Talk:Factorial/Archive 3

Archive 1

Archive 2

Archive 3

Extension to negative integers

I see that article says something about the gamma function extension generalization not holding when negative numbers are involved. What other generalization matches the extension to negative numbers?--185.53.198.164 (talk) 19:04, 14 September 2019 (UTC)

I also see that other info re the factorial of negative numbers is not present in article.--185.53.198.164 (talk) 19:06, 14 September 2019 (UTC)

No function that is defined on the negative integers can satisfy the defining recurrence, so there is no reasonable extension of the factorial function to negative integers. --JBL (talk) 19:58, 14 September 2019 (UTC)

I think it is not very clear why the defining recurrence not be satisfied on negative integers. From where follows such a negation?--185.53.198.164 (talk) 20:16, 14 September 2019 (UTC)

For instance in the case the following product of negative integers:

$(-n)!=-n\times (-n+1)\times (-n+2)\times (n+3)\times \cdots \times -3\times -2\times -1\,.$

with the particular example

$-5!=-5\times -4\times -3\times -2\times -1=-120\,.$

is the usual recurrence of usual factorial not present?--185.53.198.164 (talk) 20:24, 14 September 2019 (UTC)

You assume that $(-1)!=-1$, but this is wrong, because this would mean, from the recurrence relation, that $1=0!=(-1)!\times 0=-1\times 0=0$. Vincent Lefèvre (talk) 20:29, 14 September 2019 (UTC)

What if this aspect is circumvented by considering a convenient postulation of the factorial of -1 to be -1, similarly to the definition of the factorial of zero to be 1, thus avoiding the multiplication to zero. Does such an action break somehow the usual recurrence for integers lower than -1, perhaps a sort of a discontinuity occurs?--185.53.198.164 (talk) 20:50, 14 September 2019 (UTC)

No, the value of the factorial of 0 is not a definition; it is just a consequence of the recurrence relation (or any equivalent justification). If you define the factorial of −1 by using an arbitrary value, you can do so, but this will not be an extension of the factorial function to −1 (and the other negative integers). The recurrence relation is part of the formal definition of the factorial and must not be broken. Vincent Lefèvre (talk) 21:02, 14 September 2019 (UTC)

(edit conflict) Yes you could do this (I mean, one can always make a definition) but it's totally uninteresting because you just recover a (signed, rescaled) copy of the factorials (or maybe their reciprocals).

On another note: per WP:NOTFORUM, this page is meant for discussion of improvements to the article, not for discussion of the subject of the article -- better venues for that might include the reference desk or StackExchange. --JBL (talk) 21:04, 14 September 2019 (UTC)

If I want to add this aspect of possible consideration re factorial of -1 (why should the -1 value of the factorial of -1 be an arbitrary value?) to article I do not understand why a (signed, rescaled) version of the factorial(s) would be uninteresting?!?!--185.53.198.164 (talk) 21:20, 14 September 2019 (UTC)

Speaking as a wikipedia editor: If you want to add this to the article, you should find a reliable source that discusses it, not engage in original research. Speaking as a mathematician: It is not interesting because it is a completely artificial way to extend the sequence, with no behaviors that do not follow trivially from the behavior of the original sequence. By contrast, the gamma function extension is quite natural (it arises in many mathematical contexts, not just as an exercise in "how could I extend this?") and has many properties beyond those implied by the phrase "an extension of the factorial". --JBL (talk) 23:08, 14 September 2019 (UTC)

If some extension is straightforward and, say well-behaved with no hidden or unexpected or surprising properties, then I think it is below a certain threshold of originality which would require citing a source. Initially the factorial at its introduction was just a notation of the product of the first n positive integers.--185.53.198.164 (talk) 00:35, 15 September 2019 (UTC)

On the first sentence, I disagree: both our prohibition on OR and considerations of due weight dictate that "here's something I just made up" should not be included in articles. About the second, I also disagree: the factorial arises initially as a solution of a combinatorial problem; everything else about it, in particular the recursive and explicit formulas, are subsequent to that. It is interesting and important because it means something; that's why it gets a name and a notation. --JBL (talk) 01:36, 15 September 2019 (UTC)

So something which is a straightforward consequence of something else should not be stated, presumably to undue weight? What is due weight involvement in this? On the other hand in math topics articles I understand the ORIG threshold is not at the same level as in other areas due to WP:CALC, so straightforward consequences could be stated without problems.--185.53.198.164 (talk) 02:00, 15 September 2019 (UTC)

