It is undefined because the inverse of division is multiplication. If you multiply by zero, every answer is zero. If you try to invert that operation you can’t know which number was multiplied by zero to get zero because multiplying by zero doesn’t produce a unique answer for each operation.

Additionally, if you take the limit of 1/x as x approaches zero from the positive side the result approaches positive infinity. If you take the limit from the negative side it approaches negative infinity.

An interesting thing to think about is whether multiplication by zero really makes much sense in the concrete world. You can’t really have zero groups of something or some number of groups of zero. Zero groups of anything is still nothing. We can think of that abstractly once we have the abstract concept of numbers, but in the real world that idea is nonsense.

This clicked for me when my teacher explained it in terms of slopes.

The video here breaks it down nicely. https://virtualnerd.com/sat-math/geometry/slope/infinite-slope-definition.

So you have a vertical line with an infinite slope. There are no changes in x as you traverse the line, only changes in y. Or said differently, the line is described entirely by a single x-value which corresponds to every possible y-value.

If you think of it in terms of a function, it’s extremely problematic because you no longer have a mapping of a single y-value to each x-value. This violates the requirements of a function. It’s not possible to define the slope value when rise/run is something/zero, therefore we describe the function value as “undefined”.

But even though we can’t calculate a slope or address it with a function, it’s pretty easy to visualize and understand a vertical line. So that’s what dividing by zero represents in concrete terms.

Okay I’m not a mathologist - in fact I hate the guy - so here’s how I see it. If you have one apple, and you divide it zero times, you still have one apple. That’s it.

Dividing by zero is undefined in the realm of real numbers and basic arithmetic because there’s no meaningful answer. It’s not an abstract concept; it’s simply mathematically impossible due to the nature of division.

I guess if I were to spill over to the non-numerical and semantical or more linguistically axiomstic representation here,:

1. The divisor is a number (1)
2. The dividend is a number (0)
3. A number divided by another number consequently produces another number
4. Does mathematics line up with CS on the whole NaN designation? I might be conflating fields or… I dunno. I’m sure there’s a palpable issue here you guys can lead this ass to override whatevers going on in my head