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Comment by rsp1984 | Hacker News Reader

rsp1984

As long as lim(1/x)_x->0 = inf, 1/0 = 0 doesn't make a whole lot of sense, mathematically speaking. I might be wrong but I don't think it was addressed in the article either.

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There's a great Radiolab episode[0] that talks about divide by zero in perhaps more conceptual terms.

    KARIM ANI: If you take 10 and divide it by 10, you get one. 10 divided by five is two. 10 divided by half is 20. The smaller the number on the bottom, the number that you're dividing by, the larger the result. And so by that reasoning ...
    
    LULU: If you divide by zero, the smallest nothingness number we can conceive of, then your answer ...
    
    KARIM ANI: Would be infinity.
    
    LULU: Why isn't it infinity? Infinity feels like a great answer.
    
    KARIM ANI: Because infinity in mathematics isn't actually a number, it's a direction. It's a direction that we can move towards, but it isn't a destination that we can get to. And the reason is because if you allow for infinity then you get really weird results.  For instance, infinity plus zero is ...
    
    LATIF: Infinity.
    
    KARIM ANI: Infinity plus two is infinity. Infinity plus three is infinity. And what that would suggest is zero is equal to one, is equal to two, is equal to three, is equal to four ...
    
    STEVE STROGATZ: And that would break math as we know it.  Because then, as your friend says, all numbers would become the same number.

[0] https://radiolab.org/podcast/zeroworld

Then take 10 and divide it by -10 = -1. 10 / -5 = -2. 10 / -0.5 = -20. So from the other side of the y-axis it behaves the exact opposite. It goes to minus infinity. So at x=0 we would have infinity and minus infinity at the same time. Imho that is why it is undefined.

on computers you can have negative zeros

Negative zero is equal to zero, so it's not really a distinct number, just another representation of the same value.

It's equal (as in, comparing them with == is true), but they are not the same value. At least in IEEE 754 floats, which is what most languages with floating point numbers use. E.g., in JS:

  > 1 / 0
  Infinity
  > 1 / -0
  -Infinity
  > 0 === -0
  true
  > Object.is(0, -0)
  false

Yes.

In a language like C or Rust, you can cast your +0.0 and -0.0 to an integer, and print out the bit pattern. They are different.

I think you're misunderstanding me. They are the same value, but a different representation. The equivalence of the value can be shown with math, and has nothing to do with the implementation details of IEEE 754.

-0 is a notation for 1 / -infinity, so it is distinct. For addition it is not.

that's really just an encoding of the number to help you understand how the hell you got here

In IEEE 754 math, x/0 for x < 0 is in fact negative infinity.

  >>> np.float64(-1.)/0.
  -inf
  >>> np.float64(1.)/0.
  inf

And you're exactly right, 0/0 is NaN in 754 math exactly because it approaches negative infinity, zero (from 0/x), and positive infinity at the same time.

I always thought the answer to verbal query "let y=1/x, x=0, find y" was "Well, the answer is the Y axis of the plot". Surprising that people have to be reminded that X can be signed. I've had similar conversation IRL.

You seem to have your axes confused. The value of y can’t be “the y-axis”, which is the line x = 0 (for all values of y).

It's even worse than that. The other issue is what happens when you've got a negative number as the numerator (number on top). Then the smaller the denominator (number on bottom) the more negative the result. -10/10 = -1. -10/5 = -2. -10/2 = -20. So if you divide by zero, it's obviously negative infinity! And it's positive infinity! At the same time.

Ordinal and cardinal infinities are different. There are hierarchies of infinities.

`1/0` and `1/0 + 1` aren't meaningfully different, so it kinda does make sense for whatever notation to not make a distinction.

There's no real problem here. Infinity - Infinity is not 0; it's probably best to simply call it Infinity as well.

[ok] 1. Infinity + 1 == Infinity + 2

[ok] 2. Infinity + 1 - Infinity == Infinity + 2 - Infinity

[wrong] 3a. 1 == 2 (assumes Infinity - Infinity == 0, which is false)

[ok] 3b. Infinity == Infinity

So starting from Infinity + 1 == Infinity + 2 gets you nowhere interesting.

And that quote is a great example of what I hate about every pop-sci treatment of mathematics:

> Because infinity in mathematics isn't actually a number, it's a direction

Any time someone says "actually, in mathematics, ..." they're talking out of their ass. No matter what comes after, there is a different system of math that makes their statement false. There are plenty of branches of mathematics that are perfectly happy with infinity being a "number", not a "direction". What even is a "number" anyway?

The arguments around limits are addressed towards the end (under "Update 8/12/2018"):

> > If 0/0 = 0 then lim_(x -> 0) sin(x) / x = sin(0) / 0 = 0, but by L’Hospitals’ Rule lim_(x -> 0) sin(x) / x = lim_(x -> 0) cos(x) / 1 = 1. So we have 0 = 1.

> This was a really clever one. The issue is that the counterargument assumes that if the limit exists and f(0) is defined, then lim_(x -> 0) f(x) = f(0). This isn’t always true: take a continuous function and add a point discontinuity. The limit of sin(x) / x is not sin(0) / 0, because sin(x) / x is discontinuous at 0. For the unextended division it’s because sin(0) / 0 is undefined, while for our extended division it’s a point discontinuity. Funnily enough if we instead picked x/0 = 1 then sin(x) / x would be continuous everywhere.

Similar examples can be constructed for any regular function which is discontinuous (e.g. Heaviside step function).

It's fine. Infinity isn't a real number, so 1/x isn't continuous at 0, so it doesn't matter what the value of 1/0 is. All your open sets still behave the way you expect. Whether you choose "this function is undefined here" vs "it's impossible to ever reach the value of this function at this value, under any assumptions I'll ever care about" is purely a matter of convenience.

Yea, but conceptually it's a bit smelly to have f(x) get larger and larger the closer x is to 0 and then suddenly have it be 0 once x reaches 0.

But once you go behind, it flips suddenly anyway so you could just as well have it be intuitively “halfway between the positive and negative infinities” which is at least fun and could spawn a few “Why is 1/x suddenly go to zero” articles on HN in 2053

As others have pointed out "larger and larger" is the same when it is negative too. So I think people are just going: positive infinity + negative infinity = 0.

Intuitively nice in a sense but I honestly think '0' is misrepresenting what is going on here. I'm ok with it being ' "+ and/or -" infinity' as a new definition.

Programmatically I think it should result in a NULL or VOID or similar. I mean, by definition it has no definition.

Good point.

Well if you consider 1/z as a function of a complex coordinate it definitely makes a lot of sense to set it to infty. That identifies +infty and -infty if you restrict yourself to the real numbers.

I was also looking for this. And would like to add: lim(-1/x)_x -> 0 = -inf That is (in my opinion) the whole point why it is actually undefined. On one side of the y-axis it goes to infinity, on the other to minus infinity. I don't see a solution to this and therefore always have accepted that it is undefined.

No. 1/x^2 is undefined at 0 but has the same limit behavior, because limit behavior is not a function from "pairs of (functions from R to R, R)" to R

Infinity is not a real number.

Can limit behavior be defined as Cauchy sequences?

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