TITLE: Dividing by zero and imaginary numbers
QUESTION [2 upvotes]: If we decide to call the square root of -1 its own number, then why not do the same for dividing by zero? Just make an entirely new type of number?

REPLY [2 votes]: If you introduce a division by zero, you can only retain a subset of the usual rules of arithmetic or you run into contradictions. If $0\times x=1$ then associativity would give $$2\times 0 \times x = 0 \times x = 3 \times 0 \times x,\,2 \times 1 = 3\times 1.$$
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