If we agree that zero is a number, then there is no rational reason you can't divide by it. The result you get is not infinity. The axiom is that any arithmetic operation performed on a number always yields another number, and infinity is not a number, so infinity can't be the "answer."
The situation is a gaping fault in the structure of number theory and arithmetic that is quietly ignored by most.
I would suggest you wait until you are fully tenured before you submit papers where you divide by zero, and be sure to point out that you are aware of it.
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