I don't know much about group theory or complex numbers, so I appreciate you explaining it in detail. 
Given that you expressed § and '-' as complex numbers, does this mean that we could in theory express the same with complex numbers? It seems that equations with addition don't work with those definitions.
So your equalities of complex numbers with "-"/§ are more analogies which mean they act as if they were equal to it (with respect to multiplication)?
With your multiplication table we now have "nice" squareroots of all numbers (while with i we get a mess taking squareroots).
Actually now I get a massive idea.
If we define a sign that is associated to a number between, say, 0 and 1, let's call it °x°, then, as far as I can see, we recover the usual + and - (°0° and °0.5°), + - and § (°0° and °1/3° and °2/3°), and all imaginary numbers (expressed as x*°y° with y representing the angle; addition of complex numbers can be expressed in terms of multiplication of these numbers). Only that we now have nice addition (very important!) and neat looking roots.
Addition works as with + - and §. Multiplication works as usual (with respect to the values), and means that we add the sign value (taking it to be cylic group). Squaring means multiplicating the sign value with 2, taking the square root means dividing the sign value through 2 (third power means multiplicating with 3 etc...).
To formalize it °x°*°y°=°x+y°
°x°^y=°x*y°
But what happens if we get °°0.5°°. Do we go into the third dimension?
I would really like to see a plot of some functions of these numbers (should work similar to a plot of complex functions), and fractals?
Any good programmers here?
I think as soon as we draw a really great fractal people will be convinced that the concept makes sense
.
Given that you expressed § and '-' as complex numbers, does this mean that we could in theory express the same with complex numbers? It seems that equations with addition don't work with those definitions.
So your equalities of complex numbers with "-"/§ are more analogies which mean they act as if they were equal to it (with respect to multiplication)?
With your multiplication table we now have "nice" squareroots of all numbers (while with i we get a mess taking squareroots).
Actually now I get a massive idea.
If we define a sign that is associated to a number between, say, 0 and 1, let's call it °x°, then, as far as I can see, we recover the usual + and - (°0° and °0.5°), + - and § (°0° and °1/3° and °2/3°), and all imaginary numbers (expressed as x*°y° with y representing the angle; addition of complex numbers can be expressed in terms of multiplication of these numbers). Only that we now have nice addition (very important!) and neat looking roots.
Addition works as with + - and §. Multiplication works as usual (with respect to the values), and means that we add the sign value (taking it to be cylic group). Squaring means multiplicating the sign value with 2, taking the square root means dividing the sign value through 2 (third power means multiplicating with 3 etc...).
To formalize it °x°*°y°=°x+y°
°x°^y=°x*y°
But what happens if we get °°0.5°°. Do we go into the third dimension?
I would really like to see a plot of some functions of these numbers (should work similar to a plot of complex functions), and fractals?
Any good programmers here?
I think as soon as we draw a really great fractal people will be convinced that the concept makes sense
.