Beyond + and -

[tommy1729]

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)?

yes analogies 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).

yes however the abc formula for solving 2nd degree polynomials may fail because of lacking certain properties of these numbers and the amount of solutions =/= 2.

and thus also the concept of discriminant might need revision.

i stick with my story ; main questions are what properties and how many zero's for a certain degree polynomial.

i think considering polynomials first and other functions later makes a lot of sense.

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.

dont you mean variable when you say sign ?

are you talking about operators or logaritms ?

i didnt understand a word , number or symbol of that.

i doubt if its a good idea , but im sure it must be explained better , in fact im not sure if that is even "math talk".

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 .

since the signs have no superposition ( they cancel each other ) you have 3 lines starting from the origin ( = 0 ) rather than 3 dimensions ( that requires superposition and between 3 and 6 variables ).

fractals would thus be hard to draw.

and btw fractals might convince most ppl , but not necc most mathematicians.


regards

tommy1729