Talk:Factorial

The statement "There are infinitely many ways to extend the factorials to a continuous function." is true, but trivial and rather useless. As the cited source states, This is ridiculously easy to solve. [...] Merely take a pencil and draw some curve—any curve will do—which passes through the points. Such a curve automatically defines a function which solves the interpolation problem. This is of course true of any set of discrete points. The interesting part is that there are infinite ways to do it within certain confines, most importantly while still satisfying the recurrence relation. I therefore added the qualifier "that satisfies the recurrence relation ${\displaystyle f(x+1)=(x+1)\cdot f(x)}$ for non-integer values", which was reverted on the basis that it would exclude an interpolation based on Hadamard's gamma function. That, to me, misses the point. It also excludes interpolating the points linearly or indeed arbitrarily. I suppose it would be possible to rephrase it to make both statements at once (e.g. "There are infinitely many ways to extend the factorials to a continuous function, which remains true if the resulting function ${\displaystyle f}$ is required to satisfy the recurrence relation ${\displaystyle f(x+1)=(x+1)\cdot f(x)}$ for non-integer values."), but I don't think the bare statements that we have now is satisfactory. TompaDompa (talk) 11:36, 19 August 2023 (UTC)

The point is that you have to constrain things somehow to be able to say that Gamma is the canonical interpolation. Your edit adds half the constraint, turning that thought from "you have to constrain things" to "we have already constrained things but you have to constrain more things". What is so natural about that choice? Why not instead start with the other half of the Bohr–Mollerup theorem, and only consider log-convex functions? Or why not leave the constraints out of it until they are needed for uniqueness? I'm not embarrassed to say trivial things when they're relevant. Not every statement in our mathematics articles has to have deep reasoning behind it. —David Eppstein (talk) 13:02, 19 August 2023 (UTC)

The recurrence relation is a fundamental and defining property of factorials—it's what makes factorials factorials, so to speak. When I first heard of interpolating factorials many years ago, I took for granted that the recurrence relation would hold for the non-integer relations, because I thought that without that property it wouldn't be much of an interpolating function. The non-uniqueness of the Gamma function in this regard is, I think, very important. TompaDompa (talk) 16:07, 19 August 2023 (UTC)

Hadamard's Γ obeys a form of the recurrence relation. But it is a form with an extra term that happens to be zero on the positive integers. —David Eppstein (talk) 07:35, 20 August 2023 (UTC)

I must admit that I don't understand what you're getting at. TompaDompa (talk) 07:53, 20 August 2023 (UTC)

Hadamard's gamma function#Properties. When you generalize from integers to reals, it may be the case that part of a formula that vanish for integers becomes visible for reals. So although I agree that the usual Γ is usually the correct interpolation, I don't see the rationale for insisting that only functions obeying the integer version of the recurrence can be of any interest. —David Eppstein (talk) 09:15, 20 August 2023 (UTC)

That's not what I'm saying. What I'm saying is way closer to the Gamma function not being the "correct" interpolation—or perhaps even more to the point not the correct interpolation. Had there been infinitely many interpolations but only one satisfying the recurrence relation for non-integer values, it may very well have been the case that many interpolations are interesting for one reason or another but the only one with that property might be considered the "correct" extension to non-integer values. But that's not the case (and what I wanted to make clearer to readers). The Gamma function is the most commonly used interpolation because it has useful properties, but using it to extend the factorials to non-integer values, in general, is convention rather than correct. TompaDompa (talk) 17:20, 20 August 2023 (UTC)

I couldn't find anything about the factorial of the imaginary unit on this page. Adding the factorial of the imaginary unit can be quite useful. Bera678 (talk) 16:18, 21 December 2023 (UTC)

You probably could find a section titled "Continuous interpolation and non-integer generalization" and headed by "Main article: Gamma function". That suggests that if this material is anywhere it should be in the Gamma function article. However, I did not see anything about ${\displaystyle \Gamma (i+1)}$ in Gamma function#Particular values. Is there any reason to think that ${\displaystyle i+1}$ has any special meaning as a parameter of the Gamma function, making it significant enough to report its value in that article? —David Eppstein (talk) 16:58, 21 December 2023 (UTC)

You may want to go here to find what you’re looking for: Particular values of the gamma function#Imaginary and complex arguments 107.9.41.132 (talk) 00:42, 25 January 2025 (UTC)

Is the reference ok it' s all ok even in the version in Spanish it is https://es.wikipedia.org/wiki/Factorial#Soluci%C3%B3n_n%C3%BAmero_negativo_factorial

So, why they regressed my edition? Arrobaman (talk) 22:47, 30 December 2023 (UTC)

