Talk:Division by zero

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

That being said, we can define a function, say, ${\displaystyle f(x)=2x}$, which is a bijective function ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ and thus it has the inverse ${\displaystyle f^{-1}(x)={\frac {x}{2}}}$, but the function ${\displaystyle f(x)=2x}$ 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:

More information Google, Scholar ...

Google	Scholar	Books
division "inverse of multiplication"	1250	8990

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

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 ${\displaystyle x\to c}$ 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 ${\displaystyle {\frac {x^{2}-4}{x-2}}}$ tends to infinity as ${\displaystyle x\to 2}$. 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 ${\displaystyle {\frac {\sin x}{x}}}$ as ${\displaystyle x\to 0}$ (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, ${\displaystyle f(x)={\tfrac {1}{x}},}$ tends to infinity as ${\displaystyle x}$ tends to ${\displaystyle 0.}$ 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, ${\displaystyle f(x)={\tfrac {1}{x}},}$ tends to infinity as ${\displaystyle x}$ tends to ${\displaystyle 0.}$ 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)

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)

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)

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)