Talk:Division by zero

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)

Illogical math

The fourth paragraph starts out:

As an alternative to the common convention of working with fields such as the real numbers and leaving division by zero undefined, it is possible to define the result of division by zero in other ways, resulting in different number systems. For example, the quotient ⁠can be defined to equal zero... Literally, this means that there are 7, not 6, factors of 12; 0, 1, 2, 3, 4, 6, and 12. Any way to make this statement more flexible?? Georgia guy (talk) 11:09, 7 September 2025 (UTC)

this is the only way it could have ended Suprememachinedetonator (talk) 14:55, 21 October 2025 (UTC)

Quotient equal to zero

I haven't checked when this was added, but before my recent edit, the article claimed that the quotient of a division by zero could be defined to be equal to zero. This is true in the rather useless sense that you can make a definition to point to anything you like. But it should not be put on the same level as the sometimes-useful expedient of adding a point at infinity.

On another note, the lead should emphasize adding a point at infinity much more than it currently does. Right now it puts it off till the fourth paragraph. This is the main way in which division by zero can be given a meaning; since division by zero is the topic of the article, adding a point at infinity should be treated much more prominently. --Trovatore (talk) 18:51, 2 May 2026 (UTC)

I found the bit in the article that this may have been attempting to summarize. It's the bit about the proof assistants. This seems to be more of a coding workaround than anyone being interested in the actual structure that you get by defining 1/0=0. I don't think that deserves mention at that part of the lead, though it could go in a separate sentence talking specifically about computing. --Trovatore (talk) 19:11, 2 May 2026 (UTC)

Some people in theorem provers or the like apparently use division by zero equal to zero. Division by zero resulting in a signed infinity is also common. I agree that a single point at infinity is useful and important, but there was much discussion about this before and the compromise decision was that we should begin with an emphasis on the real number system where division by zero is undefined, since this is most likely to be relevant to a broad part of the audience, and reflects the dominant claim in mathematical books aimed at a novice or general audience. If you like, you can think of these three paragraphs as something like (1) primary school, (2) high school, (3) university and beyond. –jacobolus (t) 19:14, 2 May 2026 (UTC)

We discussed this point before at Talk:Division by zero/Archive 2 § Riemann sphere para. –jacobolus (t) 19:16, 2 May 2026 (UTC)

The theorem provers seem to be using it as essentially a workaround, not as a mathematical structure they're interested in. That's really not on the same level as the extended real or extended complex numbers. At the very least the two things should be put in different sentences, making clear that the "zero" answer is for computational purposes.

I don't really agree with your characterization of the previous discussion as a "decision". It was two against one and I got tired of arguing. That said I am not sure how exactly to make point-at-infinity more prominent. Possibly it should go ahead of the paragraph on limits. --Trovatore (talk) 19:28, 2 May 2026 (UTC)

There was also discussion about the proof assistants at Talk:Division by zero/Archive 2 § Sources for proof assistants. I don't think the folks involved would characterize this as a "coding workaround", or at least not in any way fundamentally different from leaving x/0 undefined is a "definition workaround". –jacobolus (t) 19:35, 2 May 2026 (UTC)

@Lambda Fairy maybe you can try to convince us again that this is worth discussing? Personally I'm ambivalent. –jacobolus (t) 19:36, 2 May 2026 (UTC)

I'm OK with discussing it, just not with putting it on the same level as the extended real or complex numbers. Hopefully my recent edit works. --Trovatore (talk) 19:40, 2 May 2026 (UTC)

Seems fine. –jacobolus (t) 19:47, 2 May 2026 (UTC)

• Jacobolus, please see the reference desk post that I made that brought Trovatore here. Georgia guy (talk) 19:29, 2 May 2026 (UTC)