For me, the whole point of modifying the number system is modeling different kinds of structures. Both complex numbers and projective numbers are interesting in their own right and also useful for making models of both physical reality and other kinds of mathematical patterns. Each system has its own internal logic with its own quirks.

>to eliminate these you-are-not-allowed-to-do-that kind of exception.

I'm not a mathematician by any stretch, but I never understood the inability to divide by zero to be some arbitrary rule, rather that it's just simply not possible to proof (because you can't multiply any number by 0 to get a non-zero answer). Then again I tapped out after Calculus so maybe that's just ignorance on my part.

How can something be divided by nothing? If that would give the same result as division by a whole, does that mean that nothing is something?

Issues like these are why the concept of zero had trouble gaining acceptance.

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.

Say you have one apple and want to divide it over zero people.

It can't be done - give everyone one, fifty, a million, infinity apples: you always have one apple left over.

The concept of "division" or "ratio" has different interpretations depending on context.

The Euclidean (from The Elements) concept of ratio can certainly admit a 1:0 ratio, which we can write as 1/0 or ∞ if we like, and which is equivalent to the ratio of x:0 for any non-zero number x. There’s no inherent reason why the ratio of "apples per person" should be allowed to be 0:1 but not 1:0. From first principles it’s just as problematic to divide 0 apples among x people as to divide x apples among 0 people.

When we e.g. write the values of trigonometric functions, these should ordinarily be interpreted to be among the extended reals, so that tan(π/2) = 1/0 = ∞ (i.e. the ratio 1:0) is entirely reasonable.

Is that +∞ or -∞ though? The problem with division by zero is not that it's infinite, it is that it limit could be either +∞ or -∞ depending on which side of the limit you look at. Therefor, it is not well-defined.

In some contexts +∞ = –∞ (e.g. tan ½π = tan –½π; these both represent the orientation of a line at a right angle).

In other contexts +∞ ≠ –∞ (e.g. log ±0 ≠ log ±∞; these quantities can be thought of as the two opposite ends of an infinite cylinder)

Well, in that case the parent must ask themselves: am I doing geometry on an infinite cylinder or not?

For example, when you look at a world map under the Mercator projection and you want to know where the north and south poles should be on the map, the answer is +∞ and –∞, which are not the same.