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1/0 = 0

hillelwayne.com

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Re: 1/0 = 0

#381

Post by perl4ever »

This item reminded me of how I've owned a couple different German cars with a fuel consumption meter that showed minimum consumption (or maximum mpg) when the car was idling. It seems like they must indeed have defined (fuel used ÷ distance traveled) as zero when the distance is zero. Which is annoying and grossly wrong in my opinion. For some reason, Japanese cars seem to do it correctly.

It may be true that 1/0=0 can be part of a reasonably consistent system, but that doesn't mean it's a good one. Can we define 1/0 as ∞?

Re: 1/0 = 0

#382

Post by godelski »

gjulianm wrote: post #17
luso_brazilian wrote: post #7

> But is Pony doing something unsound? Absolutely not. It is totally fine to define 1/0 = 0. Nothing breaks It actually does break something, the symmetry between division and multiplication and the many pieces of code that assume that (x / y) * y equals x. Here is a naive and non practical example, but it is not impossible to find a real world example where this simplified code manifests itself accidentally or by de…

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 field is not strictly definitely by addition and multiplication, it is definitely by two operators where one is an abelian group (addition in our case) and the other forms and abelian group over the non identity term of the first (eg multiplication is an abelian group over non zero terms). The complex numbers create a field, which is really how we do addition and multiplication in 2D space. 3D space you can't form one, so you have a ring. Which is why quaternions are so important, because they form a field.

But the author is wrong because they think division is a different operation than multiplication. It's just a short hand.

Re: 1/0 = 0

#383

Post by mcguire »

ggggtez wrote: post #24
sgillen wrote: post #4

I mean sure Pony, but as a programmer this result would surprise me quite a lot, which I tend to view as a bad thing. https://en.wikipedia.org/wiki/Principle_of_least_astonishmen... However: > One of developers of Pony reached out to me. They’re planning on writing a more in-depth explanation of why Pony chose 1/0 == 0, which I will linked when available. As I understand it, it’s because Pony forces you to handle all…

Based on the tweet, I don't expect it to be very enlightening. Since programmers write programs to solve real problems, and real problems define 1/0=undefined... then this is only going to mask bugs in those programs.

Technically, some real problems define 1/0 as illegal.

But see renormalization. https://en.wikipedia.org/wiki/Renormalization?wprov=sfla1

Re: 1/0 = 0

#384

Post by ummonk »

hpcjoe wrote: post #38

Hrm ... argumentation by reference to authority (various Ph.D. people) is a sure way to lose a scientific argument. But that isn't my only qualm. In the construction of the fields, there is a simple definition of the division function. It is intrinsically the solution of a = r * b, where r is the unknown. If a is nonzero, and b is zero, then r, the ratio, is said not to exist. Put another way, there is no real value…

>In the construction of the fields, there is a simple definition of the division function. It is intrinsically the solution of a = r * b, where r is the unknown. If a is nonzero, and b is zero, then r, the ratio, is said not to exist. Put another way, there is no real value of r that can be chosen that satisfies r * 0 = a.

Exactly. That's the definition of division. Here, he is introducing a totally different definition and claiming it isn't wrong because it is consistent. I mean, by his logic I could define division as a/b := a-b and claim it is consistent and therefore not wrong, but that is obviously not division...

Re: 1/0 = 0

#385

Post by lisael »

FWIW Pypy used to share this behaviour. It was clearly a bug, as it doesn't follow CPython convention.

It's around since 2013 and was found in a unit test coverage improvement effort in Feb. 2018.

Given that pypy is mostly used for hard-core math and scientific computations, how the hell this bug wasn't found earlier by a user ?

My bet is that ZeroDivisionErrors are very rare. Do you even remember a ZeroDivisionError in production code ? I don't.

1/0 is my dirty little trick to add a breakpoint when I'm too lazy to launch a debuger. Why does this work? Here again, because nobody never catch ZeroDivisionErrors, because they don't happen.

