I was writing a unit converter for an industrial setting. To ensure that $\frac 0 0$ and $\infty$ never show up in the user interface I made a rule that no unit conversion factor can ever be zero.
And then curiosity struck. Is there such a thing in any branch of mathematics as a unit conversion factor of zero? Is there a context where this concept actually makes sense?
The vector space analogue of "units" is "norm". Now, a norm can never have a zero conversion factor with another norm, because a norm can only be zero on the zero vector.
However, there is a generalization of norm called seminorm, which can be zero on nonzero vectors. In particular, the trivial seminorm is zero on every vector. Hence, we can convert from any norm or seminorm to the trivial seminorm by multiplying by zero. This isn't a particularly interesting or profound statement, but it happens to be true. People who study seminorms do make use of the trivial seminorm.
With respect to your context, there is no reason to disallow $0$ as the numerator in the unit converter, just that the denominator is never zero. However if your converter is reversible, then you don't want $0$ in either place.