“Now, let’s get started.” The professor drew two large ovals on the blackboard. “What’s the first thing that any mathematician learns about division by zero?”

“That it can’t be done,” one student replied.

“That’s right,” she smiled. “And why is that only half true?”

To that, there was no response.

“It’s true that the result of such an operation cannot be written or understood,” she continued her etchings, drawing a pair of dots above the oval on the right. “The notion of carving something into zero parts seems at first to be meaningless, as our species has never seen such a thing happen, but this doesn’t mean that it doesn’t happen at all. Numbers are a representation of the relationship between space and substance, but they are not a perfect mirror of the relationship itself. No matter how precise, they cannot describe what humanity does not witness, for they cannot extend beyond the language from which they are formed. Now, who here can tell me the difference between these two particular zeroes?”

Aside from the dots, they appeared to be the same. After a moment of silence, a voice came from the back: “They’re pronounced differently?”

The professor chuckled. “Well, the mathematician who made these observations to begin with was Swiss, but no, that’s not quite correct. Anyone else?”

“That’s a hot zero and a cold zero.”

“Ah, good, someone did their reading. I was starting to get worried.”  She wrote the temperatures under each oval. “Hot zeroes are zeroes that have previously been divided by, whereas cold zeroes have not yet been operated upon. Virtually any zero that you’ve ever written as an undergrad has been a cold zero: newly written, and unblemished by past arithmetic. Hot zeroes are special: they’re the used cars of mathematics. They provide a loophole by which humanity can engage with the results of a problem that it cannot solve.”

One of the students raised their hands. “If that’s what a hot zero is, then how can that be one? You just wrote it on the board. I didn’t see you divide anything by it.”

The professor nodded in acknowledgement. “Because I cannot mathematically express the solution of that division. There’s no telling what the result of a number divided by zero actually is- but we can see what’s left of the zero once it’s been done. Hot zeroes are not solutions to the problem; rather, they are its byproducts. This is a zero that someone else divided by at some point in the past. It might have been a computer, or it might have been a student like you trying to evaluate a poorly designed problem. It might have even been Euler himself. There’s no way to know for certain where it came from. Usually, when you write down a number, it is a new, generic instance of that particular number. The hot zero is an old zero, however; it contains its own history within its boundaries.”

“Why do they call them hot zeroes?”

“That brings us to the fun part of this lecture. Every hot zero contains some residue of the number that it was divided with. Given that, let’s see what happens when we divide a number by a hot zero, shall we?”

She wrote the next three glyphs slowly, and with extreme vertical precision: a serifed one, a dividing line, and a zero crowned by two dots. Then, as soon as she flanked these with an equals sign, the chalk

Mathematics doesn't always intersect reality in comprehensible ways.

Strange things can happen when the right sort of chalk is used.

Division by zero can occur naturally while rehydrating powdered water.