DRAFT: Interval arithmetic, Extended numbers and Computer Algebra Systems (CAS)

Many ambitious computer algebra systems were initially designed in a flush of enthusiasm, with the goal of automating any symbolic mathematical manipulation “correctly.” We see this kind of goal in programs like Macsyma, Maple, Mathematica. This approach often resulted in programs that implicitly used certain identities to simplify expressions. These identities, which likely seemed universally true to the programmers in the heat of writing the CAS, later needed re-examination when such systems were elaborated for dealing with unanticipated kinds of objects. In some cases problems turn up in introducing complex numbers in a system designed for real-valued variables. In the situation examined here, we consider “extended” numbers like ∞ or undefined, or intervals (say, of real numbers). We comment on issues raised by these two troublesome notions, how their introduction into a CAS may require a (sometimes painful) reconsideration and redesign of parts of the program, and how they are