Talk:Division by zero

Euclidean division by zero?

Moved to Talk:Euclidean division § Euclidean division by zero?

 – ~ ToBeFree (talk) 02:44, 6 January 2023 (UTC)

Error in Paragraph "Algebra"

In Paragraph "Algebra" there ist written:

"It follows from the properties of the number system we are using (that is, integers, rationals, reals, etc.), if b ≠ 0 then the equation a/b = c is equivalent to a = b × c. Assuming that a/0 is a number c, then it must be that a = 0 × c = 0."

This is wrong, since the assumption b ≠ 0 is already violated in the next sentence. Stephan Schief (talk) 20:15, 8 May 2023 (UTC)

The statement that the denominator of a fraction cannot be equal to zero in the number systems mentioned is a correct statement, but the sentence saying that is awkward enough so I can understand how it can be misread. I'll try to make it clearer. Rick Norwood (talk) 10:32, 9 May 2023 (UTC)

I agree that in the field structure of the reals (or rationals), if b≠0, then a/b=c implies a=b×c. But one can not use this argument to say that if a/0=c, then a=0×c. Because then one would use b=0, in contradicition to the assumption. Stephan Schief (talk) 10:50, 9 May 2023 (UTC)

As I'm sure you know, when showing something by way of contradiction, it is standard to violate the assumption, arrive at a contradiction, and so demonstrate that the assumption is correct. I've tried to rewrite the paragraph so make the hypothetical nature of the claim clearer. Rick Norwood (talk) 11:04, 9 May 2023 (UTC)

I did not realise that you were using an argument by contradiction! I am sorry about that. I like how you rewrote the paragraph, it is a lot clearer to me now. Thank you! Stephan Schief (talk) 11:13, 9 May 2023 (UTC)

Sources for proof assistants

My edit cites two sources: a blog post from Kevin Buzzard and and StackOverflow answer from Arthur Azevedo de Amorim. I'm not concerned about the authors, given that they are a Lean maintainer and an author of Software Foundations respectively. But I'm open to alternatives from more established publishers. Lambda Fairy (talk) 05:37, 2 December 2023 (UTC)

Blogs are not reliable sources. I have reverted your edit. To establish notability, such content needs more solid sources, such as textbooks or articles in peer reviewed journals. - DVdm (talk) 10:52, 2 December 2023 (UTC)

Correcting a mistake of Lambda Fairy: Kevin is not a Lean maintainer at all, but is still a subject matter expert (source: I am a maintainer for its math library).

Blogs are not reliable sources

Proof assistants are ultimately source code; would a permalink to the corresponding lines of code be sufficiently reliable? After all, there is no more reliable indicator of how a software system behaves than its source code.

I think the blog/SE post provides some interesting context beyond the simple proof that the statement is true

To establish notability

Presumably the fact that Coq and Lean have their own pages is some indication that this is sufficiently notable? Eric Wieser (talk) 17:51, 2 December 2023 (UTC)

If this has never been mentioned anywhere in a published paper or any secondary source, is it really that essential to the topic of division by zero in general? –jacobolus (t) 18:09, 2 December 2023 (UTC)

It seems just as relevant as mentioning the behavior of the Desmos graphing calculator, which is mentioned immediately above the proposed insertion point, without any references at all. If the problem is references to blog posts, would removing all references other than those to other Wikipedia pages (to match the Desmos example) suffice? Eric Wieser (talk) 20:31, 2 December 2023 (UTC)

It sounds like this section/article has become a magnet for un-encyclopedic trivia. The behavior of the Desmos graphing calculator certainly doesn't seem necessary here with or without a source. –jacobolus (t) 20:50, 2 December 2023 (UTC)

I split the section into floating point and integer subsections, and took out some of the trivia including the part about Desmos, which isn't about integer arithmetic at all. The rest of the section about integer arithmetic also needs sources. Material about proof systems probably belongs in a separate subsection, but it would be helpful to find a "reliable source" about it, not just documentation, code, or blog posts by the authors (though these could also be cited as a supplement). –jacobolus (t) 01:54, 3 December 2023 (UTC)

I think Buzzard's blog post is probably alright here, if we take a generous approach to "Self-published expert sources may be considered reliable when produced by an established expert on the subject matter, whose work in the relevant field has previously been published by reliable, independent publications". This blog post has been cited several other times, including e.g. in this peer-reviewed (but not yet published) paper https://pubs.rsc.org/en/content/articlepdf/2023/dd/d3dd00077j. It's perhaps worth having a fuller discussion of this topic, e.g. DJB criticizes Buzzard's claim here: https://cr.yp.to/papers/pwccp-20230909.pdf "in fact a mathematician is permitted to deduce b ≠ 0 from c = a/b, since a/b is undefined for b = 0. Redefining the notation to allow b = 0 breaks this. Unless it occurs to the author to state a conclusion c = a/b and a conclusion b ≠ 0, the proof assistant won’t check that b ≠ 0, whereas the reader will think that this has been checked."

There's some related material in Bergstra (2014) "Division by Zero and Abstract Data Types" and (2021) "Division by Zero in Logic and Computing", which cites Komori (1975) "Free algebras over all fields and pseudo-fields" and Ono (1983) "Equational theories and universal theories of fields". –jacobolus (t) 03:15, 3 December 2023 (UTC)

Thanks jacobulus. As for Buzzard's "work in the relevant field", I think his work formalizing perfectoid spaces is relevant, as well as his overview of schemes in Lean.