So, what a fuss for such a tiny convention. Granted, it breaks the math correctness, as do ±Infinity. CPU integer math is broken anyway. In what world does 2^31 - 1 + 1 == -2^31 ?

I took a fairly large, used and old repository, Django and I search for ZeroDivisionError:

https://github.com/django/django/search?q=ZeroDivisionError&...

guess what:

- 4 occurrences in the tests

- 3 occurences in the issues

- last, but not least 1 occurrence in the code:

   except ZeroDivisionError:
       result = '0'

Re: 1/0 = 0

#386

Post by Fomite »

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…

Yep. Having had a couple divide-by-zero errors in a complicated simulation, I'm immensely grateful that it fails hard.

Re: 1/0 = 0

#387

Post by ummonk »

todd8 wrote: post #273

Let f(x) = 1/x, then (from a mathematician’s perspective) the limit of f(x) as x->0 doesn’t converge and is representeted as being infinite, definitely not zero. As the denominator gets smaller, the result is bigger and bigger and bigger. The closer the denominator gets to zero the larger the result. Why would we want to pick an answer that is small when instead it should be larger than any number? One might argue th…

Well, you have an ambiguity as to whether it should be +INF or -INF, depending on which side you take the limit from. Arguably, 0 is more reasonably because it preserves symmetry. Of course, it would be far more reasonable to define this as an exception, because you shouldn't be able to do division by 0.

Re: 1/0 = 0

#388

Post by throwawaymath »

pron wrote: post #359
throwawaymath wrote: post #353

Earlier quoted context omitted.

At this point I've written a number of comments throughout this thread which explain why division by zero is not possible in a field - specifically, a field which contains nonzero elements. You are asking me to show you an axiom or theorem proving this, but I don't know how else to explain this to you since I've already explained it in a number of different ways. In a sibling comment I even write out the formal proof…

If you mean this, https://news.ycombinator.com/item?id=17737661 , then it contains a mistake, which I pointed out in a reply. The references you linked to are irrelevant as they refer to informal mathematics. Of course you can't meaningfully divide by zero (in the sense that the value of division corresponds with our intuitive understanding of what division is). But when it comes to formalization you must assign some…

I don't know what to tell you. You're continuing to argue but you seem to not be following my point. I've already long since told you we're in agreement that you need to account for syntactically possible but theoretically impossible algebras for computation. In so doing I have gone further and repeatedly stated that because division by zero is not actually possible in fields, the author should not have tried to use such a rigorous system to justify the practical, syntactical requirements of a programming language. It doesn't make sense to try and refute mathematical theory as the preamble of a programming language manifesto.

What exactly are you looking for me to say here?

Re: 1/0 = 0

#389

Post by gmfawcett »

sclv wrote: post #296
gmfawcett wrote: post #229

Earlier quoted context omitted.

He's not saying that it breaks; quite the opposite, he repeatedly states that 0⁻ doesn't exist. He's just defining division in a specific way. The MI property states that every element except 0 has a multiplicative inverse. He's defining division via two cases: If b≠0, then a/b = a*b⁻ (multiplicative inverse). If b=0, then a/b=0. This definition does not imply that 0⁻ exists, so there's no violation of MI.

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 isn't one that he bored into the standard definition, but a hole that the standard definition already admitted. If something is explicitly undefined, there's nothing mathematically wrong with defining it, as long as the definition doesn't lead to inconsistency.

Re: 1/0 = 0

#390

Post by starmole »

spooneybarger wrote: post #103

Hi, I'm on the Pony core team. I will be writing in more detail about this decision. A few short notes until then: 1) no one on the team has ever been happy with ending up here, understanding why the decision was made involved understand how partial functions (one that can produce errors like division by zero) are handled in Pony and interesting ergonomic issues that can result that is a large part of what my post wi…

This should be the top comment. INTEGERS are not a field. But that kind of makes the 0 choice worse? Even cpus always throw on integer divide by 0, so i don't really see the low level optimization. Does it enable some higher level optimization? Why not x/0 = x?
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