Talk:Division by zero

0/0 = x, fact

I know I've said this before but I think I might of thought of some proof that can end all of this division by zero shit forever. For those who don’t know what logarithms are, log(a) b is the same as saying "With $a^{x}$, what does x have to be to get b?". Lets look at log(2) 256 as an example, with $2^{x}$, what does x have to be to get 256? The answer is 8, because $2^{8}$ = 256. Now lets look at log(1) 1, with $1^{x}$, what does x have to be to get 1? x can of course be any number, so I will just keep the answer as x. Because log(a) b = log b/log a (with log being log(10), it still works the same with any other positive logarithm) we get

x = log(1) 1 = log 1/log 1 = 0/0

We can use a similar method to prove that 1/0 doesn't have an answer, with $1^{x}$, what does x have to be to get 10?

log(1) 10 = log 10/log 1 = 1/0

If you don’t understand the above proof you can get the same answers by asking "If you are travailing at 0mph, how long does it take you to travel 0 meters?" or simply "How many 0's go into 0?" for 0/0 and "If you are travailing at 0mph, how long does it take you to travel 1 meter?" or "How many 0's go into 1?" for 1/0. Well okay maybe 'x' isn't a suitable answer for the first one, but you have to at least accept that the answer can be any number, and that it deserves at least a small mention in this article. Robo37 (talk) 15:32, 12 August 2009 (UTC)

I fail to see how this differs from what you wrote above. The answer is the same: we want 0 divided by 0 to be a number, a single number. If we can't define it that way, it's not very useful to define it at all. You are definitely right that the correct way to define it if we thought it useful would be as an entity that assumes all possible values at once. The logarithms do not make that argument less valid (although they are completely unnecessary here), but they also do not make the definition any more useful. —JAO • T • C 18:42, 12 August 2009 (UTC)

I'm not asking for you to tell me if I'm right or not, I'm asking for something to be said about it in this article. I can't see a single bit of text in it that says that 0/0 can be any number, and since the entire article is about division by zero it would make sense to mention it don’t you think? I fail to see why it isn't important and if we all went by your "if it doesn’t produce a single number it doesn’t produce any" rule then it would be impossible for a number to have a square root. Robo37 (talk) 20:50, 12 August 2009 (UTC)

Yes, you have a point here in that there are lots of useful multivalued functions, of which the complex logarithm, and therefore the unrestricted square root function, are prime examples. Many even assume an infinite number of values in each point. I still don't see the usefulness of something that simultaneously assumes all values in its codomain, though. But if you're looking to add something like "without the usual requirement that a/b be a unique number, 0/0 could instead have been defined as an entity assuming all values at once" to the article, I wouldn't be opposed to that. —JAO • T • C 05:36, 13 August 2009 (UTC)

Please add a citation to a journal where such a way of dealing with it is used otherwise it would be original research. Please see that link for Wikipedias policy as regards that. Dmcq (talk) 10:25, 13 August 2009 (UTC)

Something like that would be fine Jao. How is that original research? Robo37 (talk) 16:55, 13 August 2009 (UTC)

For something like this the appropriate mathematical term is 0/0 is an indeterminate form. Saying 0/0=x would be removed in a flash because it is not any sort of standard mathematical notation. You thought of it yourself and therefore it is original research in wikipedia terms. The article about original research is quite specific about this, it is one of the core policies of wikipedia. Actually I do know of a reference for stuff like this, see James Anderson (computer scientist). Really I don't advise following his path as it goes nowhere. Dmcq (talk) 17:17, 13 August 2009 (UTC)

Simply saying "0/0=x" or anything like that would be stupid, I'm asking for something like "without the usual requirement that a/b be a unique number, 0/0 could instead have been defined as an entity assuming all values at once" as Jao said above or "arguably, 0/0 can, in fact, have any number as it's answer, but representing this under a single value is where the difficulty lies”. If it’s a fact (and it seems pretty clear that it is), then why shouldn’t it be mentioned? Robo37 (talk) 17:39, 13 August 2009 (UTC)

