1/0 = 0 • Hacker News Board

Re: 1/0 = 0

#411

Post by Too » Sat, Aug 11, 2018, 5:14 AM UTC

huehehue wrote: post #236

simias wrote: post #183

Earlier quoted context omitted.

I'm reading the "Pony" tweet quoted in TFA and your comment and I'm left very puzzled: is that really that common to want x / 0 == 0 ? In practice where does that crop up? You say that you frequently have to write your little shim but honestly I don't remember writing code like that in recent memory. You talk about progress bars, I suppose it makes sense if you somehow try to copy 0 elements for instance, and you end…

Practically, it's quite common to not immediately know the divisor. In cases where the divisor is initially unknown but takes an imperceptible amount of time to compute it's better to render 0%. Otherwise you might get a flash of a full progress bar (for example) while the divisor is determined. Of course, it's context dependent. As others mention, your code might be full of stuff like X / (divisor || 1) .

If you dont know the number of elements, shouldnt that be tracked in another variable, instead of reusing the total elements variable and assuming 0 elements means unknown?

Also, if loading total elements takes a significant amount of time, shouldnt this loading also be reflected in the progess bar?

Re: 1/0 = 0

#412

Post by gowld » Sat, Aug 11, 2018, 5:18 AM UTC

elwell wrote: post #233

whack wrote: post #72

My problem with "1/0 = 0" is that it's essentially masking what's almost always a bug in your program. If you have a program that's performing divide-by-zeroes, that's almost surely something you did not intend for. It's a corner case that you failed to anticipate and plan for. And because you didn't plan for it, whatever result you get for 1/0 is almost surely a result that you wouldn't want to have returned to the…

What the world really needs: "1/0 = 0" in production build, "1/0 throws error" in development build...

That makes your bugs invisible in production. You could be failing every single transaction in the real world, and never know if your tests don't tickle the unhappy case.

Re: 1/0 = 0

#413

Post by gowld » Sat, Aug 11, 2018, 5:21 AM UTC

mehrdadn wrote: post #283

eggnet wrote: post #243

Earlier quoted context omitted.

You are correct. But it really should be NaN, since 1/ε is positive infinity, whereas 1/-ε is negative infinity. Oh well :)

Nope, I'd disagree. Zero isn't an approximation of some epsilon, it's really just zero . It makes sense for the output sign to match the input sign.

What is the sign of 0?

It's 0.

Re: 1/0 = 0

#414

Post by gowld » Sat, Aug 11, 2018, 5:22 AM UTC

nanimo wrote: post #398

spooneybarger wrote: post #171

Earlier quoted context omitted.

Hillel is talking about math. Unfortunately computers don't really do "math". For example, math doesn't have overflow and underflow to deal with. The floating point standard says that division by 0.0 should be Infinity and provides a value for it. The integer math, all possible values are used for numbers, so division by zero is undefined behavior. And from there, every language is potentially going to have a mess of…

Underflow and overflow and 2's complement are all just modular arithmetic, which is definitely math.

Computers is the math you get, not the math you want.

Re: 1/0 = 0

#415

Post by gowld » Sat, Aug 11, 2018, 5:25 AM UTC

godelski wrote: post #382

gjulianm wrote: post #17

Earlier quoted context omitted.

I think OP means that nothing breaks mathematically. It is not inconsistent and not false, so you can work with it. The only issue is to deal specially with the case of division by zero, which you have to do anyways. Code that assumes that (x/y) * y = x is wrong if you don't check for y = 0, independently of what you define x/0 to be.

You do realize that division is the inverse operation of multiplication, right? Like subtraction is the inverse of addition. By defining addition we define subtraction. By defining multiplication we define division. This is where the author fails. Division is multiplication of a fractional value. This is VERY important. And just because it is mathematically a field does not mean it is particularly the right choice. A…

In a field, division by zero is not the inverse operation of anything.

Re: 1/0 = 0

#416

Post by goerz » Sat, Aug 11, 2018, 5:26 AM UTC

nostrademons wrote: post #252

simias wrote: post #183

Earlier quoted context omitted.

I'm reading the "Pony" tweet quoted in TFA and your comment and I'm left very puzzled: is that really that common to want x / 0 == 0 ? In practice where does that crop up? You say that you frequently have to write your little shim but honestly I don't remember writing code like that in recent memory. You talk about progress bars, I suppose it makes sense if you somehow try to copy 0 elements for instance, and you end…

Real example: I'm collecting some quality signals from a corpus, most of which are some form of ratio, average, weighted average, or scaled average of counting various quantities within the documents. If the elements being counted are missing, I want the term involving that quality signal to disappear from the final ranking calculation. Defining x / 0 = 0 gets this behavior for free, while leaving zero as an exceptio…

Defining a function would seem like exactly the right thing to do! There's no problem.

Re: 1/0 = 0

#417

Post by throwawaymath » Sat, Aug 11, 2018, 5:30 AM UTC

gmfawcett wrote: post #389

sclv wrote: post #296

Earlier quoted context omitted.

The problem this and the other replies miss is that the standard definition of division is multiplication by the inverse. The entire argument rests on a notational slight of hand. The property that held before -- that _when defined_ division has the inverse property -- no longer holds. Thus many equational identities that otherwise would hold do not hold.

