> Mathematicians define equality of fractions by stating that a/b = c/d if and only if a·d = b·c. This means that if we define 1/0 = 1 then 0 = 1.

That's not implied, so far as I can tell.

a / b = 1 / 0, thus

a * d = b * c => 1 * d = 0 * c => d = 0

So all you can say is that d = 0, or at most that c / 0 = 0. Is there some extra step you're taking?