0/0 could be defined as anything you want, even 1. The problem is that any definition would not follow the usual rules. In fact 0⁰ is actually defined as 1 in many circumstances even though in others it is best left undefined and is also an indeterminate form. There just is no point in your definition that I can see and there isn't a literature saying that it has a point. What would be your reason for doing so? What would be the difference between what you are saying and what James Anderson (computer scientist) wrote and why would what you wrote be any better? Dmcq (talk) 20:09, 13 August 2009 (UTC)

As an aside, I think there is a basic misconception about the possibility or impossibility of dividing by zero. If someone says "we can't divide by zero", someone else will invariably hear "nobody has yet managed to divide by zero". And since it's so extremely easy to come up with a way of dividing by zero, they think they have seen something original. (I'm not saying this is what Robo37 does here, but it certainly is the way the BBC story on Anderson's nullity was worded.) The fact, as you point out and as the article maybe needs to be clearer on, is that 0/0 can be defined to be something (and that it already has been done, several times!), it just can't be defined to be something more useful than "undefined". A useful division by zero should let us, for instance, completely solve the equation x² = x by dividing both sides by x. As far as I know, not even wheel theory accomplishes that, because it's just not accomplishable. Now, there's a gauntlet for all presumptive zero divisors (pun intended) to pick up. —JAO • T • C 05:14, 14 August 2009 (UTC)

There are a few bits missing, it links to defined and undefined which is a badly written article and it misses any mention of an indeterminate form in the calculus section. I think the article could be cleaned up a bit too. However it does mention that the value is not defined except in some special circumstances and it isn't too bad an article. It is also read by a lot of people, so overall there is a higher bar to aim at when editing it than many other articles, but of course anyone who's willing to put in the effort to try is welcome to give it a go. Dmcq (talk) 10:15, 14 August 2009 (UTC)

Why is there so much original research here? If you think you are correct, publish a paper, have it peer reviewed, accepted, and published in a journal, and then we can link to it. Or at least bring it up with the Math help desk. Talk pages are supposed to be for improving articles, not introducing original research. --Zarel (talk) 22:31, 14 August 2009 (UTC)

All I'm doing is suggesting an improvement to this article. Robo37 (talk) 23:00, 14 August 2009 (UTC)

As far as I recall you titled this section "0/0 = x, fact". That is not an accepted part of mathematics. Jao has tried to extract some sort of idea of what the problem is from what you said seeing that you haven't been able to read the article without difficulty. The precise ideas you have had though are simply not suitable for inclusion though and what Zarel said about them is correct. Dmcq (talk) 23:17, 14 August 2009 (UTC)

We have the No Original Research rule for a good reason - if we didn't, Wikipedia talk pages would be full of debate over whether or not your original research was correct or not. For instance, to me, it is obvious that your "fact" of "0/0 = x" is incorrect and an abuse of notation, but article talk pages are not a good place for the discussion of why. The "No original research" rule ensures that if you disagree, you take it up with professional mathematicians instead of us, and you can come back to us when professional mathematicians agree with you. --Zarel (talk) 10:51, 2 December 2009 (UTC)

Hmmm, now if you had said that 0/0 ∈ ℝ, you might have gotten some consensus. When you consider that division is the process of repeated subtraction, asking "how many times can you subtract the divisor (the denominator, here zero) from the dividend (the numerator, the other zero) before you have a remainder of or less than the divisor (zero)?" (the literal meaning of division) then clearly any real number answers correctly for the quotient, and all answers you can comprehend although indeterminate are members of ℝ. If you do the long division of 0 goes into 0, you'll see that no matter what you choose for your quotient, immediately upon back-multiplying and subtracting that product of your number from ℝ times zero, you have zero and thus are done. In that sense, your wanting to use "x" is understandable, but you need to precisely state x ∈ ℝ. (Remembering the definition of division is always helpful in resolving these issues, when you try an operation such as 1/0, you'll immediately see you'll be subtracting zero from 1 without bound, hence making it really clear why 1/0 is usually considered to be boundless and expressed 1/0 ∈ ±∞ .) —Preceding unsigned comment added by 130.111.163.179 (talk) 12:59, 30 September 2010 (UTC)

Creative Numbers...

