At some point last night I remembered when the whole “divide by zero” thing started for me, all the way back in grade school. We were being taught about division and it finally came up in class.

The teacher wrote some fraction, which for the first time showed “0” as the denominator. She turned to us and said “You can’t do this. If you have five cookies on a plate but give them to nobody, you could say you have divided them up for no one and think, in math, it’s like $\frac{5}{0}$ but in reality there are still five cookies on a plate. It’s not that you divided them by zero. You just did not divide them at all” and so she erased the $\frac{}{0}$ part and moved on.

But, while it was still up there, $\frac{5}{0}$ actually made sense to me — though her explanation buried it for me for a while. I started wondering about how similar that was to treating $\frac{5}{1}$ as just 5 without the $\frac{}{1}$. If no division could be equated with divided by one then something had to give. Logically, divide-by-zero and divide-by-one should not be equal, but the idea that either was implicit when neither was explicit never left me. I couldn’t come up with an answer because I didn’t really know how to ask the right question.

Whenever division-by-zero came up again, that feeling that the relationship expressed by the equation somehow made “sense” came back and compelled me to keep trying to figure out why it seemed like a sensible expression to me. I eventually realized that I had originally read it in my mind as “5 in no parts” and that is not the same thing as “5 in one part”. The latter, applied to “5 cookies on a plate” treated the quantity as a “whole” — a set which could be redistributed into equal portions.

But “5 in no parts” extends to “not even one”. The thing is, a division problem takes a given quantity and distributes it evenly between a number of recipients and the result of this process expresses the quantity in each portion. So, in any process of division where “no parts” is explicitly stated, the only possible result is zero because no part of the expressed quantity could be redistributed. All of it remains:

$\frac{n}{0} = (0_q, n_r)$ where the remainder $n_r$ is present in addition to the result $0_q$

So

$\frac{n}{0_d} = (0_q, n_r) \longrightarrow 0_q + n_r \longrightarrow n_r$ because $0_d \times 0_q + n_r = n$

This goes through the logic that first teacher bypassed by striking the $\frac{}{0}$ without addressing the $\frac{}{1}$ conflict. It is not the mathematical interpretation of “undefined” but it has its own clear and consistent logic, and resolves the paradox to my satisfaction.

If you understand it contextually, division-by-zero is both sensible and practical as a way of saying “this value exists, just not as a direct result of division”. Rather than being “undefined”, division, as a process, self-terminates at zero, where zero-division produces zero-result; in this absence of division, the dividend remains by default. Like multiplication by zero — division by zero is a null process”. For me, this resolves the question that “undefined” always raised concerning where the dividend went or what the dividend became.

Of course, that puzzle inspired me to explore a wide range of possibilities geared toward preserving something real, but inexpressible except as a displaced value or relationship. Other ways to think about it have been featured in other posts, where I used them to make sense of many other paradoxes, and most of the time, when I tapped into division-by-zero, I was not reflecting on the math interpretations at all. I was exploring the ideas that sprung from simply considering the proposition of something divided by nothing.

When blogging, the titles and topics often do not come into focus until after I’ve written down my immediate thoughts on whatever triggered the impulse to write. This means I’m thinking more about capturing a thought than making a specific point. What matters to me is that I can come back to it at some point and figure out what caught my interest, and then I’ll try to clarify where that thought was leading me.

In the past few weeks, I’ve been working to figure out what it is about divide-by-zero my mind is telling me to figure out. I know that there are a ton of rules that conflict with this interpretation of division, and I’ve got a lot of them stashed in my notes to go over again later.

Over the years, I’ve come up with many other interpretations, and most of the time I was not even considering challenging the math interpretations. I was exploring the ideas that sprung from simply considering the proposition of something divided by nothing.

For now, I’m finally able to say what leapt to mind in that almost forgotten lesson. If you have five cookies that are given to no one, it’s a bit like saying they just “magically disappear”. “No one” in this context is represented by “0”. This is exactly the conditions of a situation where you find an empty plate that five cookies are supposed to be on. Until you find out where they went, they remain in limbo. It’s a real situation, a known quantity in an indeterminate state.

An answer hiding in plain sight? Maybe. Remembering where the interest started, I wrote it down so I (hopefully) wouldn’t forget — again. I’ll keep poking at it because I am just not the kind of person who can accept a definition at face value. If you tell me something is so, I just can’t take it for granted. I need to see it for myself. It will never be enough to cite the rules to me because I learned the hard way that rules are often backed up only by “because I said so”.

Maybe my intuition was just trying to get me to pay attention to that first reading of “5 in no parts”. Maybe because, before even trying to divide it, once it was written in an equation, “5” was declared in my head and the equation required me to do something with it. It was real, even if it was simply just the “idea” of 5.If you tried to divide it, and got as far as “5 in no parts” the next logical thought is “not even one” and that implied something different from simply doing no division. Five cookies, undivided, remain five cookies. But, “divided into no parts” calls for a result, which can only be zero — zero parts of zero — then you get the remainder.

It’s not unlike doing something zero times. In multiplication, you end up with zero, not because you have annihilated its value; you get it because you don’t multiply in that instance. In division, you get what you start with because you don’t divide. Trying is just going through the motions; and getting lost along the way?

That’s just… sad.