Due to measurement uncertainty, after measuring a value of a physical quantity (or quantities), we do not get its exact value, we only get a set of possible values of this quantity (quantities). In case of 1-D quantities, we get an interval of possible values. It is known that the family of all real intervals is closed under point-wise arithmetic operations (+,-,*) (i.e., this family forms an arithmetic). This closeness is efficiently used to estimate the set of possible values for y=f(x1,...,xn) from the known sets of possible values for xi.
In some practical problems, physical quantities are complex-valued; it is therefore desirable to find a similar closed family (arithmetic) of complex sets. We follow K. Nickel's 1980 paper to show that, in contrast to 1-D interval case, there is no finite-dimensional arithmetic.
We prove this result by reformulating it as a geometric problem of finding a finite-dimensional family of planar sets which is closed under Minkowski addition, rotation, and dilation.