Also not a number isn't a mathematical construct

I know that, I was using it in the literal sense. x / 0 = not a number. It isn't a number. It is not something defined mathematically. I could just as easily have said undefined, N/A, or anything else.

conventions are just conventions. All maths is arbitrary

you cannot take an object and divide it into 0 pieces. The expression is nonsensical to begin with.