Why can’t dividing by 0 be done in a theoretical field?

Question

As a layperson who is interested in math, imaginary numbers always fascinated me. Like in the real world you taking the square of a negative makes no sense whatso ever, but in theoretical math you can just invent new imaginary numbers, make it so that *i*^2 = -1 and suddenly you have just revolutionized math. If this is useful, why can’t you break other rules and account for them with new imaginary symbols?

So let’s pretend that we call them made up numbers and use *m* to represent them. Why is *m*=1/0 impossible when something like *i*^2 = -1 is not?

In: 27

Answer 1

We could define m to equal 1 / 0, but the problem is, this new number forces us to add weird exceptions to some of the rules of arithmetic.

For example, if m = 1 / 0, then presumably, 0 x m = 1. But this breaks the rule that 0 x [anything] = 0. We also have to sacrifice one of the rules used in the following argument: 2 = 1 + 1 = 0 x m + 0 x m = (0 + 0) x m = 0 x m = 1.

Either 0+0 isn’t 0 any more, or a x c + b x c isn’t always (a + b) x c, or 2 isn’t 1 + 1…

It would be fine to sacrifice those rules if the benefits of having this new number were worthwhile, but the ability to divide by 0 doesn’t actually gain us all that much.

When Complex numbers were introduced, they faced a lot of skepticism – hence their derogatory names: “complex numbers” and “imaginary numbers” as opposed to “real numbers”

Gradually, though, people found more and more uses for complex numbers, until they’re an indispensible tool for mathematics. We do lose some of the normal rules of arithmetic, for example, we no longer have sqrt(a x b) = sqrt(a) x sqrt(b) always: otherwise 1 = sqrt(1) = sqrt(-1 x -1) = sqrt(-1) x sqrt(-1) = i x i = -1. But we gain others: for example, an n-degree polynomial always has n roots (some maybe repeated), instead of some number between 0 and n. Or this: if a function is differentiable once, it’s differentiable as many times as we like.

Complex numbers are useful and interesting enough that it’s worth getting used to a new set of rules. Not so much with allowing division by 0.