There is also a peer-reviewed source on n/0=0 here: https://arxiv.org/pdf/1506.04205.pdf, page 8. (Though that only says how it's defined, not why.) --Lambda Fairy (talk) 12:58, 19 January 2024 (UTC)

Do you want to take a crack at re-incorporating this topic? I think it's fine to cite Buzzard's blog post, but I don't think this should go in the "Computer arithmetic" section, but would probably be better to put into the "Alternative number systems" and/or "Higher mathematics" sections. –jacobolus (t) 15:12, 19 January 2024 (UTC)

Added it back, cheers!

Re placement: I think it does belong in "Computer arithmetic", actually, because IMO it's more an implementation detail than a fundamental change. Allowing n/0=0 does not change the overall "shape" of a proof; it merely shifts the proof obligations from the statement of a theorem to its body. The mainstream ring theory textbooks still apply. This is unlike e.g. replacing epsilon-delta limits with filters, which has much mathematical content, enough to fill a volume of Bourbaki.

Re DJB: I'm hesitant to include DJB's opinion alone—while his opinion is justified, it's not representative of the consensus among experts in proof assistants (who are largely supportive). --Lambda Fairy (talk) 12:46, 21 January 2024 (UTC)

Relatedly, regarding https://en.m.wikipedia.org/wiki/Special:MobileDiff/1197667226; Lean advertises itself both as a programming language and a proof assistant, so I'd weakly argue it does belong in this table. Eric Wieser (talk) 10:11, 22 January 2024 (UTC)

It's not about whether it technically satisfies some (made up) criteria; the question is whether including it there is on net helpful to readers, when we are already saying the same thing directly in the article text and when it's at least somewhat misleading to compare. (Frankly I think the table has questionable value and I'm considering entirely scrapping it.) –jacobolus (t) 16:19, 22 January 2024 (UTC)

@Eric Wieser I just took the table out altogether, and rewrote/reorganized the section a bit. What do you think? –jacobolus (t) 18:13, 22 January 2024 (UTC)

Research paper tag

the article reads like an academic paper or a teaching aid. it speaks to the reader as if the reader is a student. sections such as Elementary arithmetic epitomise this. The article needs to be rewritten to avoid being a research paper or a teaching aid as opposed to be encyclopedic. It should not posit that the reader considers examples. The Fallacies section goes into details of mathematical examples, which treats the section as an academic exercise instead of an encyclopedic article. PicturePerfect666 (talk) 17:56, 7 December 2023 (UTC)

To clear things up Wikipedia states as follows on what Wikipedia is not " [articles should not ] presented on the assumption that the reader is well-versed in the topic's field." the flede begtins as follows"

In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as ${\textstyle {\tfrac {a}{0}}}$, where a is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is no number that, when multiplied by 0, gives a (assuming ${\textstyle a\neq 0}$); thus, division by zero is undefined (a type of singularity). Since any number multiplied by zero is zero, the expression ${\tfrac {0}{0}}$ is also undefined; when it is the form of a limit, it is an indeterminate form. Historically, one of the earliest recorded references to the mathematical impossibility of assigning a value to ${\textstyle {\tfrac {a}{0}}}$ is contained in Anglo-Irish philosopher George Berkeley's criticism of infinitesimal calculus in 1734 in The Analyst ("ghosts of departed quantities").

- directly from a maths teaching textbook or aid.

This is confusing and dense and challenging for the reader with limited initial knowledge of the topic to understand the subject of the article, hence it reads like something from academia. PicturePerfect666 (talk) 19:32, 7 December 2023 (UTC)

This article does not read like an academic paper, but like a (poorly written) encyclopedia article. The banner is not appropriate. It is entirely fine (indeed, extremely helpful) for encyclopedia articles to include concrete examples of abstract concepts and situations. If you would like to rewrite one or another section for style and clarity, please do so. –jacobolus (t) 18:09, 7 December 2023 (UTC)

I think a greater level of explanation as opposed to simply going the 'article does not read like an academic paper', is needed here. It is stated clearly by you that you believe this is a poor-quality article. Just stating that it is a poor quality is unhelpful, expand on why. Also, the shifting of burden here is something to be avoided, if you believe it is a poor quality article be bold and improve it. I am not a mathematician and this reads to me someone outside the heft of mathematics as an academic paper. You can disagree with that but assume good faith. PicturePerfect666 (talk) 19:20, 7 December 2023 (UTC)

The banner is facially absurd. The article does not read remotely like a research paper. PicturePerfect, quit trying to force this. If you have specific examples of language that needs to be improved, feel free to bring them up, or even correct them. --Trovatore (talk) 18:22, 7 December 2023 (UTC)

The banner is not 'Factually absurd" and stating so misses the points raised. the article does to the person who added it (me) fully believes this is the feel of academic. If you have expert knowledge on this topic fine, but I have not got a clue how to get through the academic density here with complex mathematical examples and mathematical theory in the article. PicturePerfect666 (talk) 19:20, 7 December 2023 (UTC)

Note, "facially absurd" (not "factually") means self-evidently incorrect and therefore immediately dismissible.

The "complex mathematical examples" begin with the integral division of a pile of cookies, then proceed to nontechnical historical material, then some middle school level algebra manipulation, and then some high school level calculus. Eventually we hit some discussion of the topic as it applies to undergraduate level and beyond, and some discussion of the computer engineering aspects of the topic.

No part of any of these is written anything like a research paper, which is why several editors here think that {{research paper}} is an inappropriate banner.

It is expected, and valuable, for Wikipedia articles to include relatively comprehensive coverage of a topic, even though some aspects of that coverage are only accessible to a relatively narrow audience (we should try to make them as broadly accessible as practical). Less-technical readers can easily skip past parts with advanced prerequisites, but more-technical readers often need to look up or find out about advanced material and Wikipedia is a fine place for it. –jacobolus (t) 20:10, 7 December 2023 (UTC)

it is not "self-evidently incorrect and therefore immediately dismissible" which assumes bad faith on your part.

