Is there such a thing as 2.3 variables? Is there such a thing as 2.3 variables? for example there are system of equations of n variables, but what about when n is not an integer , when n is a real number or even complex number? What about matrix? 
Is there anything in math that generalises the  notion of variable? or even defines it?
I have known about fractional calculus for 25 years, but never came across the topic of generalisation of number variables. 
I really want to tag this question "What am I looking for"
 A: Context is everything in "soft questions".  Sticking to a context of formal languages, no, there is not a possibility of "a system of equations in $n$ variables... when $n$ is not an integer".  For the sake of having validity of formulas be effectively decidable, formal languages are limited to finite strings drawn from "alphabets" of distinct symbols.  Therefore in this context, a non-whole number of variables cannot be realized.
However if we are trying to think outside the box, then one possible tack is the notion of fractional degrees of freedom as discussed over at Cross Validated (Stats.SE).  The connection is fairly straightforward:  In statistical estimation the notion of degrees of freedom typically corresponds to the number of independent parameters available in optimizing a fit.  For example, the sample mean (as an estimator of the population mean) using $N$ points has $N-1$ degree of freedom (because samples sharing a common mean are constrained by that one degree).
Formulas involving a degrees-of-freedom value are important in computing confidence intervals and thus in justifying tests of significance.  In some cases, such as variance of non-chi-square distributions, the appropriate "degrees of freedom" value is not an integer.
This suggests an analogy to the context suggested in Comments above, by @Ivan, for topologies with fractional dimensions, e.g. with sampling in a space where "fractional" numbers of coordinates are needed to specify a point. In both cases it might be objected that we are taking liberties with terminology to make a connection with the number of variables in a system of equations.  However the same can be said about the number of derivatives taken and the fractional calculus.