The other aspect about the significance of the factorial is that it is a notation that happened to be included in a combinatorial calculation problem. It is a shorter notation that that involving ∏ notation of a product. The center of meaning of the factorial is given by the product itself. The factorial appears beside combinatorial topics also in number theory subjects.--185.53.198.164 (talk) 02:12, 15 September 2019 (UTC)

WP:DUE does not consider things that can be derived with WP:CALC unless they are also reported on by reliable sources, rendering your proposal ineligible for inclusion in the article. WP:NOTMADEUP applies especially to defining new things in math: definitions are made for a reason, and there are no sources that provide a motivation for defining (-1)! = -1. I could invent the "Jasper numbers" as the numbers which (e.g.) divide any of the numbers in my birthdate's month, day, or year, but that would not be suitable for inclusion here. By the same token, (-1)! = -1 is not suitable for inclusion on Wikipedia.--Jasper Deng (talk) 06:22, 15 September 2019 (UTC)

The comparison you mention is doubtful. Indeed defining and also naming a new type of numbers would be OR, but something which is straightforward like a product of (negative) integers can NOT be included to something MADEUP.--185.53.198.147 (talk) 14:13, 20 September 2019 (UTC)

It is clear that there is a consensus against adding anything about this without reliable sourcing. If you want to add it to the article, you should find a reliable source that discusses it. --JBL (talk) 15:26, 21 September 2019 (UTC)

I see that the subsection re negative numbers is unsourced (beside other subsections of this article) and explains an immediate consequence of the definition for positive integers which is not valid for negative ones. I think that in this situation another direct unsourced short explanation can be added as well.--185.53.196.88 (talk) 15:29, 27 September 2019 (UTC)

It is clear that there is a consensus against adding anything about this without reliable sourcing. If you want to add it to the article, you should find a reliable source that discusses it. --JBL (talk) 20:31, 27 September 2019 (UTC)

I think that your repetition of the same sentences is not necesary. The consensus you mention is not quite clear, I'd say that there is at most NO consensus. The NO consensus should not be confused with a supposed clear consensus against adding something unsourced in an already NON-sourced section which explains a straightforward consequence.--185.53.197.26 (talk) 18:35, 3 October 2019 (UTC)

Please see: Consensus is not unanimity. –Deacon Vorbis (carbon • videos) 18:46, 3 October 2019 (UTC)

I have taken a brief look at the mentioned link, but I think it is not applicable to the context of this discussion.--185.53.197.26 (talk) 18:55, 3 October 2019 (UTC)

Where there is an unsourced simple explanation I think another unsourced simple explanation is acceptable. The insistence to cite a source in a non-sourced neighbourhood is at least problematic.--185.53.197.26 (talk) 19:00, 3 October 2019 (UTC)

Consensus is clear, you are the only one supporting your position and several experienced editors have told you what is wrong with with it. Your last statement that one wrong implies that a second one is okay just does not hold water. --Bill Cherowitzo (talk) 19:10, 3 October 2019 (UTC)

I'd say that the noticed fact of missing sources should NOT be considered as automatically wrong as you label it. Some simple explanations based on definitions are not very stringent to be based on sources if WP:CALC is reasonably used, the burden of sources citation is just smaller in such a context.--185.53.197.26 (talk) 19:19, 3 October 2019 (UTC)

On the other hand I think I may have encountered some source re the factorial of negative integers somewhere on the internet.--185.53.197.26 (talk) 19:30, 3 October 2019 (UTC)

185, you should give up since it is totally impossible that you will ever get this material into the article. Not only those who have tried to make you understand here, but many many others including me will remove it on sight. You can expect essentially unanimous opposition; you should accept that and move on. McKay (talk) 23:33, 3 October 2019 (UTC)

Unsourced sections in this article

I see there are quite many sections and subsections which contain NON-sourced statements about simple explanations and consequences. In this context I think is not very stringent to really use a source for another simple explanation like one mentioned above in a previous section.--185.53.197.26 (talk) 18:50, 3 October 2019 (UTC)

There's a big difference between explaining simple consequences of mathematical facts and suggesting a new definition for which there is no mathematical backing. The factorial is not defined for negative integers simply because it does not play well with established mathematics; it's not a useful concept. It's akin to trying to define 1/0 = 0; it's possible to do so, but it's not useful because it bears no relationship to 1/x for other x close to 0. Stop wasting your time. -- Elphion (talk) 04:59, 4 October 2019 (UTC)

