Likely no (the peculiar distributive laws seem unusable to me). But I have come across similar rules that I think may have a real usage. It sometimes seems to me that it might make more sense to keep 'factors' attached to 0s, such that x-x = (1-1)x = 0x might be 'sound', and to keep track of 'powers' of 0s by having factors of 0 attached to 0s: (0)0 = 0^2 != 0, etc.

If you keep factors like this, then you could implement L'Hopital's rule without the result only being true if considered under a limit: say, lim(x->0) (5x^2/ 3x^2) = 5/3 could be computed as (5)0^2 / (3)0^2 = 5/3.

This is not like Wheels though; it requires that any power of 0 be a distinct number. I of course have no idea if it is sound or meaningful, but I do find myself thinking about it a lot.

Those "factor" zeros look like infinitesimal forms, which are well studied.

There is a similarity, but no one claims infinitesimals are _literally 0_, and certainly no one would claim that 2-2 = 2 epsilon.