As discussed earlier, the extension of intervals to complex numbers is fraught with (currently unsolvable) problems.
Complex numbers as _pairs of real numbers_ "works", but isn't particularly useful, as the precision isn't very good. Note that in this incarnation, the only complex sets are Cartesian products of real intervals.
For example, you can't represent just the set {-1-i,1+i}; you can only represent the larger set {+-1+-i} = ([-1,-1] union [+1,+1]}x([-i,-i] union [+i,+i]).
Therefore, I'm ignoring complex numbers in my current prototype.
At 11:16 AM 2/12/2013, Stavros Macrakis wrote:
>I don't think there is any conceptual difficulty in defining arithmetic on sets of intervals, including open/closed endpoints.
>
>The challenge is implementing them uniformly across all of Maxima's functionality reasonably cleanly and reasonably efficiently. Several approaches have been proposed in the past, but no one has actually implemented any of them completely.
>
>Any approach should generalize to other kinds of "generalized number", such as complex numbers in rectangular form, complex numbers in polar form, complex neighborhoods (circular, rectangular, or other), partial fractions, quadratic numbers, etc. etc. (Most of these are closed under elementary arithmetic, but not necessarily under other functions.) Not the specifics, of course, but the mechanism.