I'm just curious if there are any physical calculators out there that deal in the hyperreal number line or with multidimensional (i.e., complex) numbers.
It would be a fun project to do, but I'm curious if anyone has done one already. Searches of this sort are difficult, as google always returns "calculators" on websites, not physical calculators.
On
Eh, part of the problem is that you'd have to be able to do computations with its set of natural numbers, which in general is impossible.
That being said, I bet you could work in a proof system axiomatically. You could also use a regular computer algebra, and then use the transfer principle.
(In particular, I imagine something like mathematica or the wolfram language, argumented with a "variable" $H$, along with assumed facts "$H \ge 0$", "$H \ge 1$", "$H \ge 2$", $\dots$ would be pretty powerful.)
The hyperreal field is fairly large and not easily implementable on a computer but a smaller field called Levi-Civita field has been implemented by a group around Shamseddine, see e.g.,
The implementation is accessible here.
The Levi-Civita non-Archimedean field is probably sufficient for most applications that a calculator user may be interested in.