> Reusing symbols like +, *, or / to define operations that aren't the + or the / you're used to is pretty common in math. It's just notation.

Reusing symbols in a different context is pretty common; taking a symbol that is already broadly used in a specific way (in this case, that `a/b` is defined for elements in a field as multiplying `a` by the multiplicative inverse of `b`) is poor form and, frankly, a disingenuous argument.