1/0 (opinion)

Browse up the internet and you’ll find scores of students wondering what, if anything is the result of dividing 1 over 0. There’s a second, less numerous but still large group of people who think they can “solve” the matter. If any of them posted here on E2 the only possible answer to them would start with the words “Your radical ideas about defining new math…” or something similar.

So let’s tackle the question head on. If lots of things in mathematics are conjured out of thin air (like that pesky thing i whose square is -1) why is it that mathematicians don’t “invent” something that settles the value of 1/0 once and for all?

Useful tip for anyone who deals with mathematics: any time you wish to extend a definition—or explore an old one—see if it “plays” nicely with the old ones. This usually leads to a refinement of your new radical idea.

Let’s see an example. Suppose that I define 1/0 as a number that we will call H. For now we don’t care what it is exactly, but I’ve defined it as so.

Victory! What next?

Well, that number will only be useful after it’s put to work, and to do that we need to see how it behaves. And for that we need a small interlude.

Interlude: Long ago, people defined the multiplication as an operation, that is, an “action” that acts upon two things. It has a few rules:¹

1. If I take two things out of the set S and multiply them together, the result should also be an element of S
2. The order in which I multiply two things shouldn’t matter.
3. There should be one thing (called the identity) that makes the multiplication useless. In other words, this “identity” when multiplied with “something else” will always result in “something else”
4. If I have a “thing 1” there should be a “thing 2” so that when I multiply them, I get the “identity” as a result.
5. The above pairs of “things that multiplied together give the identity” should be unique.

That might seem like kids play, right? Some of you might already see that the above rules are how multiplication works, that my “identity” is the number 1 and the “things” are regular old numbers.² Why is this important?

Let’s go back to my number H. How does it behave?

If 1/0 = H it follows that 0 × H = 1 right? So far so good… except this multiplication should work with the above rules. We know that anything multiplied by 0 gives 0; but this H multiplied by 0 gives 1. How?

“Andy, what if H = ∞?” you might ask. That doesn’t work, because the funny squiggle ∞ is not a number. That merits another essay, but let’s leave it there.

If that still doesn’t convince you, let’s see this: how about 2/0? Well, it’s the same as 2 × 1/0 = 2 × H. With a bit of algebra, we see that if 2/0 = 2H it follows that 0 × 2H = 2 right? But we have the same problem: we are multiplying something by 0 and getting a result that is not 0. How? Moreover, given our multiplication rules, we have 0 × 2H = (0 × 2)H=0H.

At this point, some might be tempted to suggest backpedaling and rewriting the multiplication rules. Fine, we can do that, it’s math. We can add clauses and sub-clauses, and special cases… But the more we do that, the more we have to check what else are those rewrites doing.

This is precisely the problem with retconning and traveling to the past: the more you go back to change things, the more things change in the present. In media, this ends up with a lot of weird questions. In math, you end up with useless definitions. A rule that has more exceptions than applications is a lousy rule.

And that’s the final answer. Mathematicians don’t “just invent” a new number that satisfies 1/0 because it tends to screw up lots of things, it doesn’t “play nicely” with existing concepts and as such ends up being a nuisance. Contrast this with complex numbers—the ones with a real and imaginary part—turns out that they do obey many preexisting properties: they can be added and multiplied very much like 2-vectors.

If you want to try it for yourself, write a convincing argument of why 0! = 1. Look for definitions and examples of the factorial process for help.³

1. Mathematicians, please understand that this is a simplification for educational purposes. I know I’m skipping over a lot and I’m leaving some properties of multiplication over the reals out for ease of use.

2. Real numbers in this case.

3. For instance, remember that n! = n × (n − 1)!