Topics Mathematics

Question In mathematics, it is commonly accepted that it is impossible to divide any number by zero. But I don't see why this necessarily has to be the case. For example, it used to be thought of impossible to take the square root of a negative number, until imaginary numbers were invented. If one could create another set of numbers to account for the square root of negatives, then what is stopping anyone from creating another set of numbers to account for division by zero.

Accepted:
August 31, 2017

Comments

It's actually easy to invent

It's actually easy to invent a system of numbers in which division by zero is possible. Just take the usual non-negative rational numbers, say, and add one new number, "infinity". Then we can let anything divided by zero be infinity. Infinity plus or times anything is infinity. Infinity minus or divided by any rational is still infinity. We have a bit more choice what to say about infinity minus infinity or divided by infinity. But we can let those be infinity, too, if we like. So infinity kind of `swallows' everything else. (Oh, any rational divided by infinity should be 0.)

Note, however, that many of the usual laws concerning multiplication and division now fail. For example, it's true in the usual case that, if a/b exists, then a = (a/b) x b. But (3/0) x 0 = infinity, not 3; of course, you can carve out an exception for 0, if you wish, but there's no way to make that work in all cases. This is not a fatal flaw, though. In the reals, a x a is always positive; not so when we add imaginary numbers. So we would expect some old generalizations to fail in the new case.

The real question is: Is there anything useful we can do with these new numbers? So far as I'm aware, the answer is "no". There are, in fact, good and useful theories of infinite numbers, but there doesn't seem to be much use for a notion of division involving them.