185, you are quite correct that there are things in this article that are unsourced but should be sourced. The cure for that is to provide them with sources, not to add additional unsourced things. If you want to help with improving the article, finding sources is one way to do it. McKay (talk) 05:37, 4 October 2019 (UTC)

quote vs cquote

Alright, let's waste some time, I guess. =/

An editor (User:DESiegel) changed the format of a a quotation, changing an invocation of {{cquote}} to the more plain {{quote}}, that is, changing

“ Now the nature of these methods is such, that the changes on one number comprehends [includes] the changes on all lesser numbers ... insomuch that a compleat Peal of changes on one number seemeth to be formed by uniting of the compleat Peals on all lesser numbers into one entire body.[reference] ”

to

Now the nature of these methods is such, that the changes on one number comprehends [includes] the changes on all lesser numbers ... insomuch that a compleat Peal of changes on one number seemeth to be formed by uniting of the compleat Peals on all lesser numbers into one entire bodyy.[reference]

That's his perfect right per WP:BOLD, so no problem. I changed it back, with a an edit summary of

Restore format per WP:BRD, take it to talk and (or, better, MOS talk) and get consensus for change, absent that it's just pointless roiling of format, just as good before as after.

However, another editor -- User:Deacon Vorbis -- changed it back, although WP:BRD says not to do so if its not an emergency, which it isn't, so here we go. Let's work it out. User:Deacon Vorbis is supposed to open a thread on the talk page, but no prob; I'll do it. Here to help.

It is true that the MOS says not to use {{Cquote}}. {{Cquote}} says the same thing, putting it in logical and functional agreement with the MOS. Although the Wikipedia goes beyond basic HTML for most of its presentation, we do use and recommend the raw HTML tags <blockquote>...</blockquote> for long quotes (there's also a {{quote}} template which I think is just a rapper for <blockquote>...</blockquote>), and -- in the MOS -- no other, and in fact others are specifically deprecated. Why? Well, it's an artifact really. <blockquote>...</blockquote> is the HTML tag used for quotes, HTML is widely used, and it got imported over here. There's been a lot of back-and-forth on the question of long quotes over the years, and a lot of ink spilled.

It's a complicated issue, and I suppose a good place to start getting a handle it would be looking at one of the more recent and longer of the very many discussions that have occurred around this issue over the years: Wikipedia talk:Manual of Style/Archive 184.

Read that? OK good, continuing on...

My take on all this is this: the Wikipedia is not rule-bound. Rules follow (or are supposed to follow) common practice and good practice. Only the basic key rules -- NPOV, V, and a few others -- are not up for discussion. The Wikipedia is not a bureaucracy, not a battlefield, and not a typical hierarchical organization, and per WP:IAR our first duty is to bestify the encyclopedia.

As to how long quotes should be formatted, there are a number of "camps". Some people like {{quote}}, some people like {{quotebox}}, and some people like {{cquote}}, and there may be other quote templates. Some people like more than one. Some people think that having more than one template is chaotic or sloppy, some people think that having only one is overly rigid. Some people think that {{quote}} should be used sometimes and {{quotebox}} at other times, depending on the page layout and what you're trying to do and so forth. Some people don't think that long quotes should be used at all, and that all quotations should be in-line (with only quotation marks to differentiate them from normal text). Some people like one of the templates but aren't into micromanaging what other editors do. Some people are just annoyed by what a time sink this has proven to be over the years. And so forth.

As a matter of practice, a lot of editors have "voted with their feet" to create a de facto consensus that its OK to use different templates. You can tell by counting the transclusions: a lot of editors have transcluded {{quote}}, a lot of editors have transcluded {{quotebox}}, and a lot of editors have transcluded {{cquote}}.

Some people are inclined to favor a very plain presentation even at the possible cost of some readability and other problems, and so they favor plain <blockquote>...</blockquote> (or its wrapper, {{quote}}), and have the MOS to hammer on and generally annoy people who don't agree with them, and so forth.

There isn't really a consensus for any of this. It's hard to get consensus. But you can try if you want. Want to you want to do is take some hours or days to study the matter and its history, bone up on your knowledge of ergonomics, page layout, industry practices (particularly for other encyclopedias), and so forth, formulate a CENT RfC (probably preceded by a series of RfCs so that the core issue(s) can be clarified and presented succinctly), and like that.

Or, of you wish, you can make a case here that this particular article has some special nature such that templates other than {{quote}} won't work in this particular article. Can't imagine what's unique about this article, but willing to be educated.