The rest of what you have said after this is do not quite fathom as it appears to be a non-sequitur, and more personal comments, which are again not relevant and seem to be cherry-picking of sections which you believe make things accessible, but It could equally not be something which makes smoothing accessible. This is user bias on the part of the examples listed. PicturePerfect666 (talk) 21:27, 7 December 2023 (UTC)

No, the article doesn't read "like an academic paper". It has its problems, but that's not one of them. XOR'easter (talk) 18:36, 7 December 2023 (UTC)

Please explain how it does not do what you are stating it does not do, simply saying "it does not", does not address the concerns raised. also if the article has other problems please expand on them so this article can be improved, don't be secretive or shy about the problems you think the article has. PicturePerfect666 (talk) 19:20, 7 December 2023 (UTC)

I can't "address" concerns that make no sense. Read academic papers, and read this article; the experience is not the same. (On the flipside, if you don't have experience reading them, then don't say it resembles one.) XOR'easter (talk) 19:26, 7 December 2023 (UTC)

You have stated above "It has its problems" - what are these problems you have identified then? I cannot read your mind to know what they are unless you state what they are. PicturePerfect666 (talk) 19:28, 7 December 2023 (UTC)

I'm not sure why you're going for such a confrontational tone; that won't actually help anybody. The article is already tagged with needing citations on multiple specific places, which is more helpful than slapping banners across the top of it or griping here. XOR'easter (talk) 19:31, 7 December 2023 (UTC)

If you believe I am being confrontational that is not my intention here and not the tone being aimed for...but remember to assume good faith. If you think that I am not acting in good faith then please consider why you believe this and re-evaluate, as I can assure you my actions are exclusively in good faith. I am simply asking for you to state what your concerns with the article are. You have said you think the article has problems, yet won't state what these are, so no one else can attempt to improve the issue you see the article as having. I may have used direct language, but this is because you won't tell anyone what you see the issues with the article are. Please share what you think the issues with the article are. No one here can read your mind. — Preceding unsigned comment added by PicturePerfect666 (talk • contribs)

No one here can read yours. You opened this thread; it's on you to say what you mean. (I have already pointed to one obvious problem.) What exactly are you finding too difficult to understand? Remember, this topic will naturally have a gradation in difficulty: some aspects can be understood by schoolchildren, while others might not be encountered until university. Ideally, the opening sections should be more widely comprehensible than the more demanding ones that follow, while the paragraphs before the first section break need to summarize everything that follows. XOR'easter (talk) 19:42, 7 December 2023 (UTC)

Indeed, as best as I can follow, your complaints are self-contradictory. How is the article supposed to be more accessible "for the reader with limited initial knowledge of the topic" without giving basic examples to help such a reader? XOR'easter (talk) 19:48, 7 December 2023 (UTC)

I am going to disengage from the discussion with you as it has become circular. Others are making improvements to the article and you will not state the issues you think the article has. I have provided examples above as to the issues I have and this discussion with yourself and myself is no longer helpful to the improvement of the article in its current form. If you would like to continue discussions please do so by stating clearly the issues you have with the article as opposed to assuming bad faith on my part and essentially attacking the content I have [put above in a manner which is unconstructive. Also please refrain from patronising statements such as the following "This topic will naturally have a gradation in difficulty: some aspects can be understood by schoolchildren, while others might not be encountered until university" as it assumes the person you are discussing with is an idiot, which is an assumption of bad faith and potentially a personal attack. Please withdraw or reword that section. PicturePerfect666 (talk) 19:46, 7 December 2023 (UTC)

@PicturePerfect666 about half of this article has been substantially rewritten since your previous comments. Did that address your concerns? –jacobolus (t) 18:56, 22 January 2024 (UTC)

List of general references tag

The end of the article contains a simple list of "Sources" which are not referenced in the article. It is therefor challenging to know what these refer to specifically or how they can be scrutinised as verifiable or simply padding. This needs to be updated and improved. PicturePerfect666 (talk) 18:03, 7 December 2023 (UTC)

Please feel free to read the article and hunt for sources about each section and/or read the provided sources to figure out which specific part each one discusses. If you are not willing to do that, then just pasting an eyesore banner that says "someone else please fix this for me" is not accomplishing anything except distracting and annoying readers. –jacobolus (t) 18:08, 7 December 2023 (UTC)

Not being a mathematician, this is like asking a Chinese Person who has never encountered Arabic or English to translate into English the Qur'an. Not everyone has the technical expertise to wade through the density of mathematics in this article. This is a very technical and dense topic and one I am not even remotely qualified at being able to decipher here on this platform or wade through even denser published research papers. PicturePerfect666 (talk) 19:24, 7 December 2023 (UTC)

Why would a Chinese person be trying to tell English speakers that they need a big disclaimer about the poor quality of the English translations of the Qur'an? To the extent this were really true that this article was totally over your head and conceptually impossible for you to make any sense of, you should probably go work on some other one and leave it to experts.