The reason given for the first revert was "not an improvement, broken citation, technically not make reader understand". The reason given for the second revert was "This was recently reverted. Please do not reinsert it without first discussing it on the talk page". Another reason would be that the standard extension of the factorial to numbers other than the non-negative integers is given by the gamma function and that function diverges to infinity rather than having a finite value at all negative integers. Additionally, the link you give cannot be used as a reference (Wikipedia cannot be used as a reference for itself). —David Eppstein (talk) 22:59, 30 December 2023 (UTC)

That is not true the gamma function is for $(n -1)!$ and the equation that I make reference find value for $(-n)!$ and the reference is to this paper https://figshare.com/articles/journal_contribution/beta_SM_project/24901614 Arrobaman (talk) 23:06, 30 December 2023 (UTC)

That is also not a reliable source. And do you really think there is a difference between ${\displaystyle (n-1)!}$ (for ${\displaystyle n}$ a non-positive integer) and ${\displaystyle (-n)!}$ (for ${\displaystyle n}$ a positive integer)? They are both expressions for the factorial function at negative integers. Besides being incorrect (for the standard extension of factorial to gamma) this material appears to be original research, forbidden on both the Spanish and English Wikipedias. —David Eppstein (talk) 23:27, 30 December 2023 (UTC)

Yet another problem is the equation

${\displaystyle {\displaystyle {\displaystyle {\displaystyle {\frac {n+1}{n!}}=(-1)^{n}(-n)!}}}}$

uses ${\displaystyle (-n)!}$ before it is defined.—Anita5192 (talk) 23:36, 30 December 2023 (UTC)

I remember you and import limit of the gamma function is that he can' t represent negative factorial that' s why the limits are infinity to 0 if can resolve also negative factorials will be to infinity to -infinity.

And what you say of original research if you read a little bit more you can see "material—such as facts, allegations, and ideas" is some of this the paper I making reference no so it not a original research.

And what Anita say is just the equation you have to isolate $(-n)!$ having the solution

$(-n)! = \frac{n +1}{n! (-1)^n}$ Arrobaman (talk) 23:44, 30 December 2023 (UTC)

Can I make the modification? Arrobaman (talk) 11:59, 31 December 2023 (UTC)

These equations are mathematically incorrect. Please do not insert them again.—Anita5192 (talk) 13:35, 31 December 2023 (UTC)

Why is incorrect? Explain to me please Arrobaman (talk) 13:45, 31 December 2023 (UTC)

Please read the lead carefully and you will see that the factorial function and the gamma function are not defined at all for negative integers. Thus the equations you inserted make no sense at all.—Anita5192 (talk) 15:14, 31 December 2023 (UTC)

Do you even read the paper that I' m making reference? Look the name of the paper is Beta SM project in the introduction say the objective of the project is literally this "This project want to resolve all the problems or functions that are calculations with a difficult solution or they are limits of the basis of mathematical" so yes is solving a problem of the maths is the point of the paper.

Before we continue talking please read the paper that I' m making reference and please also read this Wikipedia:Edit warring#The three-revert rule Arrobaman (talk) 15:30, 31 December 2023 (UTC)

Your above cites are not WP:reliable sources and cannot therefore be used here. Otherwise, I fully agree with Anita192. D.Lazard (talk) 16:14, 31 December 2023 (UTC)

You are right figshare isn' t a very good reference I will comment the problem that you say to the mail of the author Arrobaman (talk) 16:51, 31 December 2023 (UTC)

Why isn't the Pi Function discussed? Pi(x)=Gamma(x+1)=x!=the integral from 0 to infinity of t^x * e^-tdt. Derek Verduijn (talk) 11:04, 22 November 2024 (UTC)

I don't think the Pi function may be included, as it basically the same thing as changing the gamma function with its input is by adding 1: ${\displaystyle \Pi (x)=\Gamma (x+1)}$, which seems to be redundant. The article should provide the understanding, not the technical confusing symbols per WP:TECHNICAL. Here, the article is focus on factorial only: its basic concept, definition, and its appearance in different fields. Dedhert.Jr (talk) 11:14, 22 November 2024 (UTC)

Discussing the Pi function here would require reliable sources establishing that this variant of the Gamma function is still in common use. This seems not the case. Moreover, using both Gamma and Pi function would be confusing, because of the need of distinguish many formulas from similar formulas resulting from a shift of 1 of the variable. In Gamma function#19th century: Gauss, Weierstrass and Legendre, you will find a discussion on the historical choice of Gamma over Pi. D.Lazard (talk) 11:59, 22 November 2024 (UTC)

Should there be a section on why negative factorials (like -1! = undefined) are calculated as undefined? Alimsts (talk) 20:14, 16 May 2025 (UTC)

