Why Can't You Divide by Zero?

A teacher marked one of my solutions wrong. The working held up until a step where I had divided by zero, and that step invalidated everything after it. I was furious, which in hindsight is the interesting part. I did not want the explanation. I wanted the rule to be wrong.

So I went home and asked why it was there.

The idea I had

My guess was that division by zero might mean something in a place we cannot picture. Torque was my analogy. A two-dimensional creature living in a plane could feel things rotating but could never picture the axis they rotate around, because that axis points out of its world entirely. I thought dividing by zero might sit in the same relationship to us.

I still like the instinct, even though it did not survive contact with the actual mathematics.

How I went about it

Badly. I would rather say that plainly than dress it up.

I had a theory that new ideas come from absurd places nobody thinks to look, so I pushed an AI at the problem from every direction I could invent. Quantum physics. Mechanical engineering. At one point classical music. I was not really doing mathematics. I was hoping that if I forced enough unrelated fields together, something would fall out of the collision. I called the thing I was building Fluxion. I can no longer tell you what it did.

Two weeks of this produced nothing that held. Some of what came back looked genuinely interesting to me at the time, and I thought one or two pieces might have been worth writing up properly. I have since lost them, which tells you how carefully I was keeping records.

I did not do most of the work here, and I am not going to present it as though I did.

Talking to a professor

I took it to Dr. Balbir, a professor of mathematics. We met three times, and each time he sent me away with something new to look at.

That is where it ended, though not because he told me to stop. He kept opening doors, and behind every one of them was an established field with decades of work already in it. Division by zero is not an oversight somebody left lying around. In a field it is ruled out by the axioms themselves, and the structures that do make it work, like the Riemann sphere or wheel algebra, pay for it by giving up properties you wanted to keep. People have been careful about this for a very long time.

What it was actually worth

Not much as mathematics. I would put it near the bottom of the things on this site.

What it gave me was a sense of scale. Before this I assumed there were open doors everywhere and the trick was being bold enough to walk through one. Afterwards I understood how much has already been done, how thoroughly it has been done, and that being annoyed at a teacher is not a research method.

It was fun. It is here because it happened, not because it is impressive.