But frankly this is a misleading analogy. The most important parts of this article needing the most improvement are or should be accessible at a high school level, and are related to mathematical history, pedagogy, philosophy, and the most basic structures taught to schoolchildren and ultimately very formally specified in introductory courses for undergraduate students requiring no special prerequisites. You could certainly read and follow many relevant sources, and you shouldn't declare yourself a priori incapable of understanding them. –jacobolus (t) 19:54, 7 December 2023 (UTC)

The analogy is to describe the difficulty of understanding the complexities in this article. It is an analogy not something to be taken literally. Please also stop going on about education levels it is not relevant to the discussion. i have also not "declare yourself a priori incapable of understanding". i have simply said it assumes a lot of knowledge as opposed to be accessible from the ground up. Wikipedia article shroud not rely on users going to complex and dense academic papers or specialist resources to understand an article. This is what I am saying this is not an article for mathematicians, it is a general access encyclopedia article. It also assumes that all mathematics globally is taught in the same way, which is a centric-view of wherever the person asserting the claim of this is at this level and this at this level etc. A form of unconscious bias if you will. Please stop making assumptions as this is not benefiting the clarity or understandability of the article. PicturePerfect666 (talk) 21:23, 7 December 2023 (UTC)

Wikipedia [articles should] not rely on users going to complex and dense academic papers or specialist resources to understand [them]. This particular article doesn't require familiarity with "dense academic papers" to follow, though some sections assume familiarity with introductory textbook material about various topics, some of which are relatively advanced. But even if it did, Wikipedia articles should cover their subject reasonably comprehensively, including advanced material, while staying as accessible as practical in each section. Sometimes arcane and difficult prerequisites are necessary; we can't include the multiple years' worth of technical coursework in the middle of every article which would be necessary to make every part completely accessible to a lay audience. What we can do is cover the lay-accessible parts of the subject as clearly and completely as we can, organize them toward the top of the article if possible, and then strive to make more advanced parts accessible to the broadest audience we can without going excessively off topic to do so. –jacobolus (t) 23:43, 7 December 2023 (UTC)

Lead section

I'm trying to rewrite the lead section for clarity, but I don't think I've done an amazing job so far. Does anyone want to take another crack at it, or maybe polish up some of my sentences? For example, can someone improve this vague sentence I wrote without making it too long or technical? "When a real function involves division by a quantity that can become zero for some values, that is a type of singularity." @XOR'easter? –jacobolus (t) 19:47, 7 December 2023 (UTC)

Is that sentence necessary? I don't think the body text follows up on it in any serious way (unlike the concept of a limit in the next sentence). XOR'easter (talk) 20:03, 7 December 2023 (UTC)

Similarly, I think we should try to get the historical coverage right in the main text and then do a summary in the lead, rather than polishing lines in the lead that aren't elaborated upon later. XOR'easter (talk) 20:10, 7 December 2023 (UTC)

I like discussing the limit of $1/x$ in the intro before bringing up indeterminate forms. XOR'easter (talk) 21:03, 7 December 2023 (UTC)

"Analysis" section

I think this section would be better titled something like "other number systems", since the focus is the extended real numbers and the projectively extended real numbers, and the material about Wheel theory could be moved there.

It should IMO be preceded by a section (perhaps titled "Calculus") discussing the treatment in calculus and real analysis of the limits of functions with real numbers as the domain/codomain, where infinity is often treated as a limit but not a number per se. This section would discuss infinite singularities, and possibly contrast them with other kinds of singularities, and would then include material about indeterminate forms and L'Hospital's rule.

It might be worth adding a section about complex analysis and the idea of zeros and poles and meromorphic functions. –jacobolus (t) 23:20, 7 December 2023 (UTC)

I think "Calculus" is a more illuminating section heading than "Analysis". The former refers to a specific math subject (and even people who haven't studied it have heard of it as a math thing), whereas the latter has a general meaning of "thinking about" that could be confusing in this case. XOR'easter (talk) 00:12, 8 December 2023 (UTC)

Riemann sphere para

In regards to Jacobulus's edits, latest here: This article is about division by zero. Therefore acknowledgment that there is sometimes such a thing as division by zero should not be deferred to the fourth paragraph. I am also not sure why Jacobulus preferred to speak of the projectively extended real line rather than the more-used Riemann sphere (when the real line is extended it is more common to add a +∞ and a −∞, which doesn't allow division by zero because it isn't clear which one to pick). --Trovatore (talk) 00:22, 8 December 2023 (UTC)

(As an aside, the "Riemann sphere" is a confusing name to use for the extended complex numbers, and in my opinion extended complex numbers and Riemann sphere should be split into two separate articles, since the former number system is also commonly used to model flat and hyperbolic phenomena, not only spherical ones. But that discussion is off topic here.) The reason I think the projectively extended real numbers rather than extended complex numbers should be used as an example in the lead section here is that they are simpler, approximately as widespread, easier to draw pictures of, and don't involve as many conceptual prerequisites to make sense of. I absolutely think we should discuss both the affinely extended real numbers, the projectively extended real numbers, and the extended complex plane in this article about division by zero, along with a more extensive discussion of which types of geometric phenomena and other more abstract situations they are each good models for. (As another aside: there is also a reasonable complex analog of the affinely extended real numbers: the complex logarithm should arguably have the extended complex numbers as its domain, and a complex cylinder as a codomain, where $\log(\infty )=+\infty$ and $\log(0)=-\infty ,$ two distinct values at opposite infinite ends of the real axis.) –jacobolus (t) 00:40, 8 December 2023 (UTC)

Confusing or not, I'm pretty sure Riemann sphere is the usual name. The reason to call it a "sphere" is that the structure itself is topologically a sphere, not that you'd apply it on a sphere. I really don't think the one-point compactification of the reals is nearly as used (or useful).