Negative factorials are not "calculated as undefined". They are not defined. Similarly, the factorial of ⁠${\displaystyle {\sqrt {2}}}$⁠ is not defined. It is silly to try to explain why people do not define something. This is already sufficiently difficult to explain why common definitions have been chosen as they are. D.Lazard (talk) 20:34, 16 May 2025 (UTC)

Well, if we can find a high-quality source that properly explains why they are undefined (not so much because nobody has bothered to define them, but because it is impossible to define a numerical value that fits the recurrence equation) we could add a brief note to the definition section. My own searches were not promising, though, turning up lots of low-quality sources that instead said that the factorial of a negative number was defined as infinite, or defined as some integer multiple of the infinite value of –1!, or some such. —David Eppstein (talk) 20:42, 16 May 2025 (UTC)

(edit conflict) This being said, the factorial is formally defined by ⁠${\displaystyle 0!=1}$⁠ and ⁠${\displaystyle n!=n\cdot (n-1)!}$⁠. In the case of ⁠${\displaystyle n=0}$⁠, this would give ⁠${\displaystyle 1=0!=0\cdot (-1)!}$⁠, that is, ⁠${\displaystyle (-1)!={\frac {1}{0}}}$⁠. For the generalization of the factorial to values that are not natural numbers, you may see Gamma function. D.Lazard (talk) 20:51, 16 May 2025 (UTC)

Should we add a small section about "Terminal" under "Related sequences and functions" and show an example similar to factorial (since it is a related function like:

n?=1+2+3...+(n-2)+(n-1)+n Eszett-Enjoyer (talk) 16:37, 10 December 2025 (UTC)

You mean termial, better known as a triangular number. It was briefly mentioned under exponential factorial but I added an entry about this. I don't think it needs elaboration with an example. —David Eppstein (talk) 18:47, 10 December 2025 (UTC)

This choice matches the gamma function ${\displaystyle 0!=\Gamma (0+1)=1}$, and the gamma function must have this value to be a continuous function.

I removed everything after the comma and my edit was reverted. I am curious what this text means. You wouldn't say "${\displaystyle x^{2}}$ must take the value 0 at 0, because otherwise it wouldn't be continuous." — xo Ergur (talk) 06:52, 28 December 2025 (UTC)

The definition of the gamma function is by analytic continuation of a formula that works only for values with positive real part. Saying that it must equal 1 at 0 is one step of this continuation process. "To be a continuous function" is maybe an oversimplification but not one that is important in this context. You are instead trying to write the article to say "it must be defined to have this value because we have defined it to have this value", which is just circular reasoning. —David Eppstein (talk) 07:12, 28 December 2025 (UTC)

We are talking about the gamma function at 1. It is given in the article as an integral over reals. — xo Ergur (talk) 09:24, 28 December 2025 (UTC)

Other methods suitable for its computation include memoization, dynamic programming, and functional programming.

As far as I am aware, all these methods also just use iteration or recursion, which are mentioned earlier in the section. Also, memoization and dynamic programming are essentially the same thing, unless it is supposed to be bottom-up dynamic programming which would just be iteration (the sources support this). Also also, going by the sources I take it that "using functional programming" means "using fold." This like saying "you can use for-loops or while-loops or do-while-loops et c." — xo Ergur (talk) 07:50, 28 December 2025 (UTC)

You are completely missing the point.

When a topic is frequently used for something, describing that something is due for its article.

The factorial is frequently used as an example of different programming styles.

Therefore, describing the fact that it is frequently used as an example of different programming styles, with a discussion of the styles for which this has been done, is due for its article.

The fact that these are all easy is not relevant. In fact, I suspect that it has been used as an example so much precisely because it is easy: that way, the style becomes the more obvious thing about the examples. —David Eppstein (talk) 15:57, 28 December 2025 (UTC)

Indeed, I was missing the point. I missed that all those examples where in reference to the sentence The simplicity of this computation makes it a common example.... In my defense, I think these sorts of misunderstandings would be less likely to happen if the paragraph break would be moved from after that sentence to before it.

I have tentatively made this edit. Feel free to revert if you don't agree; I won't fight you on it any further. — xo Ergur (talk) 15:23, 5 January 2026 (UTC)

User:Quondum recently made an edit that, when I scanned it, appeared to consist entirely of changing the spacing of template arguments and changing the formatting of mathematics markup to a template that wraps the mathematics markup in a template for little to no reason other than wrapping it in a template. I found the new version of the source code difficult to read and (because it requires understanding another template and in some cases using additional markup to work around its idiosyncracies) likely difficult to maintain, and I don't think it's useful to be making mass cosmetic edits, so I reverted it. In a subsequent null edit, Quondam claimed without detail that the edits included substantive changes (that I somehow missed among all the cosmetic ones) and that my revert was somehow WP:POINTy. So maybe we should have a discussion here: what was the actual substantive change and what is the actual benefit of changing <math> to {{tmath}} and of changing the spacing around template arguments? And is it appropriate to mask substantive changes by placing them amid hundreds of cosmetic changes so that other editors trying to follow their watchlists can't even determine what has changed? —David Eppstein (talk) 19:00, 13 March 2026 (UTC)

