) ThirdDimension(new_number)];
%v = [1 0 0]; % or v = [0 1 0];
%phase = atan2(norm(cross(u,v)), dot(u,v));

%To be continued: 3D polar representation of new number:
%new_number = R*exp(phase*3DUnitVector)

%You may try graphically  

%disp('For many works dedicated to only you, pay me salary in natural number to my real account number for work in on the specific subdomain of hyperreal analysis domain or other custom.')

%Developed from needing help code by
    %https://independent.academia.edu/PMazniker
    %[phone]
    %https://diag.net/u/u6r3ondjie0w0l8138bafm095b
    %https://github.com/goodengineer
    %https://orcid.org/0000-0001-8184-8166
    %https://join.skype.com/invite/oXnJhbgys7oW
    %https://willwork781147312.wordpress.com/portfolio/cp/
    %https://www.youtube.com/channel/UCC__7jMOAHak0MVkUFtmO-w

[Carlos Javier]

Thank You!