Well, why not? The usual answers boil down to either an appeal to arithmetic intuition (e.g. treating division as repeated subtraction, you can never subtract zero to get anywhere) or definition (you can't divide by zero because zero is defined not to have a multiplicative inverse, i.e. "because I said so"). But the former is mathematically naïve and the latter is unsatisfyingly circular. So let's dig in to why we want to define zero out of division.
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