In any case I don't think this material should be pushed so far down. --Trovatore (talk) 00:46, 8 December 2023 (UTC)

It's called the "Riemann sphere" because stereographic projection provides a bidirectional link between spherical geometry and complex analysis. For example it is common to adopt chord length or arc length on the sphere as an alternative metric for measuring distance between complex numbers, or in the other direction to use complex numbers and their arithmetic as a representation of points on the sphere so that the geometry of spherical objects can be treated numerically. The topological aspect is only one part of that.

don't think the one-point compactification of the reals is nearly as used it is used ubiquitously throughout most areas of mathematics and applications to science and engineering, but as a quirk of history is unfortunately rarely explained in a clear or systematic way. –jacobolus (t) 00:55, 8 December 2023 (UTC)

Hmm, I think you have an origin story for the name that I have never heard. It's shaped like a sphere; that's a good enough reason to call it a sphere, without needing to make any detailed connection to spherical geometry.

I'd need to see examples for your claim about the one-point compactification. Usually when you're working in the reals you know whether things are positive or negative. (In fact the structure [0, +∞], which is fundamental to measure theory, is in my experience more useful than the "projectively extended reals"; unfortunately it doesn't seem to have a snappy name.) --Trovatore (talk) 01:01, 8 December 2023 (UTC)

shaped like a sphere – yes, exactly, it's shaped like a sphere. For example, the natural setting for "Möbius transformations" (which, contra Wikipedia's article, should fundamentally be defined geometrically, as general transformations generated by reflections and circle inversions) is the "inversive plane" a.k.a. Möbius plane, which can be naturally modeled either by the Euclidean plane with a single extra point at infinity or by the 2-sphere; as it happens the inversive geometry of these, because all of the relevant relationships are preserved by the stereographic projection, is the same. If we want to represent Möbius transformations numerically, one natural representation is using complex numbers.

I'd need to see examples for your claim For example, any time you see the trigonometric tangent function (or in general the slope of a geometric line) you are properly dealing with the projectively extended real numbers, which is the natural codomain there. –jacobolus (t) 04:05, 8 December 2023 (UTC)

About the ordering: I think it's clearer to roughly order the article (and the lead) in an order from less to more advanced material, which means talking about the conventional number system taught to every high school student first, including its appearance in introductory calculus, and then talk about relatively niche alternative number systems not seen until the middle of an undergraduate pure math degree until after that. I also was aiming to make the lead section as legible and jargon-free as I could. Maybe this paragraph could be rephrased to make the alternative number systems seem more important or to explicitly mention more of them though. –jacobolus (t) 00:46, 8 December 2023 (UTC)

@Trovatore: Maybe it would be clearer for this paragraph to lead with something like, "As an alternative to the convention of leaving division by zero undefined, ...". –jacobolus (t) 09:22, 9 December 2023 (UTC)

@Jacobolus: Just looked at this again, and I still think it's entirely unacceptable to defer discussion of when division by zero makes sense to the fourth paragraph. I hope we can come to an agreement here. --Trovatore (talk) 17:20, 5 January 2024 (UTC)

Do you have a concrete idea for how the lead should go? Feel free to paste a draft in here, in a subpage of this talk page, in user space, or the like. (Or, as always, boldly edit the article.)

The reason I think this is fine is because division by zero is typically treated as undefined / nonsensical in most of mathematics (and pretty much universally at the high school / early undergraduate level), and it's worth being clear about that in this article and its lead. I would not expect readers to be confused or misled by the current text. [I'm not against division by zero. The projective real numbers are one of my main mathematical interests.] –jacobolus (t) 19:47, 5 January 2024 (UTC)

I think the ordering of the lead is pretty much fine as it stands now, on the general philosophy that stuff one doesn't see until the later years of a math major can wait until after the basic arithmetic. XOR'easter (talk) 21:18, 5 January 2024 (UTC)

I still think the structures in which the topic of the article makes sense are not prominent enough. It might be a solution to expand the first paragraph with a sentence that says that in the most usual contexts, the expression is undefined, but that contexts in which it makes sense do exist. Then we can elaborate later in the lead section. --Trovatore (talk) 21:29, 5 January 2024 (UTC)

Can you make a concrete proposal? I was trying to do more or less what you are suggesting, but at a bigger scale: start with describing why division by zero is considered undefined for real numbers, then later discuss alternative structures where it is defined, and then elaborate on that further down the article. –jacobolus (t) 21:49, 5 January 2024 (UTC)

One thing that might be helpful could be rewriting the "meaning of division" section for clarity, and expanding it to include other kinds of places where division (or ratio) is used, including those where a ratio like 1 : 0 can be meaningfully interpreted. –jacobolus (t) 21:53, 5 January 2024 (UTC)

I'm thinking something like adding two sentences to the end of the first para, so it reads something like:

In mathematics, division by zero, division where the divisor (denominator) is zero, is a unique and problematic special case. Using fraction notation, the general example can be written as ${\tfrac {a}{0}}$, where $a$ is the dividend (numerator). In ordinary arithmetic, for example on the real numbers, the result of this operation is undefined. However, there are other mathematical contexts in which division by zero is allowed.

