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


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

As a computer engineer that became an educator, one of the things you need to realize is that math is just a language, but it’s the most precise way of saying things. It’s a way of saying most precisely exactly one thing. No matter how many words you use, it doesn’t make sense to take nothing, and divide it into smaller pieces of nothing and no amount of nothing becomes something. That’s what nothing is.

We could, there just hasn’t been any proven benefit in doing so. We use imaginary numbers because they allow us to do useful things (mostly involving rotation). If you can find a practical application for *m*, go for it.

The reason is that the basic operations that we define on real numbers and complex numbers (and fields in general – there are more than just these two) can’t fulfill the basic properties we want them to have if m=1/0 was defined.

It’s not too hard to see this:

What’s 0? 0 is the element that you can add to any number without changing it (the “neutral element”): For all numbers a, a+0=a

From this and our idea of how addition and multiplication should be compatible, it follows that any number multiplied by 0 is 0 again: For all a, a*0 = 0 (you can see this by multiplying the above equation with any number b: ab + a*0 = a*b, so a*0=0).

But now, m=1/0 can’t be defined, because m*0 should be 0, but if m=1/0, m*0= 1/0*0, it should also be 1, and that’s not possible at the same time.

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.

The one easy way to illustrate why dividing by zero is problematic is to literally see what happens to 1/x as it approaches 0.

If you approch from the right (positive numbers), it races off to positive infinity. If approached from the left (negatives) it races off to negative infinity. In other words, it’s two literally opposite things at once. A contradiction. It just doesn’t behave nicely.

If I recall properly, there are attempts to define 1/0 as a new number similarly to i, but it keeps causing problems. I honestly don’t know the details, though.

In comparison, i has well defined properties and complex numbers not only don’t introduce contradictions, but solve a few other problems along the way.