Perhaps a more neutral and substantive framing is appropriate than that given above. There are multiple points here that can be debated, and they should be considered separately. I will not list these all, but I will point out that this is a MOS discussion, not an article discussion, and that this is obviously not the appropriate forum for a MOS discussion. How purely cosmetic edits should be handled is also a matter for debate; reverting another's edits seems rather a pointy approach, as there is no direct justification aside from "punishment" (or expression of displeasure at another's well-meant efforts); IMO any reasonable person would actually point to the MOS instead: "Keep in mind that reverting a cosmetic edit is also a cosmetic edit. If the changes made in a cosmetic edit would otherwise be acceptable as part of a substantive edit, there is no reason to revert them."

I will make one observation relating to {{tmath}} that may have been missed, and relates to a change made in this article by me: replacing <math> with {{tmath}}, when there are adjacent non-breaking characters, serves to prevent a possible break that occurs with <math>. Previously, one solution was to include the adjacent characters as LaTeX aty the cost of a font mismatch, but {{tmath}} obviates this. The other solution is to enclose everything in {{nowrap}}. Would there me MOS consensus for the double wrapper, such as (the sum {{nowrap|<math>x+2<\math>),}} and versus (the sum {{tmath|x+2}}), and as perhaps being more readable, which seems to be the claim being made by David Eppstein? {{math}} is already in extensive use with good effect, so the claim that "it requires understanding another template" needs explanation. —Quondum 19:34, 13 March 2026 (UTC)

You have not answered the request to substantiate your claim that you were making substantive changes to the article. As for your example: for short mathematical expressions it usually considered bad style for them to start at the beginning of a line; one would also want to prevent a break before the expression as well as after. So really the comparison should be between (the {{nowrap|sum <math>x+2<\math>),}} and versus (the sum {{tmath|x+2}}), and. It is also the case that tmath appears to introduce extra restrictions in the formatting of the LaTeX mathematics coding that it accepts: It does not work to reformat <math>2^{2^{\alpha}}</math> (${\displaystyle 2^{2^{\alpha }}}$) as tmath without adding some extra spaces between the brackets. —David Eppstein (talk) 20:49, 13 March 2026 (UTC)

Examples requested (each eliminating a line wrap point, something that the source of this article seems to be careful about):

• <math>\vert\!\underline{\,n}</math>, → {{tmath| \vert\!\underline{\,n} }},
• <math>n!\pm 1</math>, → {{tmath|n!\pm 1}},
• <math>1^1\cdot 2^2\cdots n^n</math>. → {{tmath| 1^1\cdot 2^2\cdots n^n }}.
• <math>n</math>th factorial → {{tmath|n}}th factorial
• <math>n</math>th [[triangular number]] → {{tmath|n}}th [[triangular number]]

A reformatted comma to fit the surrounding font:

• <math display=inline>\tbinom{n}{n} = \tfrac{n!}{n!0!} = 1,</math> → {{tmath|1=\textstyle \tbinom{n}{n} = \tfrac{n!}{n!0!} = 1 }},

Your restyling of the example to include the   is not general: an expression as long as a word should not be prevented from breaking at an adjacent space, and such expressions are common. I do not want to argue style opinions here, nor your choice to apply a rule of thumb to this template particularly when the "maintainability" issue universally applies to templates. If you dislike my choice of styling, it is a general issue, and opinion should be sought in a general forum about style. Especially if you wish to convince me. —Quondum 23:27, 13 March 2026 (UTC)

Surely the WP:ONUS should be on the person trying to change the choice of styling to convince me that the change is worthwhile? Also I would not call any of those changes substantive (actual changes to content), although it is possible that changing a comma in one font to a comma in another font might actually reach the level of being a visible change. —David Eppstein (talk) 06:24, 14 March 2026 (UTC)

This is the first time that you are mentioning general replacement (change) of style that I can see, and I regard it as the only valid objection that you have raised so far. You have unfortunately given the impression of grabbing at straws to bolster a weak position. Your point on my choice of the word "substantive" is weak – why make it? Those examples were all edits that were worthwhile in their own right. Yes, the number of cosmetic edits grew to make checking difficult, but that is not something that I planned.

I am not trying to convince you that the change is worthwhile. I am simply pointing out that choice of style is a matter of individual choice despite your preference, that there is no established reason to prefer one (aside from the retention of an established style), and that your approach has not been helpful. I lost interest in any actual changes the second my edit was reverted. I was merely responding to your comments. —Quondum 17:28, 14 March 2026 (UTC)