Giving the answer that is relevant to the situation is very sensible. Saying that "this is the only thing division can mean in mathematics" is evidently false, though (I take it that you agree with me on this now?), and false in a way that is very relevant to the original question, which did not specify a context of abstract algebra and seemed to me to be very interested in expanding the mathematical horizons of the questioner, not restricting them.
The extended complex plane is a great example in my opinion, because it shows that yes there are reasons to extend the numbers in various ways, that can give useful structure, but you may have to give up something else in order for that make sense. In my opinion that is a much more complete answer to the deeper question. (Similarly for the reals mod 1, which do have the property that x + 1 = x).
The extended complex plane is a great example in my opinion, because it shows that yes there are reasons to extend the numbers in various ways, that can give useful structure, but you may have to give up something else in order for that make sense. In my opinion that is a much more complete answer to the deeper question. (Similarly for the reals mod 1, which do have the property that x + 1 = x).