Til then, respect WP:CONSENSUS and its enforcement arm, WP:BRD. The floor is open. We are all ears and willing to be convinced. Herostratus (talk) 16:06, 20 December 2019 (UTC)

I have read your comment (not following any links) and see no answer to the following question: why do you think cquote is better here? (Ideally, I hope for a reply of one or at most two sentences.) --JBL (talk) 16:12, 20 December 2019 (UTC)

User:Joel B. Lewis, sure, there are a few reasons why this change in this place is not something to be done without discussion

1) On the merits of the user experience, when leading into a quote, it's important for the reader to understand that they are reading a quote. The large pastel quote symbols do this well, slightly better than just indenting. Or not; that's my guess, but it's hard to prove either way, without controlled-study data (which may exist somewhere) and anyway we're mostly talking about milliseconds here. But the large quote marks surely aren't worse for this purpose. The reader's experience is by far the most important thing, but:

2) On the merits of the editorial environment, it's not good to micromanage editors. Somebody went to the trouble of finding that quote and a good source for it and formatting it an putting in in the article -- in other words, the actual work of the project. They chose to use {{tl}cquote}} and you have to respect that. Well, you don't have to, but all things being equal you should, if you're wanting to build an organization made of motivated volunteers.

3) On the merits of the Wikipedia process, since quotes are used (in various formats) on thousands and thousands of pages, trying to get a WP:LOCALCONSENSUS for each one is not the best way forward in my opinion. If you want consensus to have the right and permission to go around changing other editors contributions to a form more pleasing to you personally, go get that permission -- at an MOS RfC, not on a page-by-page basis -- and write a bot or make it a personal goal or put together a team to fix it. It's either a problem in which case it needs to be fixed everywhere, or it's not a problem and it doesn't need to be fixed anywhere.

4) If it's a matter of aesthetics (and don't be silly, of course it is), well... assuming that aesthetics is entirely a matter of personal taste (not exactly true, but close enough), then that's insufficient reason to roil the text. It just ends up as a time sink. I certainly don't go around changing your formats to my preference. If you write "He graduated from Harvard" I don't go around changing that to "He was graduated from Harvard" even tho that's how I'd write it. And so forth.

So I mean that's four right there. I can think of more but that's a start. Herostratus (talk) 00:53, 21 December 2019 (UTC)

I guess you didn't read the part about one or at most two sentences? Also, trying to avoid local consensus is an argument for going with what the MOS says (no decorated quotes), not for going against it. —David Eppstein (talk) 01:13, 21 December 2019 (UTC)

MOS:BLOCKQUOTE and the documentation for {{cquote}} already deprecate its usage in mainspace. The onus is thus really on you to get consensus for using it, and this isn't the place for doing so. Simple transclusion count is a useless metric; if someone sees a template being used, they'll often just copy it without realizing that it's not appropriate. –Deacon Vorbis (carbon • videos) 16:21, 20 December 2019 (UTC)

• It seems to me that the use of cquote, rquote, or other forms of decorative quotes, or boxed quotations, generally gives undue weight to a quotation. Perhaps in an article with a specific quotation is the topic, sauch as Ain't I a Woman or I Have a Dream. I note however that neither decorative quotes nor boxed quotes are used in the former article, while one boxed quote 9but not a cquote style) is used in the I Have a Dream article. Outside of such articles that are actually about a speech or quotation or text, it seems to me that the use of cquote is pretty much always going to give undue weight, and should, inline with the MOS, never be used. I removed it here for that reason. DES ^(talk)_{DESiegel Contribs} 17:21, 20 December 2019 (UTC)
  • I note for the information of those discussing here that discussion was already ongoing at Template talk:Cquote#Proposed changes, a discussion i started and Herostratus has participated in. I expect to post a formal RFC at WT:MOS in the not too distant future, but i am not ready to do so yet. The history at Template talk:Cquote may also be of interest. DES ^(talk)_{DESiegel Contribs} 18:58, 20 December 2019 (UTC)
• The consensus on MOS not to use cquote or other decorative quotes is clear. This is not the place to fight it. —David Eppstein (talk) 17:33, 20 December 2019 (UTC)
• Procedural comment WP:BRD says that after an action is reverted, a discussion should be opened, and more specifically says After someone reverts your change, thus taking a stand for the existing version or against the change, you can proceed toward a consensus with the challenging editor through discussion on a talk page. While discussing the disputed content, neither editors should revert or change the content being discussed until a compromise or consensus is reached. BRD does not say that no other editors my reinstate the challanged change. Indeed it strongly implies that doing nso is a legitimate part of the cycle. BRD also says: BRD is never a reason for reverting. Unless the reversion is supported by policies, guidelines or common sense, the reversion is not part of BRD cycle. and RD is not an excuse to revert any change more than once. If your reversion is met with another bold effort, then you should consider not reverting, but discussing. Therefore it seems to me that this edit is not one properly applying BRD, and is arguably edit-warring although not a 3RR violation. Also, note that BRD, while widely respected and followed, is an essay, not policy. DES ^(talk)_{DESiegel Contribs} 19:09, 20 December 2019 (UTC)
• I am not seeing in the above discussion any actual reason to prefer the use of cquote, or of decorative quote marks or set-off quotations, beyond the standard indetation which the blockquote tag implements. The only reason I see is "lots of editors have done this in the past." Academic and scientific publications never use such styles, nor do most serious non-fiction books. But beyond that, there is a pretty clear consensus here not to do that, documented in the MOS provision. There should be a good reason for departing from or changing any such consensus, and I don't yet see that here, nor can I think of one. Why should cquote be used here? DES ^(talk)_{DESiegel Contribs} 19:15, 20 December 2019 (UTC)