Then we could probably trim some of the language in the second para as redundant. Later in the lead, I would prefer to speak explicitly of structures rather than just conventions, but that's a second-order point. --Trovatore (talk) 07:13, 6 January 2024 (UTC)

I think a turnabout in the very first paragraph would confuse more people than it would help. XOR'easter (talk) 15:40, 6 January 2024 (UTC)

Not sure what you mean by a "turnabout". I think it is essential to acknowledge in the first paragraph that the topic of hte article does sometimes make sense. --Trovatore (talk) 20:30, 6 January 2024 (UTC)

@Trovatore nearly every mathematics source aimed at laypeople or students for the past ~100 years says something like "in mathematics, division by zero is not allowed, because it leads to absurdities". It's potentially more confusing if we lead by saying "In mathematics, division by zero is sometimes not allowed, but sometimes it is". –jacobolus (t) 20:38, 6 January 2024 (UTC)

That's just going to confound people who have barely grasped the real numbers and never gone beyond that, i.e., most people who are looking up the topic of "division by zero". XOR'easter (talk) 20:38, 6 January 2024 (UTC)

This sounds like you're trying to make things easier for middle-school math teachers. That's not what we do. We don't tell lies to children. The first paragraph should give a fair top-level summary of the whole situation. --Trovatore (talk) 20:43, 6 January 2024 (UTC)

First, making life easier for middle-school math teachers is a good idea. Heaven knows they've got it hard enough. Second, it's not a "lie to children" to hold off on the projective real line and the Riemann sphere until the fourth paragraph. XOR'easter (talk) 21:26, 6 January 2024 (UTC)

It's something like a "lie to children by omission" if you let people read the first para or two and come away with the impression "you just can't divide by zero, period", because that's just not true.

As for the teachers, sure, they do a hard job and should be valued for it, but it sounds like you're worried about a scenario where Jenny raises her hand and says, "Mr. Simmons, you told us you can't divide by zero, but I looked it up in Wikipedia and it says sometimes you can, and now I'm confused".

Well, that's productive confusion. It's not our job to try to blunt the confusing nature of reality for the benefit of learners. Maybe Jenny will be bothered enough by it that she'll go and learn something on her own. Or maybe not; either way it shouldn't affect our writing. We should summarize the whole picture early in the article, while leaving the details for later. --Trovatore (talk) 21:41, 6 January 2024 (UTC)

If someone can't be bothered to read a few short paragraphs, I can't see how a vague half-sentence is going to help them come to a better understanding. –jacobolus (t) 21:42, 6 January 2024 (UTC)

My objection is that the omission reinforces the false claim, which as you note is standardly taught to youngsters, that division by zero is never allowed. --Trovatore (talk) 21:45, 6 January 2024 (UTC)

From what I can tell the current lead section doesn't make any false claims. It just first discusses arithmetic of fields, and then later discusses alternative arithmetics. –jacobolus (t) 21:49, 6 January 2024 (UTC)

That's why I said "by omission". Lies by omission aren't technically lies at all, but we still should want to avoid them. --Trovatore (talk) 21:51, 6 January 2024 (UTC)

Nothing's being omitted. It's just being organized in a way that's probably somewhat better for the most likely audience. XOR'easter (talk) 21:58, 6 January 2024 (UTC)

It's not better if they can come away thinking division by zero is never allowed. --Trovatore (talk) 21:59, 6 January 2024 (UTC)

One reason I think it's okay to hold off is that arguably treating projective quantities as representing "division" per se is already an abuse. Really the desired concept here is ratio, and what we want is a ratio 1:0, not a "number" ∞, but division has been shoehorned in because modern mathematics has all but done away with ratio as a separate concept, especially at the high school / undergraduate level, instead putting all its eggs in the "real numbers" basket. This streamlining is a trade-off: fewer overlapping abstractions, but not quite so well tailored to the specific circumstance. –jacobolus (t) 21:41, 6 January 2024 (UTC)

I mean, that's a possible perspective; I'd have to think about it more. In any case the sources do call it division. --Trovatore (talk) 21:47, 6 January 2024 (UTC)

@Trovatore I added a bit about slopes of lines in the "meaning of division" section. Does that help at all? –jacobolus (t) 20:36, 8 January 2024 (UTC)

Not for the point I'm trying to get at, which is that the possibility of division by zero making sense should be addressed early in the lead. --Trovatore (talk) 20:40, 8 January 2024 (UTC)

Overlooked proof

Let imagine a graph (1/x) is divided into 2 halves where the negative half is x<0 and the positive half is x>0. The primary proof against 1/0 being a defined value is that the two halves directly contradict one another (negative half shows -∞ but positive half shows ∞) so it must be an undefined value right? No, so lets change our way of thinking. We will be adding a new rule which states: "Any division that might return a negative result shall be written in a way in which a negative number will never be dividing another or divided against.". Since this only applies to HOW we write down or calculate the equations, this does not change anything ((wrong) 2÷-5 = -0.4 --> (correct) -(2÷5) = -0.4). By applying the rule, we would never actually divide by the negatives which completely removes the negative half. This will leave only the positive half with nothing contradicting it, which could only mean x/0 = ∞.

I don't study math like a mathematician but this is easy. I need yall to say im wrong rn holy moly. Also the rules are made ambiguous, think of the best case scenario of my explanation as the primary one. SussusMongus (talk) 04:52, 16 December 2023 (UTC)

It's not "wrong" per se. You are welcome to define your own new number system that behaves however you want. It's just that most mathematicians and other technical professionals have settled on a few conventional choices of number system which were found to be the most convenient in the broadest class of existing uses, and your proposed system is not one of them. (Your number system is quite limiting in that you have to constantly check that you aren't dividing by negative numbers, which is even more burdensome than checking to not divide by zero, and there are many kinds of algebraic manipulation that have the potential to introduce illegal divisions which you will have to either disallow or be very careful about. Many of the most convenient and useful tools of algebra are going to be made much more troublesome, if they aren't entirely broken.)

