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Comment by somat | Hacker News Reader

The thing about division by zero is that it does not make sense only because you don't know the math to make sense of it.

Math has a long honorable tradition of things that did not make sense until somebody invented the math to describe them.

A inane short list of stuff invented just to deal with results that did not make sense from a person who is terrible at math.

Subtract 5 from 3, the result does not make sense, someone had to invent a whole new category of numbers to handle screw ball results like this.

The square roots of negative numbers. What sort of black magic is this.

divide something by nothing. get out of here you witch. The interesting thing is that the mechanics of computation(which surprisingly has little to do with actual math) has to do something with the operation. My favorite is this one. https://www.youtube.com/watch?v=7Kd3R_RlXgc (curiousmarc)

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There are some contexts in which division by zero is defined in a meaningful way. The most common such situation is the real projective line or its complex analogue, the Riemann sphere [0]. We’ll focus on the latter, since it has a richer structure. The Riemann sphere is essentially just the complex plane “closed up” (compactified) by adding a point at infinity. The metric is defined such that points of very large norm are all close to infinity, and hence to one another.

Any nonzero number divided by zero is infinity. Zero divided by zero is undefined. The operation of adding 1 has a parabolic fixed point at infinity.

The Riemann sphere is not a group, but it still has many very nice properties. For instance, the set of analytic maps from the Riemann sphere to itself is precisely the set of rational functions (together with the constant map infinity).

A much more profound statement is Thurston’s characterization of rational functions [1], which states that every orientation preserving branched covering map F from the sphere to itself is equivalent (topologically conjugate) to a rational map on the Riemann sphere, unless F satisfies a certain computable property.

In other words, maps on one of the most ubiquitous topological spaces, namely the sphere, are often best understood by introducing a structure where division by zero is allowed.

[0] https://en.m.wikipedia.org/wiki/Riemann_sphere

[1] https://pi.math.cornell.edu/~hubbard/ThurstonRatMaps.pdf

While I enjoy your overall tone and enthusiasm, there's no guarantee doing a weird operation will create a new useful mathematical object. The imaginary unit, i, follows associative/commutativity laws etc and is so enshrined. However, defining something like 1/0 as the "zero divided unit" it doesn't follow any of those laws to make it a group etc. So it can't be manipulated like a normal quantity in your algebra. Though it could be manipulated in other algebraic structures, such as wheels. However, if you haven't thought through this before, I think you totally have the right attitude, the right way you want to approach mathematics.

I don't really have the discipline to sit down and actually learn some useful math, however I enjoy the subject at a superficial level, so... that is, everything I say is wrong.

There was an interesting comment in the CouriusMarc video I linked. Quoted in full.

"Folks, no need to argue about colorful alternative mathematical theories - the thorny problem of division by zero was solved for good over 100 years ago by the rigorous development of infinitesimal calculus. Which says: division of a positive non-zero constant by something that tends to zero, tends to infinity [added note: dividing "zero by zero", or more exactly, two things that tend towards zero, is more complicated: it can give zero, infinity, or anything in-between, but that's for another time...]. So the calculator sort of gives the right answer, using almost the correct method: trying to fit an infinitesimally small number into a big one, and finding it fits so many times it goes to infinity. I would put it in the category of happy mechanical accidents."

Which fails to entirely satisfy, I suspect this is because it focuses on the mechanics of one view of the problem and leaves many other views still conflicting. The heart of the problem, that is, why the operation tends to be left as undefined, is that to be mathematically rigorous you have to satisfy all views as to what division actually means.

Yeah, the limit view is only defined in specific cases. For instance, if you have ((x+3)(x-5))/(x-5), this expression evaluates at x=5 to 0/0, but approaching it from either side, evaluates to 8. However, (x+3)/(x-5) at x=5 will be approaching -inf on one side and inf on the other.

I have an anecdote here too. When I learned about automatic differentiation, I very excitedly told one of my formalist math professors about the quantity e, e^2=0, e!=0, and how it could be used to compute the derivative of a function. He couldn't understand and essentially asked me to motivate the number and sketch the proofs that it's a well-behaved extension, which I couldn't do at all at the time, and he dismissed the whole thing which is actually the basis for some pretty important machine learning these days, among other things. So sometimes you can do pretty weird operations to get an extension, but without an intuitive grasp of how it works - either with a proof it is a well -behaved extension or by analogy for example, it's really hard to tell how useful such a concept is to a mathematician, how well it plays by the rules.

Being mathematically rigorous, you will have to define what the views you are talking about. In the one input function limit case, you need to check the limit from both sides. In the 2 input case, you need to check all possible approach paths to prove the limit exists. It gets out of hand. However, these are different views within a formal mathematical system. We don't need to reach consensus among different formal systems which happen to use the same symbols. For instance, I can define the set {cow,0} and equip it with operator '/' such that cow/0= and 0/cow=0 etc. This might be isomorphic to Z mod 2. Then cow by zero is defined, yet we can still agree that division by 0 is undefined, because when we are in general talking about division by 0, we're talking about an operation in a relatively specific context - how numbers behave. So, there really can be a very wide set of perspectives, and being rigorous must also specify the perspectives that count. Also - the cow case is pretty trivial, though this point could also be made with more complex structures, like wheels, that may also have useful physical interpretations, and they still wouldn't affect the collective agreement here because it's a contextually isolated case.

Thanks for the intro to “e” notation for differentiation!

Formally, it’s just e = dx, with implicit dx -> 0, semantically, but seems so much easier to think about as just e^2 = 0, in practice.

Never came across that.

I guess they use epsilon, not e. https://en.m.wikipedia.org/wiki/Automatic_differentiation

Do you have a source for e = dx perspective? I haven't seen it explained this way before.

Derivative of f = (f(x + dx)-f(x))/dx as dx -> 0

Which is to say:

1) create the function f(x + dx)

2) ignore (subtract away) the terms not proportional to dx

3) of the terms proportional, only care about terms minimally proportional (i.e. divide by dx, keep terms no longer proportional to dx, eliminate any still proportional to dx.)

The last step corresponds to eliminating terms proportional to dx^2 or higher

> The thing about division by zero is that it does not make sense only because you don't know the math to make sense of it.

No, that's inaccurate.

Division by zero cannot be defined in a consistent way in the domain that most people talk about it, which is arithmetic.

In other domains, sure, there are useful other definitions of division by zero (as you alluded to in another comment, calculus is one such domain).

But none of those other domains will impact arithmetic, which cannot have a consistent concept of division by zero.

You could just as easily have said: You can’t subtract large numbers from smaller numbers because that’s not “natural math”. Extending beyond that will not impact “natural math”.

I am pointing out that calling something “arithmetic” in a narrow sense, then declaring that extended concepts can’t be handled within your previous narrow definition is just a tautology of terminology choice.

Yes, a change is a change. And an extension is an extension.

Just use geometry to explain these things. Vectors. In a 2D plane. If that isn't intuitive enough for a person I don't know what is.

You think anything productive can come out of reinventing NaN and calling it "nullity"?

Even if the modern math notation with a minus sign is a fairly recent invention, subtracting 5 from 3 has made sense to human beings since the dawn of civilization, because that is how debt works.

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