The article uses two example theorems to justify this: a * (b/c) = b * (a/c), which can be extended to cover the c=0 case, and a/a = 1, which can be extended to

The article uses two example theorems to justify this: a * (b/c) = b * (a/c), which can be extended to cover the c=0 case, and a/a = 1, which can be extended to cover the a=0 case. I can see how this can be useful in some situations, but it's not natural in the same way as e.g. analytical continuations of complex functions are.