I'd recommend you take this kind of question to Wikipedia:Reference desk/Mathematics, since it's not really about improving this article per se. –jacobolus (t) 05:05, 16 December 2023 (UTC)

I am going to reply since i already posted this and the rule actually NEVER specified about when you check, you just need to know that before you actually divide it. If it's a variable then just look if it have a sign. Afterall, the variable itself might be negative but the value might NOT be negative (or vise-versa) and the rule never specified any of that. This reply is the primary example of why I left the phrase "The rules are made ambiguous" there (i knew my initial post isnt going to be accurate at conveying what i literally meant, especially the little details). SussusMongus (talk) 06:28, 19 December 2023 (UTC)

Note: SussusMongus copied this discussion over to Wikipedia:Reference_desk/Mathematics#Overlooked proof. Seems fine to let the conversation live over there.–jacobolus (t) 07:26, 19 December 2023 (UTC)

0/0 = 0

if a/b=c then cb=a, right? then, 0/0=0 as 0*0=0. anyone find any sources for this?49.37.202.122 (talk) 16:07, 5 January 2024 (UTC)

I realise my mistake. 0/0 is 0, 1, i, and every number. 49.37.202.122 (talk) 16:14, 5 January 2024 (UTC)

going from cb = a to a/b = c is only valid if b<>0. Dhrm77 (talk) 16:15, 5 January 2024 (UTC)

rewritten "meaning of division" section

I split this section up and rewrote the first part (the second half, now titled 'Inverse of multiplication') still needs a rewrite). Does it make sense to folks? This kind of informal discussion about concrete interpretations of division and the way zero might fit in seems valuable to me to include up front, but I'm just one person here. Hopefully it doesn't seem like belaboring the point or getting to far into the weeds. I'll try to find and insert a few relevant reliable sources if I can. –jacobolus (t) 01:41, 7 January 2024 (UTC)

@XOR'easter (or anyone else reading along) do you mind giving the first half of this article another once-over? I tried to add some more accessible examples and informal discussion, but I can't promise it makes more sense to anyone other than myself. –jacobolus (t) 01:40, 10 January 2024 (UTC)

Looks OK to me. XOR'easter (talk) 03:23, 10 January 2024 (UTC)

A reason why "Division is the inverse of multiplication" is an incorrect sentence

Multiplication of two real numbers $x$ and $y$ is a function $f:\mathbb {R} ^{2}\to \mathbb {R}$. Does it have the inverse? No, because it's not a bijection.

That being said, we can define a function, say, $f(x)=2x$, which is a bijective function $f:\mathbb {R} \to \mathbb {R}$ and thus it has the inverse $f^{-1}(x)={\frac {x}{2}}$, but the function $f(x)=2x$ is not a multiplication (of two numbers), it's a multiplication by a specific number.

I hope this explains why we can't say "Division is the inverse of multiplication", because this sentence doesn't specify the specific number. For example, we could say something like this: "Division by 2 is the inverse of multiplication by 2", but I haven't seen such usage (it's much simpler just to write the function in such cases). — Preceding unsigned comment added by Robertas.Vilkas (talk • contribs) 21:35, 24 February 2024 (UTC)

The word "inverse" in this context is not being used to mean the same as "inverse function". Calling division the "inverse operation" as multiplication is widely accepted by reliable sources, including by mathematicians, scientists, schoolteachers, mathematics education researchers, etc. If anyone cared enough the term could be precisely formally defined. I think it's fine to just use the plain-English meaning; no claims here depend particularly strongly on making the term's definition precise. Edit: however, I added a quick inline definition. –jacobolus (t) 22:17, 24 February 2024 (UTC)

Division is the inverse of multiplication in the same way that subtraction is the inverse of addition. This language is standard. Rick Norwood (talk) 11:01, 25 February 2024 (UTC)

Pretty standard indeed:

Google	Scholar	Books
division "inverse of multiplication"	1250	8990

- DVdm (talk) 11:38, 25 February 2024 (UTC)

Calculus paragraph

I am hoping someone can fix the following issues in the intro paragraph starting with "Calculus":

It's not correct that a positive ratio of functions whose denominator tends to 0 tends to infinity, because the numerator could be going to 0 faster.

There is no reason to require a "positive fraction". (It would even be OK for it to be complex; one just needs the denominator to be nonzero in a punctured neighborhood. Can someone figure out a nice way to say this without getting bogged down in details?)

We should avoid conflating "becoming arbitrarily large" and "tending to infinity". The function sin(1/x)/x as x approaches 0 becomes arbitrarily large but does not tend to infinity.

It is 0/0 that is the indeterminate form, not the quotient of functions. So the last sentence of the paragraph needs to be rewritten. Ebony Jackson (talk) 21:31, 9 March 2024 (UTC)

@Ebony Jackson I don't think it's all that important to make these two sentences completely airtight as far as exact correctness is concerned. We're just trying to introduce the subject, not make a full formal specification. It's probably fine to also cut the note about sign expressing ${\textstyle f(x)\to \pm \infty }$ as $x\to c$ as "tends to infinity" (we can leave "from the right" in the image caption). I think it's entirely fine to conflate this with "arbitrarily large", since we are talking in an informal loose way here. Likewise, specifying that the numerator can't tend to zero is wordier and more confusing (than just not mentioning the numerator) without really much benefit in my opinion. We can be more precise in the dedicated section below. I don't think there's any advantage in mentioning complex numbers in this paragraph. –jacobolus (t) 22:42, 9 March 2024 (UTC)

If you want to try to add a caveat in § Calculus about limits of fractions with functions in the denominator which tend to zero but whose sign continues to oscillate even in arbitrarily small intervals, so that in the affinely extended real numbers there is no well defined limit, that would be fine with me, though also doesn't seem totally necessary; I'm generally not that excited about pedantically emphasizing the most unusual obscure counterexamples, though I know some mathematicians really enjoy it. I expect anyone who is curious about this is going to be able to find their way to e.g. essential singularity. –jacobolus (t) 22:50, 9 March 2024 (UTC)

