We can divide by negatives though? They are just as unreal as zeros, so the only difference seems to be that division by zero is undefined, while division by negatives is well-defined. I think the real difference is that dividing by zero is not useful, so it has never needed defining.

> I think the real difference is that dividing by zero is not useful, so it has never needed defining.

Incorrect. The result of dividing by zero is provably "not well defined" - for standard mathematical definition of what is and what isn't "well defined".

Specifically, Limits (1). for 1/x, as the x approaches zero from above and below, the two limits are not the same, therefor the limit is not well-defined (2)

As x tends to zero from x > 0, 1/x tends to a limit of +infinity. Calculate 1/x for values of x = 2, 1, 0.5, 0.1, 00.1 and so on as you approach zero: the results are larger and larger positive numbers, without bound.

As x tends to zero from x < 0, 1/x tends to a limit of -infinity. Calculate 1/x for values of x = -2, -1, -0.5, -0.1, -00.1 and so on as you approach zero: the results are larger and larger negative numbers, without bound.

Even allowing "infinity" as a valid number, this isn't converging to the same answer. (2)

I suppose that you could "define" the answer in much the same way that we define i as "the imaginary square root of -1" and thus get complex numbers.

The question that would follow though, is: If we e.g. define "q" as the limit of 1/x as x -> 0, this is a number that is both positively and negatively infinite at the same time, what then? What can we do with it? It turns out that complex numbers are useful (3) and look a bit like quarter-circle rotations. But this q, the "self-contradicting infinite number" doesn't really seem to be useful.

1) https://byjus.com/maths/limits/

2) https://www.mathway.com/popular-problems/Algebra/200474

3) https://issuu.com/harrowhongkong/docs/final_scientific_harro...

"q the self-contradicting infinite number" is so useful that it sits at the foundation of the whole subject of projective geometry.

Which we can use for such lovely practical applications as rendering the CGI baddies we want to blast in 3D shooter games.

This "number" is also of fundamental importance to complex analysis. See https://en.wikipedia.org/wiki/Riemann_sphere https://en.wikipedia.org/wiki/Zeros_and_poles https://en.wikipedia.org/wiki/Meromorphic_function

Explain?

Yet another not a mathematician here. Division by zero never converges (as a limit of any generating process, numerical or symbolic) in contrast to division by anything else. There is no amount of zeroes that can add up to a whole number, by definition of zero. Not an infinite amount, just “no” amount.

If you want to handle infinities as numbers, you have to maintain a “progress” somehow, or you end up comparing different stages of processes and get something like 2=1. Much easier to not dig a pit to fall into.

My comment may not have been clear.

I'm saying I understand the idea that we can't divide by zero to make perfect sense. There is nothing arbitrary about it, which the parent comment seems to say. We don't just have a special rule saying "You can't divide by zero because we say so". It's a mathematical proof.

My mistake! I see what you mean now.