How to add and multiply on fractional vector space How to add and multiply on fractional vector space
Please, answer in layman terms. I don’t understand the notation of Supersimetry and Super vector spaces.
If a fractional “vector space” (or his fractional equivalent) has dimension between 1 and 2:  $1 \,\leq\, (d=1+\frac{1}{n}) \,\leq\, 2 \;\;\;\; n\in \mathbb{N}$, and a generic vector is $(r,f)$ with $r \in R$ and  $f \in \mathbb{R}$, being r the real part, and f the fractional one:
How are the vectors added? What is the formula? Is $(r_1,f_1)+(r_2,f_2)=(r_1+r_2,f_1+f_2) $?
How are those vectors multiplied? Like a complex number? What is the fractionary imaginary unit $\omega$?
That would make them the same as a complex number, so I guess that's wrong.
I guess that the fractional number is redundant, like it is periodic.
Can this be $r*e^{\omega. \theta}=r(cos(\Omega.\theta)+\omega.sin(\Omega. \theta))$
so, $\Omega$ would change the periodicity of the number, so the same number will be repeated with faster frequency than in complex numbers. Is that appropriate? what would be $\Omega$ as function of $1+\frac{1}{n}$?
I know that the question is vague. I do not really know exactly how to ask it.
I imagine that numbers with a fractional dimension $1 \leq \alpha \leq 2$ belong to a space which is a surface whose area grows with a non integer power of the radius. Like complex belongs to a plane whose area grows with the square of the radius.
 A: If we concern ourselves with number systems rather than vector spaces, then potentially some different notion of dimension can be defined.  However (and this is already argued somewhat by the OP) there is no number system between the real numbers and the complex numbers with a definition of addition and multiplication operations consistent with their definitions on the real numbers.  So the simplest possible example of number systems of "fractional dimension" you envision does not exist.  
See Haudorff-Besicovitch dimension for a definition that allows fractional values, but this is for metric topological spaces and not for "number systems" generally.  However "Hausdorff dimension" (as it is usually called) does generalize the notion of dimension of a vector space, since when we apply it to a (finite) $n$-dimensional real vector space with the product topology induced by the usual metric topology on the real numbers, we get the same number $n$ for the Hausdorff dimension.
As far as explaining this in layman's terms, the discussion seems to require some degree of mathematical preparation in order to get "fractional dimensions".  It is true that the concept of fractals has penetrated into popular culture somewhat, mainly in the form of attractive pictures, but this hardly constitutes an explanation in mathematical terms. 