The second sentence of the paragraph currently claims, for example, that ${\frac {x^{2}-4}{x-2}}$ tends to infinity as $x\to 2$. Surely we would want to correct this? This is not some obscure counterexample! WP:PROVEIT requires an inline citation to a reliable source for questionable statements, whether or not they are in the lead, and we're not going to find a reliable source supporting false statements like this. Ebony Jackson (talk) 04:26, 10 March 2024 (UTC)

It emphatically does not currently claim that. That is an aggressively pedantic intentional misreading. (There's even a clarification about precisely this case in the immediately following sentence!) –jacobolus (t) 04:47, 10 March 2024 (UTC)

Hi, sorry that we don't seem to be understanding each other yet. Just to make sure we are talking about the same sentence: I am referring to

"When a real function can be expressed as a fraction whose denominator tends to zero, the output of the function becomes arbitrarily large, and is said to tend to infinity, a type of mathematical singularity."

Are you arguing that it would be unreasonable for a reader to think that the hypothesis

"When a real function can be expressed as a fraction whose denominator tends to zero"

applies to a function such as ${\frac {\sin x}{x}}$ as $x\to 0$ (to give another example)?

I do see that two sentences later, there is a separate statement about a quotient of functions both tending to zero, but as written it doesn't seem clear that it is meant to restrict the applicability of the earlier statement above. Ebony Jackson (talk) 08:44, 10 March 2024 (UTC)

No, I am arguing that: (a) The immediately following sentence which you keep cutting out of your quotation/discussion directly clarifies this point. Here's the whole paragraph in its current form:

Calculus studies the behavior of functions in the limit as their input tends to some value. When a real function can be expressed as a fraction whose denominator tends to zero, the output of the function becomes arbitrarily large, and is said to "tend to infinity", a type of mathematical singularity. For example, the reciprocal function, $f(x)={\tfrac {1}{x}},$ tends to infinity as $x$ tends to $0.$ The quotient of two functions which both tend to zero at the same input is called an indeterminate form, as the resulting behavior depends on which functions are being considered.

Maybe the last sentence could be reworded to more closely parallel the first though, e.g. "If both the numerator and denominator of the fraction tend to zero at the same input, ..."

I am also arguing that (b) making a fully precise statement here is going to be awkward and excessively detailed for the context of this lead-section summary, and readers trying to understand what this sentence means are entirely capable of looking a few sections down to a (hopefully) clearer and more precise description in the relevant topical section. Your replacement version wasn't in my opinion appropriate to the full intended range of the audience of this article, which could plausibly include e.g. middle school students. –jacobolus (t) 16:31, 10 March 2024 (UTC)

I've reworded this to:

Calculus studies the behavior of functions in the limit as their input tends to some value. When a real function can be expressed as a fraction whose denominator tends to zero, the output of the function becomes arbitrarily large, and is said to "tend to infinity", a type of mathematical singularity. For example, the reciprocal function, $f(x)={\tfrac {1}{x}},$ tends to infinity as $x$ tends to $0.$ When both the numerator and the denominator tend to zero at the same input, the expression is said to take an indeterminate form, as the resulting limit depends on the specific functions forming the fraction and cannot be determined from their separate limits.

Is that any better? –jacobolus (t) 19:40, 10 March 2024 (UTC)

Seems to be an error.

In the "Elementary arithmetic" section; "The meaning of division" - there is, and I quote: "Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 1:0}" - I suppose a mathematical function was put there, but has failed for reasons beyond my comprehension. I hope someone may be able to fix it? Or was it only on my browser? I used two browsers and both showed this error.

Althuios (talk) 14:58, 29 April 2024 (UTC)

These errors are transient, and there's nothing editors can really do about them. –jacobolus (t) 15:15, 29 April 2024 (UTC)

0/0 is not undefined

In reference to the last sentence in the 2nd parapraph, 0/0 is not technically undefined. It fits the definition of division given in the beginning of the 2nd paragraph. It is indeterminate which is just as bad. Unisonshock (talk) 21:51, 3 October 2024 (UTC)

The expression 0/0 is indeed "technically undefined" in many (most?) contexts, especially in the context of school mathematics which is the context most relevant to many of this article's readers. There are contexts where division by zero, or even 0/0, has a definition, for instance in wheel theory. If you e.g. try putting 0.0/0.0 into a computer program based on IEEE floating point arithmetic, the result will be not a number. Our article also already discusses the concept of indeterminate forms when taking limits. Saying "0/0 is indeterminate, not undefined" is not an a priori meaningful statement, and does not seem like a useful distinction to draw in the lead section of this article; instead, it's likely to just confuse readers, especially novices. (You also didn't provide a source for the claim.) I reverted the change. –jacobolus (t) 22:15, 3 October 2024 (UTC)

0/0 is defined by the definition given earlier in the same paragraph. That's the source. 170.64.78.10 (talk) 22:46, 3 October 2024 (UTC)

That's fine, but expressions like "30/2", "6/8", or "0/5" are well-defined in the sense that they each represent a particular rational number, whereas the expressions "4/0" and "0/0" are not well-defined in that sense, and are thus both called "undefined" by common sources (e.g. elementary algebra textbooks, papers in math teachers' journals, introductory analysis textbooks). This article devotes quite a bit of effort to elaborating about various contexts in which these expressions or concrete situations they represent might be absurd or sensible, and why. Making a distinction between "undefined" vs. "indeterminate" is not something common in sources, and in my opinion does not seem helpful to readers in the 2nd paragraph. –jacobolus (t) 22:55, 3 October 2024 (UTC)