My math is rusty but 1 / 0 := 0 implies that 0 * 0 := 1. This contradicts the definition of a binary field [1], let alone a field of real numbers.

[1] https://en.wikipedia.org/wiki/GF(2)

> "It is totally fine to define 1/0 = 0."

No, it's not, at least not useful. If you define it that way, you will not have a field. Then you don't have +, -, *, and / operations with the commonly assumed behavior. However, it is definitely possible to define some operation such that 1 `op` 0 := 0, just not the inverse operation of multiplication.