Creative numbers have purported to solve the problem of division by zero. I'm no mathematician but perhaps someone who understands the subject matter more critically could check out http://science.mistu.info/Math/Numbers/Creative_numbers_and_division_by_zero.html and see if it warrants expanding the article. 71.207.183.254 (talk) 18:22, 4 January 2013 (UTC)

Its original research, with no rigorous proof or peer review. It doesn't merit inclusion here. Mindmatrix 18:40, 4 January 2013 (UTC)

I guess you would know better than I would. But, it makes sense to me as a layman. 71.207.183.254 (talk) 19:31, 4 January 2013 (UTC)

I know how to properly divide by 0 and do 0 math in general, so I reviewed the reference for you

 No.
Victor Kosko (talk) 19:03, 3 June 2018 (UTC)

Grade seven revisited

Long ago, way back when a four-function calculator was consider high tech, I got into a sparring match with my grade seven math teacher, which did not end well so far as I was concerned.

He made completely valid points, but he still seemed to be missing something essential about how the problem appears to the novice as yet unencumbered with mathematical convention. After our interaction, I remained as inarticulate about what my math teacher had skipped over as I was before, so I found his arguments correct, but nevertheless unsatisfying.

And so I suddenly get the urge to see how this is old chestnut plays out on Wikipedia, and "whoa! time machine". Is that you Mr W? Are you editing Wikipedia from your retirement cabin on the lake? Because this is the same damn thing.

A compelling reason for not allowing division by zero is that, if it were allowed, many absurd results (i.e., fallacies) would arise.

...

The fallacy here is the assumption that dividing 0 by 0 is a legitimate operation with the same properties as dividing by any other number.

What that passage actually demonstrates is that defining 0 as its own multiplicative inverse (0/0 = 1) instantly leads to the absurd. I quickly refreshed myself on the axioms of field theory, and the axioms pertaining to a/b exclude the cases where b = 0. Defining 0/0 to some quantity does not appear to violate the existing axions of field theory, although we quickly ruled out 0/0=1 as leading to absurdity. By the same argument we can quickly rule out 0/0=a for any non-zero a.

But what about defining 0/0=0?

This isn't inconsistent with the axioms of field theory based on my quick review. Maybe it's inconsistent with a theorem of field theory, such as x=x^-1 having a single solution (er, a single non-negative solution) as proven from the existing axiom which excludes b=0 from any quantity a/b.

But that's kind of narrow-minded about axioms, and we ought to know better than to proceed on the basis of this kind of axiomatic narrow-mindedness because of non-Euclidean geometry.

So why don't we define 0/0=0 (and 0/x=0 and x/0=0 all around) and amend the theorem to x=x^-1 having a singular solution (the multiplicative identity) for x strictly positive.

While we're at it, we can also thoroughly subordinate division under multiplication, so that we can't actually divide both sides of an equation by any quantity; whereas we can multiply both sides of an equation by any chosen reciprocal. In this formalism, you can't divide an equation by zero, you can only multiply by 1/0=0, as defined in the previous paragraph. And you really can't get into much fallacy trouble multiplying both sides of an equation by 0 (be careful with strict inequalities such as a < b, however).

So why don't we do this? Here are my two primary guesses:

1. it's uniformly infelicitous to multiply both sides of an equation by 0 (wherever you were going, this doesn't help you get there, even if you salvage your proof later—leaving behind some foolish but non-fallacious deadwood).
2. when progressing to infinitesimals, this peculiar pointwise discontinuity clutters your statements and proofs to no useful effect

It appears to me on superficial review that defining 0/a=0 and a/0=0 for all a is not precluded by fallacy, nor by necessary rather than whimsical field theory axiomatization, but by pure infelicity: it simply doesn't buy you anything you can finally wield to your advantage; refusing to extend the definition of division in this way votes a certain kind of deadwood off the island at first point of contact.

I have far more background in computer science, and in this realm it is surely not a win in many contexts to lose your shit over deadwood that isn't causing any real trouble; better to let the computation proceed uniformly and deal with any issues that arise at the other end. What's a NaN here or there between friends that probably disappears again at a later step anyway? Did we really need to raise a bat signal to the overlord of the galaxy over one piddly NaN (the Toyota model where any task station can stop the entire assembly line in a heartbeat with one tug of a nearly rope, for 10,000,000 parallel arithmetic units spread over a 1000 distinct systems)? Perhaps exception handling at scale is already hard enough without the hair trigger?

