I just love the rational arithmetic.
A possible new companion to it might be interval arithmetic.
While it might be a little tricky to implement in some areas, it would be really fascinating. And it could be extraordinarily useful for error analysis.
An interval number would have 2 parts -- a minimum value and a maximum value. We're running out of syntactic joining characters, but perhaps d could be used for interval Delta. For example, 1.0d1.3 would mean a number of about 1.2, with a possible error range of 0.3.
The parts would propagate through all calculations, and perhaps the gap would get wider, indicating a growing uncertainty.
There are some interesting ramifications, too.
For example, an argument could be made that
0÷0 ←→ ¯∞d∞
and that constant might have the nickname of Anything.