I have to disagree -- this isn't sleight of hand. The standard definition isn't being violated here, because standard division isn't a total function. The denominator's domain in Hillel's function is a proper superset of the standard domain: when restricted to the standard domain, the two functions are precisely equivalent. Therefore, every standard identity still holds under Hillel. The hole that he is filling here…

> If something is explicitly undefined, there's nothing mathematically wrong with defining it, as long as the definition doesn't lead to inconsistency.

The definition does lead to inconsistency...you can't look at the field axioms, observe that 0 has no multiplicative inverse, then proceed to define a special, one-off division rule that doesn't involve multiplicative inverses for that one element. Either your division rule is pathological and breaks a fundamental field property or you've introduced a division rule which is just a syntactical sugar, not a real operation (in the latter case you've introduced confusing notation, not a new division function). Why do you think mathematicians explicitly state that the real field with the augmentation of positive and negative infinity (which allow division by 0) is not a field?

I don't understand why there is so much resistance to this idea in this thread, but the simple fact remains that if you define division by an additive identity (0) in any way, the field containing that unit ceases to be a field. This is because all elements cease to be unique. You can quickly prove that every element is equal to every other element, including (critically) the additive and multiplicative identity elements. Fields are defined by closure under the operations of addition and multiplication, and that closure requires uniqueness of their respective identities. Upend that and your entire field structure breaks down, because all you're left with is a field with a single element 0.

Stating that you've defined division by 0 using a one-off case that permits all other field identities to remain consistent is like saying you've turned the complex field into an ordered field using lexicographic ordering. You haven't, because i admits no ordering, much like 0 admits no multiplicative inverse.

Onlookers reading these comments probably think those of us harping on this point are anal pedants with a mathematical stick up our ass. But this thread is increasingly illustrating my central point, which is that the author shouldn't have tried to justify numerical operation definitions in a programming language using field axioms of all things.

Re: 1/0 = 0

#418

Post by rrampage » Sat, Aug 11, 2018, 5:33 AM UTC

godelski wrote: post #382

gjulianm wrote: post #17

Earlier quoted context omitted.

I think OP means that nothing breaks mathematically. It is not inconsistent and not false, so you can work with it. The only issue is to deal specially with the case of division by zero, which you have to do anyways. Code that assumes that (x/y) * y = x is wrong if you don't check for y = 0, independently of what you define x/0 to be.

You do realize that division is the inverse operation of multiplication, right? Like subtraction is the inverse of addition. By defining addition we define subtraction. By defining multiplication we define division. This is where the author fails. Division is multiplication of a fractional value. This is VERY important. And just because it is mathematically a field does not mean it is particularly the right choice. A…

> Division is multiplication of a fractional value

Thus, division by 0 is multiplying by (1/0). Does such a fraction exist? It can go along two potentially different paths depending on the limit we take.

Alternatively, there is information lost when you multiply something by 0: a x 0 = 0, b x 0 = 0. When you perform an inverse by dividing, will you get back a or b (or any number)? Thus, it is not invertible at least at 0.

Disclaimer: Not a mathematician

Re: 1/0 = 0

#419

Post by mortdeus » Sat, Aug 11, 2018, 5:37 AM UTC

there is a pretty good episode on numberphile that explains why dividing by zero is undefined.

Basically the reason why 1/0 = 0 is wrong is the fact that you cant reproduce 1 by taking the answer 0 and multiplying it by 0 to get the number being divided. Which goes on to break a bunch of other fundamental rules of mathematics.

That is to say, if you took 6 / 2 = 3 you can reverse the division by doing 3 * 2 = 6.

However if you have 6 / 0 = 0 and you were to try and reverse it then you would have 0 * 0 != 6. Then you have bizarre circumstance where everything decided by zero logically equals the same thing. Where (2 + 2) / 0 all of a sudden equals (Einsteins laws of gravity) / 0. And there is no logical way to really explain what that is supposed to mean.

Also if you try to use limits to find a solution of f(x) = 1/x where x is defined as the limit of 1 and -1 as it approaches 0 from both sides. You end up with the answer that is even more confusing when you try to graph it.

That is x = +inf and -inf. Which on a graph is represented as two curves where the limit as 1 -> 0 from the positive side of the x axis curves up along the y axis to infinity without intersecting where x=0. And from the negative side where we approach 0 from -1 we end up with a curve that moves down the -y axis without intersecting at 0.

So frankly, n / 0 = 0 just doesn't make sense because just trying to approach it from both sides suggests that the closer you try to move to zero from both sides the further apart the answer moves away from converging together at the origin. Which would be expected if n / 0 = 0 was actually true.

Re: 1/0 = 0

#420

Post by anoplus » Sat, Aug 11, 2018, 5:45 AM UTC

phyzome wrote: post #60

Some people say "oh, that's easy, 1/0 is +Infinity". So the real fun is at 0/0. The limit of x/y as x and y go to zero depends on which path across the xy plane you take towards the singularity. Along one approach, the limit is 0, along another approach the limit diverges to infinity, along yet another the limit is 17. I'm not kidding! Go to https://www.geogebra.org/3d and enter "x/y" and spin the graph around. The "…

I like your point here. I think 1/0 = Infinity+ is a satisfying expression. Its a clear concept which can be visualized in a simple graph.

I think 0/0 is a different concept than 1/0. It's a different expression. 0/0 doesn't explicitly express a particular "path" on the graph.

We can call it (0/0) different names if we want. They can say 0/0 = "undefined". I am currently satisfied with 0/0 simply equals to 0/0, or simply "undefined".

If we all agree it is "undefined" we ironically defined it.