You can add a non-zero number which squares to zero to your number system [edit: but this doesn't let you divide by zero]. This results in the "dual numbers" and is practically useful for "automatic differentiation", where we represent quantities in the form x + x'ε, and have the rule that ε² = 0.

https://en.wikipedia.org/wiki/Dual_number

https://en.wikipedia.org/wiki/Automatic_differentiation#Auto...

Quantities which square to zero are also implicit in spacetime. A "lightlike" vector which has equal displacements in space and time between two spacetime “events” (e.g. the displacement between two points along the path of a photon) has a squared length of 0, compared to "timelike" vectors with negative squared length and "spacelike" vectors with positive squared length. (conventions about signs vary from source to source)

https://en.wikipedia.org/wiki/Spacetime#Spacetime_interval

* * *

In other contexts it makes sense to define 1/0 to be the quantity ∞. There are two relevant models here, with different practical applications. One model where we add a single number ∞ which is also equal to –∞, and makes the number line "wrap around" into a circle. Another model where we add two separate numbers +∞ and –∞ at the two ends of the number line.

https://en.wikipedia.org/wiki/Projectively_extended_real_lin...

https://en.wikipedia.org/wiki/Extended_real_number_line

Adding some kind of infinitesimals to the reals can be useful, as is done with hyperreal numbers, dual numbers, and other kinds of hypercomplex numbers, but it doesn't allow you to divide by zero; there is no hyperreal or hypercomplex number x that solves x = 1/0.

As I understand it, if you add 1/0 = ∞ you can't treat that ∞ as a quantity in the usual algebraic ways. In particular you can't multiply ∞ by 0 and get back some well-defined quantity such as 1; if you allow that, you quickly find that you can prove that all finite numbers are equal. The standard tricky example of this is in https://www.math.toronto.edu/mathnet/falseProofs/first1eq2.h...:

Let a = b.

Then a² = ab, a² + a² = a² + ab, 2a² = a² + ab, 2a² - 2ab = a² + ab - 2ab = a² - ab = 2(a² - ab) = 1(a² - ab).

If we then divide both of these last expressions by a² - ab we get 2 = 1. This is only invalid because a² - ab = 0. Adding a different ∞ₙ for each value of n/0, as lisper suggests, doesn't help.

So, if you want to extend your number system with 1/0 = ∞, you either need to throw out some of the standard laws that permit algebraic manipulations like that, or you end up with all finite quantities being equal.

I am not at all a mathematician, but I just can’t wrap my head around how 1/0 isn’t infinity. 1/1=1, 1/0.5=2, 1/0.25=4, etc. The closer we get to 0, the larger the number. I just can’t accept that if we actually get to 0 in our progression, it is anything besides infinity. Perhaps it becomes its own class of number, like what happened with imaginary numbers. That is, infinity times zero is 1n (where n is a bookkeeping marker like i in the square root of negative numbers), or like with limits, where you have a “+ C” in there.

> closer we get to 0, the larger the number

From above. What if you approached 0 from below instead?

1/-1=-1, 1/-(1/2)=-2, 1/-(1/4)=-4.

So you just proved that 1/0 equals both positive and negative infinity, and therefore you also just proved that positive and negative infinity are equal. This is one of the reasons why giving 1/0 a definition is rarely seen as useful.

Well, that's not so bad; they are equal on the real projective line: https://en.wikipedia.org/wiki/Real_projective_line. Making 1 = 2 is a bigger problem.

Haha, ok this blows my mind a little more now. This definitely helps me conceptualize the problem a bit better, even if I’m still not satisfied with it. I still feel like there is a better solution, like giving 0 a sign or something.

That is in fact what we do in IEEE 754 floating point! We have +0 and -0; 1/+0 = +∞ and 1/-0 = -∞. However, this means that IEEE 754 floating point violates substitutability, the most basic property of equality, because +0 == -0 but in some cases f(+0) ≠ f(-0).

You may enjoy Tom 7's paper/movie about this: http://tom7.org/nand/

Okay, but then you'd show that 1/-1=-1, 1/-0.5=-2, 1/-0.25=-4, etc. The closer you get to 0, the more negative the number. Then you just wouldn't accept that if you actually got to 0 in your progression, it is anything besides negative infinity. And then you've shown that negative infinity is equal to infinity.

Sure, I definitely agree that 1/0 doesn't give you a finite quantity. What I'm saying is that it doesn't give you a number at all unless you're willing to sacrifice major parts of algebra. And I'm not much of a mathematician either, so I might have gotten something wrong.

The problem with any finite number of infinities is that you can't maintain the identity (a/b)*b = a. To make that work you need a different number to represent a/0 for every value of a, including all of your infinities, so you end up with an infinite hierarchy of infinities.

You already can’t maintain the identity (a/b)b = a when you have a zero element, except by declaring "division by zero is illegal". Adding ∞ is not fundamentally different, but just takes a slightly modified list of exceptions.

In many contexts this is worth the hassle.

Well, yeah, but the whole point of inventing a new kind of number to represent a/0 analogous to inventing a new kind of number to represent sqrt(-1) is to eliminate these you-are-not-allowed-to-do-that kind of exception. With sqrt(-1) it turns out to be straightforward. With a/0 it isn't, which is why sqrt(-1) is a thing in math and a/0 isn't.

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.

No, but there are algebraic structures that allow for this, like the Riemann sphere. The proper way to talk about this is the concept of "zero dividers". For Z the only zero divider is zero, 0/0=0.

The closest thing to what you describe is the the dual number (that together with the imaginary and hyperbolic numbers make the geometric numbers), which has zero dividers, and is defined as k^2=0 (where k is not in R).

This is a very interesting number and it can help clean up a lot of problems when using complex numbers.

No, or at least not in a field. From the field axioms you can directly proof that 0x=(a-a)x = ax - ax =0, and therefore 0 doesn't have a well defined inverse. You can look at other algebraic structure, but those behave less like numbers.

Yeah. Great mind teaser. I can imagine an imaginary number (-/+) "inf" for divisions by number approaching zero from left/right, yet the algebra would not be possible to define properly because you can approach zero with a different rate.

The undefined cases, where the left/right limits are not equal coulf get imaginary number "shrug" because it would be even less useful.

Of is anyone able to define a useful algebra for these? I'm really curious.

yes, check out Yaglom's "Complex Numbers in Geometry" for a fairly complete treatment of this topic. You can extend the dual numbers and double numbers with their own special infinities, similarly to how the complex numbers can be extended to include a single point at infinity.