Beyond + and -

[Benny]

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 dont think any living being on this planet understood that.

Sorry, the concept of using an meta-operator that is assigned a sign to produce an infinite numbers of operators is quite new, so that's why it might be confusing.

As we added a new direction from + to + and - and from + and - to + and - and §, we can obviously do that again and again (always using the same simple addition rules based on "exlusive direction"). Then we get the 4th roots of unity, the 5th roots of unity, etc... So we can extend the concept to infinity, but then we can't use unique signs, because we can only define a limited amount of them.
So we simply use a number within the "meta-operator" (°°) to express which operator we mean.
Really it doesn't matter which numbers we use, so I suggested 0 to 1. (with °0°=°1°=+).
Then °0°=°1°=+, °0.5°=-, °1/3°='-', °2/3°=§.
Given that we then have infinitely many directions, we can use that to map them onto the 2-d plane, so we have a description of a point on it using "°operator-value° number-value". We could convert it to an angle and a length, or to complex numbers.
Of course we still have the advantage of addition (or rather °°-tion) working the same for all angles (in contrast to imaginary numbers were -1+1=0 and i+1!=0) and having nice roots.

Actually, maybe we could extend that to the third (or n-th) dimension using a meta-operator with two arguments (°x,y°) - x representing the angle with respect to + in the second dimension, y representing the angle with respect to + in the third dimension. Or maybe even with nested operators °x°y°°?