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1/0 = 0 (2018)

hillelwayne.com

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Re: 1/0 = 0 (2018)

#71

Post by moomin »

lupire wrote: post #14
moomin wrote: post #5

Honestly this hurts my head but Hillel is inevitably correct. You can define an explicitly undefined operation to do whatever you like. But what’s the point? There’s no new mathematics you can do with it, no existing behaviours you can extend like this. Normally, when you divide by a small number, you get a large number. Now for some reason it goes through zero. Why not five? Why not seven? Just because it’s formally…

Because exceptions are expensive, and functions with holes are dumb. "Dumb" is purely a matter of aesthetic preference. Calling things "dumb" is dumb. > Normally, when you divide by a small number, you get a large number. Now for some reason it goes through zero. Zero is not a "small" number. Zero is the zero number. There is no number that is better result than 0 when dividing by 0; "Infinity" is not a real (or comp…

Consider that lim -> inf does not mean “It goes to infinity”. Its actual definition has nothing to do with infinity. So your argument about infinity is a red herring.

Or try it the other way, tell me what mathematics works better if 1/x=0 than 1/x=5. If there’s an aesthetic preference displayed here, it’s for mathematics as a tool for reasoning.

Re: 1/0 = 0 (2018)

#72

Post by beala »

mjcohen wrote: post #68
beala wrote: post #62

Earlier quoted context omitted.

Under what definition of division is (a/b)*b=a true for all values?

If 0 is not an allowable value for b is necessary but not generally sufficient.

Can you say more? If "0 is not an allowable value for b", then it seems to me that (a/b)*b=a isn't true for all values. Specifically, it's false when b=0.

IIUC, codeflo is arguing that the division operation defined in the article isn't "actual division" because (a/b)*b=a isn't true for all values. But I can't think of a definition of division that satisfies that criteria.

Re: 1/0 = 0 (2018)

#73

Post by dang »

One megathread and a couple small ones. Others?

1 / 0 = 0 (2018) - https://news.ycombinator.com/item?id=42167875 - Nov 2024 (8 comments)

What is the best answer to divide by 0 - https://news.ycombinator.com/item?id=40210775 - April 2024 (3 comments)

1/0 = 0 - https://news.ycombinator.com/item?id=17736046 - Aug 2018 (570 comments)

Re: 1/0 = 0 (2018)

#74

Post by dccsillag »

I find it odd that all of the mathematicians cited at the end are actually pretty much CS people, working on proof assistants. Kinda renders that section pointless, IMO (though the comment by Isabelle's author was interesting).

IMO, whether something like this makes sense is a separate matter. Personally I always just think of division in terms of multiplicative inverses, so I don't see how defining division by zero helps other than perhaps making implementation easier in a proof assistant. But I've seen people say that there are some cases where having a/0 = 0 works out nicely. I'm curious to know what these cases are, though.

Re: 1/0 = 0 (2018)

#75

Post by sidpatil »

ilius2 wrote: post #50
ilius2 wrote: post #48

Earlier quoted context omitted.

That's nonsense. a/b is float in Python 3, and even in other languages a/b gets closer to it's actual value as a and b get bigger (the "limit", which is the basis of Algebra). So four operations in programming generally do agree with foundations of Algebra. But a/0=0 is %100 against Algebra. And it's very unintuitive. It's basically saying zero is the same as infinity, and therefore all numbers are the same, so why b…

If you were to define a/0 the most logical choice would be a new special value "Infinity". The second best choice would be the maximum supported value of the type of a (int, int64 etc). Anything else would be stupid.

What if a is negative?

Re: 1/0 = 0 (2018)

#76

Post by clipsy »

tux3 wrote: post #26
codeflo wrote: post #12

This article invents a new binary operation, calls it "division" and uses the "/" operator to denote it. But the article repeats multiple times that this new operation isn't a multiplicative inverse, so it's not actually division. For example, (a/b)*b=a isn't true for this new operation.

Reusing symbols like +, *, or / to define operations that aren't the + or the / you're used to is pretty common in math. It's just notation. At the end of the day, the / that we have in programming has the same problem as this article's /, almost all programming languages will return 5/2 = 2 when dividing integers, even though 2 * 2 is not 5! Division is not defined for all integers, but it's just convenient to exten…

> Reusing symbols like +, *, or / to define operations that aren't the + or the / you're used to is pretty common in math. It's just notation.

Reusing symbols in a different context is pretty common; taking a symbol that is already broadly used in a specific way (in this case, that `a/b` is defined for elements in a field as multiplying `a` by the multiplicative inverse of `b`) is poor form and, frankly, a disingenuous argument.

Re: 1/0 = 0 (2018)

#77

Post by itishappy »

d_tr wrote: post #30

Whatever as long as the name does not imply that these are integers, because then it is just wrong. The same holds for overflowing results being clamped or resulting in smaller or negative values due to wraparound. These are not integers. There is only one correct behavior for something named "int". Give the correct result or throw an error.

Agree `int` is the problem. This implies we're doing math over all integers, when in most languages what we're actually working with are bounded integers. (There's some counter-examples, Python and Haskell come to mind.) Calling them sane names like `i32` and `i64` makes it clear that overflow exists.

Re: 1/0 = 0 (2018)

#78

Post by MathMonkeyMan »

Not that anybody asked me, but I think about it like this:

You have a field (a set of "numbers"). Multiplication is defined over the field. You want to invent a notion of division. Let's introduce the notation "a/b" to refer to some member of a field such that "a/b" * b = a.

As Hillel points out, you can identify "a/b" with a*inverse(b), where "inverse" is the multiplicative inverse. And yes, there is no inverse(0). But really let's just stick with the previous definition: "a/b" * b = a.

Now consider "a/0". If "a/0" is in the field, then "a/0" * 0 = a. Let's consider the case where a != 0. Then we have "a/0" * 0 != 0. But this cannot be true if "a/0" is in the field, because for every x we have x * 0 = 0. Thus "a/0" is not in the field.

Consider "a/0" with a=0. Then "a/0" * 0 = 0. Any member of the field satisfies this equation, because for every x we have x * 0 = 0. So, "a/0" could be any member of the field. Our definition of division does not determine "0/0".

Whether you can assign "1/0" to a member of the field (such as 0) depends on how you define division.

Re: 1/0 = 0 (2018)

#79

Post by sgentle »

TrianguloY wrote: post #33

I've always wondered what would happen if we defined /0 as a new symbol, for example 'z'. The same as we define sqrt(-1) as 'i'. So if you can do 4*sqrt(-1)=4i, you could also do 4/0 = 4z. These two seems similar, as in taking something that should not exist, and just letting it exists in a totally different and orthogonal domain. I tried once to investigate the implications, but it quickly became far more complex th…

You might be interested in the hyperreal numbers, which sound a bit like the avenue you were exploring.

Re: 1/0 = 0 (2018)

#80

Post by epidemian »

recursive wrote: post #25
WithinReason wrote: post #16

Earlier quoted context omitted.

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
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