Back to math, what I don't want in response is people crawling all over me because I'm lacking some vital pinprick of sophistication here. I'm deliberately standing in for a clever 14-year-old who will not be even slightly impressed with an answer that amounts to this: after you have worked within field theory for another decade, it will make you sick to your stomach to even think about defining a/0=0, because so much older and so much wiser.

Worst. Possible. Answer. Abominably unacceptable to clever, independent-minded 14-year-old.

Can we make this more usefully explicit, or am I smoking a crack pipe here?

What Mr W. ought to have told me: the seemingly obvious 0/0=1 leads to an instant quagmire, the weirder 0/0=0 (augmented with a/0=0) leads to no particular quagmire, but it's completely infelicitous and doesn't buy you a a darn thing as a working mathematician (proceed directly to tautology, do not collect $200).

My other perspective comes from quantum mechanics where destroying information is regarded as pissing away useful energy. Multiplying by zero is special in that way: it's the only multiplication that destroys information.

This is why fields don't promise for all a: a*b*b^-1=a when b=0 (b=0 is at least implicitly ruled out in conventional axiomatization because 0^-1 is undefined).

In the infelicitous extension of field theory where 0/a=0, a*b*b^-1=a except when b is an information-destroying black hole (b=0).

[*] See The Black Hole War (2008) concerning the controversial non-existence of information-destroying black holes in the real universe; only science book I've ever read where I got halfway through and said to myself "hmmm, better luck next time" as I glazed over into a perfect non-absorbent mirror. Coincidentally, next time is circa next week, because it's presently en route from my library for a second kick at the cat. — MaxEnt 00:49, 3 May 2021 (UTC)

This is clearly a subject that interests you. There are many books that will answer your questions. Essentially, we do not define 0/0 = 0 and abandon, for example, the other axioms of Field Theory which would make this "definition" fail to be well defined, because a great deal of interesting mathematics has already been written with the standard definitions. We could, of course, define something new, call it a quasi-Field, and try to come up with consistent axioms that would allow your definition. In fact, many people, ranging from the advanced mathematician to people almost totally ignorant of mathematics, have written extensively on this subject, and I know at least one grade school teacher who teaches your definition to her class, has for years, and isn't about to stop. Rick Norwood (talk) 11:43, 3 May 2021 (UTC)

Feel free to correct me, if I am wrong.

Wouldn't it be simpler to use examples such as Foucault's pendulum, as a real world example of dividing by zero to get an observable result? Sin 0 being at the equator. At the Equator, 0° latitude, a Foucault pendulum does not rotate. In the Southern Hemisphere, rotation is counterclockwise.

Or is this page simply concerned with the arithmetical concept? 49.185.200.59 (talk) 04:35, 18 May 2022 (UTC)

This is solely about the arithmetic concept, but it does address division by zero as a limit of a function. Your example is the limit where the period of precession approaches infinity because the precession rate approaches zero. –LaundryPizza03 (dc̄) 04:45, 18 May 2022 (UTC)

Wiki Education assignment: Computer Science Principles

This article was the subject of a Wiki Education Foundation-supported course assignment, between 19 September 2022 and 9 December 2022. Further details are available on the course page. Student editor(s): Annie.nguyen0811 (article contribs).

— Assignment last updated by Annie.nguyen0811 (talk) 21:33, 27 October 2022 (UTC)

Euclidean division by zero?

File:Gcd exercise.gif

Gcd exercise

In case I'm not asking this question in the right place, I'll be asking this same question at Talk:Euclidean division.

I'm curious why Euclidean division by zero is never discussed?

It seems like such a simple thing to assume that any dividend divided by zero yields a zero quotient plus a remainder equal to the dividend so that the multiplicative inverse is also true, that: the zero quotient times the zero divisor yields zero plus the remainder yields the original dividend. This may seem useless, yet it is not. For, it is useful to postulate division by zero as modulo zero when GCD factoring using a potentially limitless expansion of the GCD in two-dimensional format. -- Vinyasi (talk) 12:21, 5 January 2023 (UTC)