If you know anything about math, you’ll almost certainly have heard that division by zero is undefined.

Now, to the math savant, this probably makes perfect sense. To the math simpleton (*raises hand*), this is a record scratch in the brain.

Math is a(n annoyingly abstract but somehow ridiculously accurate) tool for describing real world phenomena. (Well, at least sometimes. Some types of math describe non–real world phenomena, but never mind that right now. Stay with me here! XD)

If you have a guava† and your friend has a guava, put them together, et voilà! You’ve just “mathed” 1 + 1, and now you have this lovely thing known as 2 guavas! (What a concept! XD)

Then when you and your friend eat your guavas, and nothing’s left, math has a word for that, too. It’s called 0 (zero). (Another mindblowing concept‡, I know. XD)

And, okay, if zero is defined, why is dividing by it not? Why is dividing by zero literally defined — by which I mean, the answer that you get in math textbooks everywhere when you look up this whole “division by zero” term in the glossary or index — as “undefined”?

Whenever I pressed my math teachers on this, they invariably told me that division by zero just wasn’t allowed (and to stop bothering them [with questions they couldn’t answer] -.-). So, for a long time, I thought “undefined” in math just meant “verboten”. That is, the Math Police declared it illegal to divide by zero for Ineffable Reasons™.

But, like a pedestrian who feels compelled to question the cop ticketing them for the oh-so-heinous crime of standing around doing nothing “loitering” in a public space, I feel compelled to ask, “Then why ‘undefined’ ?” I mean, wouldn’t it be more accurate to just declare:

division by zero = 🚫

At least then it would be clearly marked as a no-go zone!

But, no, that’s not what we get. Math textbooks definitively say division by zero is “undefined”.

. . . Which, to my mind, is definitively clear as mud. -.-

After all, what does saying that something (anything, not just division by zero) is “undefined” actually mean? What does that represent in the real world? How can something just be undefined in the real world??

And, of course, asking the question feels foolish (or, at least, it makes me feel foolish — far be it from me to speak for math simpletons everywhere! XD) because the word itself suggests that trying to get an answer to this question is intrinsically a futile endeavour. After all, you can’t define something that’s undefined — that is, not defined — by definition! Right?

But, but, but — !

To me, math simpleton ordinaire that I am, that feels like a copout answer. That’s a don’t-bother-with-your-questions “answer”. (I can hear my old high school math teachers glaring at me from here. XP) And I refuse to accept this lying down. (“But, officer, I’m not doing anything — I’m just standing here!”) Because how shall I ever become a math savant extraordinaire if I can’t even get a straight answer to one of the most fundamental underpinnings in all of math??

Well, fast forward from high school, and now, I hope, I can at least give you, dear readers, a (slightly ‘^^) better answer than my math teachers gave me. You see, while I confess that my math chops have not gotten much better in the interim, I daresay my chops in other areas have developed quite a bit more — like in learning foreign languages.

And doesn’t math look like a lot like a foreign language?

Aside from all the foreign-alphabet-esque squiggly and angly bits that show up in equations and all that, math (at least some kinds, anyway), remember, is a tool for describing the real world — and one of the best things we have for describing things is — spoiler alert! — language.

In other words, it’s helpful (to me, anyway) to think of math, not as a real thing in and of itself (though it might be: philosophers of math have been debating math’s “realness” for a long time, and I hear the jury’s still out XD), but as a language that helps us better understand real things.

You, your friend, and guavas — all real things. The addition of said guavas — also a real thing. The subtraction of said guavas into nothingness via the highly technical process known as nomming — real again.

Language helps us talk about these kinds of things. These things exist in the real world, and their symbols and signifiers exist in the language we use to describe these real world things.

But just as there are some things in the real world that don’t exist because the real world follows these pesky things known as the laws of physics, there are some things that don’t exist in language, either — because it, too, follows certain rules. (Enter the Language Police. XD)

Some things don’t exist in language for the simple reason that they don’t make any sense. Noam Chomsky, a famous linguist, came up with the now well-known sentence, “Colorless green ideas sleep furiously” (1957, 15), as an example of how a perfectly grammatical sentence can be complete semantic gibberish. Ideas cannot be green, which in itself cannot be colourness; nor can ideas sleep, much less furiously. The entire sentence is nonsense; it has no sense. It’s meaningless.

In the same way, I think that that’s what “division by zero is undefined” means in this wonky language about the world known as math. It means it’s meaninglesss. That is, division by zero is just a nonsensical concept in math. And to try to make it have meaning, to try to forcibly define it, would be to render the entire project of math completely useless. To make dividing by zero mean something would break math, just as making “colorless green ideas sleep furiously” mean something would break English.

And division by zero is not just “verboten”, like I thought — though it is (because, again, the whole breaking-all-of-math thing) —, in the same way that “colorless green ideas sleep furiously” is not verboten. Chomsky’s sentence is perfectly legal in the grammatical sense. (Language officers, please stand down!) There’s nothing stopping you from trying to parse it, just as there’s nothing stopping you from trying to divide something by zero. It’s just that, when you do, everything falls apart.

Some things, then, are — and must be — left undefined because to define them would be to undefine everything else.

†Why guavas and not apples, considering the long association between apples and teachers and, you know, generic items most people are familiar with that work well as examples of easily enumerable items and all that? Simple: I don’t like apples, and I do like guavas. =P

‡As it turns out, it actually did take rather a long time for humans to figure out that zero is a thing — because, after all, it’s not terribly intuitive to think of needing something to represent nothing, right? — and then even longer to accept it. If you’re at all interested in the history of zero, let me recommend Charles Seife’s Zero: The Biography of a Dangerous Idea.

References

Chomsky, Noam. Syntactic Structures. Berlin: Mouton, The Hague, 1957.