Why are a and b defined as i-j and i+j?
What is the geometrical meaning behind it?
c=i^2-j^2 or is it
c=i^2+ij-ji-j^2
Keep in mind j is always < i
What does a and b MEAN in this context?
Imagine conversion to base>1 as a case of division.
We have some value c and write it in some base, it is equivalent to dividing the number by base.
The result is a weighted sum of same digit (digit = base) + the remainder: think of c=a+ib, where a is remainder, i is sum of b(base).
And knowing only (a,i) is enough for us to reconstruct c. a,i are unique result for any b. Meaning there is only one way that c could be written by division.
Can we extend this to multiplication? Maybe 0<base<1.
When we multiply(change base<1) a number the result is no longer unique as there would be multiple way for us to reach the same number by a different multiplication (base).
In multiplication the result is not a weighted sum of the same digit (not equally split) but rather a combination of all digits. In case of multiplying by 10 (base1/10) I'd imagine it would be a weighted sum of 0,1,1/2,1/3,1/4,…1/9.
The result in this case is not unique and cannot be written by just two variables.
Can multiplication be represented as fractional base? If it could be lets define negative base as 1/base: b=-10 would be base10^-1.
Lets make a spreadsheet.
c on x axis (c-c..c-2,c-1,c+0,c+1,c+2..c+c) and base b on y axis (c…2,1,0,1,2…c).
And each cell containing r=remainder (first digit) or some other data.
The goal is to find a way around this grid from the origin (base=b, c=c+0) to a cell where r=0 (base=factor of c). Can we use fractals to help?
Just some thoughts. Will try to make more sense of fractional bases later and write a script. Maybe its an equivalent problem to what VQC is proposing.