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1/0 = 0 (2018)

hillelwayne.com

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Re: 1/0 = 0 (2018)

#31

Post by rsp1984 »

lupire wrote: post #8
rsp1984 wrote: post #3

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.

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.

Re: 1/0 = 0 (2018)

#32

Post by tmtvl »

IshKebab wrote: post #13

> It’s saying that Pony is mathematically wrong. This is objectively false. Pff. The author wants to show off their knowledge of fields by defining a "division" operator where 1/0 = 0. Absolutely fine. I could define "addition" where 1 + 2 = 7. Totally fine. What I can't do is write a programming language where I use the universally recognised "+" symbols for this operation, call it "addition" and claim that it's tot…

> What I can't do is write a programming language where I use the universally recognised "+" symbols for this operation, call it "addition" and claim that it's totally reasonable.

As a programmer, you're right: we have standard expectations around how computers do mathematics.

As a pedant: Why not? Commonly considered 'reasonable' things surrounding addition in programming languages are:

* (Particularly for older programming languages): If we let Z = X + Y, where X > 0 and Y > 0, any of the following can be true: Z * I haven't yet encountered a language which solves this issue: X + Y has no result for sufficiently large values for X and Y (any integer whose binary representation exceeds the storage capacity of the machine the code runs on will do). Depending on whether or not the language supports integer promotion and arbitrary precision integers the values of X and Y don't even have to be particularly large.

* Non-integer addition. You're lucky if 0.3 = 0.1 + 0.2, good luck trying to to get anything sensible out of X + 0.2, where X = (2 ^ 128) + 0.1.

Re: 1/0 = 0 (2018)

#33

Post by TrianguloY »

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 that with 'i' and never went far. Still intrigued if this is somewhat interesting or a total time loss though.

Re: 1/0 = 0 (2018)

#34

Post by akira2501 »

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…

Sure.. but there are infinite series that sum to a finite value. Perhaps a pertinent example would be summing all the distances between each successive reciprocal of 1:

    Sum[1/x - 1/(x+1), {x, 1, ∞}] == 1
You do actually need infinity to arrive at that 1.

Re: 1/0 = 0 (2018)

#35

Post by Asooka »

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.

Those are all integers. https://en.wikipedia.org/wiki/Modular_arithmetic - "The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801." They have been integers for over 200 years now.

Re: 1/0 = 0 (2018)

#36

Post by [deleted] »

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…

[deleted]

Re: 1/0 = 0 (2018)

#37

Post by yarg »

eqvinox wrote: post #20

Note 1/0 (or x/0 with x>0) isn't undefined or an exception in 754 FP math, it's +infinity. It's 0/0 that's the problem. Defining 1/0=0 isn't really helpful imho.

> Defining 1/0=0 isn't really helpful imho

But it's quite a nice way to mask program bugs.

Re: 1/0 = 0 (2018)

#38

Post by jvanderbot »

lupire wrote: post #28
jvanderbot wrote: post #7

Q on this post: Is the field rule "Every element Except Zero has ... " (the 9th rule) defined with respect to the additive identity "zero" or the magical other undefined "Zero" that is the number we're all familiar with? If so, how weirdly arbitrary that the additive zero is omitted for all multiplicative inverse definitions. (At least it seems to me). I always figured this was a consequence of our number systems, no…

I am not familiar with any "magical other undefined Zero". Please familiarize me :-) What is your, uh, definition of this undefined* number you are familiar with?

> Every element EXCEPT 0 has a multiplicative inverse, a⁻, such that a*a⁻ = 1.

What is "0"? It's not defined in the axioms other than additive zero. Or is it multiplicative zero? (1?). Is it the number zero?

If it is the additive zero defined in axiom (3), then it just seems weird to me that additive zero is undefined for multiplicative inverse for all fields always and forever.

If it is the number zero, then how does that generalize to other fields?

If the answer is "Numbers are the first field and all fields generalize that", then I suppose we are referring to the number (0), and that's fine, as other fields are welcome to define their own larger definition of zero that includes the number (0) ... ?

Re: 1/0 = 0 (2018)

#39

Post by ashton314 »

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.

Multiplicative inverse happens to be a convenient way to define division in the reals, but there are cases when multiplicative inverses do not correspond to any notion of division. E.g. take a finite ring of integers, like what you’d use for cryptography or heck any operation on an `int`!

It’s all just definitions. Always has been.

Re: 1/0 = 0 (2018)

#40

Post by tbrownaw »

CharlieDigital wrote: post #9
rsp1984 wrote: post #3

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.

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…

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.

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