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Nothing is impossible in mathematics. Rather, it's all a matter of how you define your axioms, and what sorts of axioms are most useful to define.

So, I should say up front: Yes, it is impossible to divide by zero within the field of complex numbers and obtain a unique complex number. The definition of real and complex numbers depends upon the fact that division by zero is impossible within this field.

That said, there are other fields in which division by zero is possible.

Most trivially, the null field, the field containing only one number: 0. In the null field, it is possible to add, subtract, multiply, divide, exponentiate, and whatever else you want to do to zero, and you're guaranteed to get back zero. It's all perfectly consistent-and completely useless.

There is however another, more useful field (actually there are infinitely many, but this one is particularly worthwhile) of numbers in which division by zero is possible: it's called the transreal numbers, and though you may not realize it, it more or less forms the foundation for modern computer calculations.

In the transreal numbers, everything works much as it does in the real numbers, except for the following rules:
1. 0 and -0 are not the same number; when tested by the "=" operator, they evaluate the same, but they are stored as distinct binary codes.
2. 1/0 = infinity, 1/-0 = -infinity.
3. 0/0 = nullity.

Why do this? Well, on a computer, every possible instruction has to return SOME value, even if that value would not be mathematically meaningful. So when asked to divide 0/0, a computer will produce an error message "Division by zero" or "Not a number" (it's not standard practice to actually call it "nullity"), but it will produce *something* nonetheless. "Infinity" is usually given as an "overflow" message, and similar error messages for division of 1/0 or -3/-0 and so on.

It is in this sense only that division by zero is possible; if you're hoping we'll discover how to do it in ordinary mathematics, don't hold your breath.

Learn more about this author, Patrick Julius.
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