Right, I laid this out above, where you can look for details, but as an overview, the existing status should be kept because:

1) It's better for the reader (arguable)

2) It's better for the community (arguable, I suppose, if you squint)

3) It's poor process to do this case-by-case, needs a centralized discussion

4) Assuming its really mainly about personal aesthetic taste (which, probably), that is insufficient cause for undoing other editors' work. 01:47, 21 December 2019 (UTC) — Preceding unsigned comment added by Herostratus (talk • contribs) 20:47, 20 December 2019 (UTC)

This is not,Herostratus a matter primarily of personal aesthetic taste it is a matter of compliance with our hose style, and of lending undue weight to some quoted, thereby violating WP:NPOV. As a polixy issue, aesthetic judgemnts are not relevant, and the "reader exprience" is not highly relevant either, althoguh i disagree with your reasoning there. If it is "important that the reader realize that this is a quotation": they why supply decoratice quotes for only soem quotations? Obviously to emphasize thsoe quotes over other quotes and the rest of the articel text, and that is undue weight, clear as day. DES ^(talk)_{DESiegel Contribs} 02:09, 21 December 2019 (UTC)

While coming late to this discussion, I'd still like to add my two cents. I have found the use of those big bold quotation marks to be quite jarring visually and this in turn leads to undue weight being given to the quoted material. While there may be some rare instances where this can be useful, I do not see any need for them in any mathematical (actually, any academic) article. We should follow the WP:MOS and not use them here.--Bill Cherowitzo (talk) 20:01, 20 December 2019 (UTC)

• Issues related to this are now being discussed at WT:MOS# RfC: Use of Large Quotes in article space, and the Cquote template. Those who participated in this discussion (and others) may find that RfC of interest. DES ^(talk)_{DESiegel Contribs} 16:54, 2 January 2020 (UTC)

Underlined exponent "k" in Double Factorial section

This notation is also used in the Double Factorial article. To whoever did this: Nobody knows what that is. Please do not assume that the audience of Wikipedia has a graduate level math degree when choosing notation. This is becoming a general problem with Wiki articles, but that's another conversation. 74.140.199.156 (talk) 06:18, 26 May 2021 (UTC)

Number of prime factors of $n!$

Dear everyone. I have a suggestion of what can be added to the factorial Wikipedia page, which is sadly missing under the 'Number Theory' section in the article. Most of the interesting number theoretic properties of the factorial, including Wilson's theorem and Legendre's formula are already mentioned. There is, however, one quite natural question that is not mentioned anywhere. Namely: How many prime factors does n! have, when counted with multiplicity, on average?

This is a known result, and the answer to the question is the following:

For any positive integer $k$, let $\Omega (k)$ denote the number of prime factors of $k$, counted with multiplicity (I.e., for $1944=2^{3}\cdot 3^{5}$, we have $\Omega (1944)=3+5=8$).
Then, for every integer $m\geq 1$, one has

$\Omega (n!)=n\log(\log(n))+\theta n+\sum _{r=1}^{m}C_{r}{\frac {n}{(\log(n))^{r}}}+O\left[{\frac {n}{(\log(n))^{m+1}}}\right],$

as $n\to +\infty ,$ where $\theta ,C_{1},C_{2},\ldots$ are constants. Specifically

$\theta =M+\sum _{p\;{\text{prime}}}{\frac {1}{p(p-1)}}$, where $M$ is the (Meissel-)Merten constant, and $C_{r}=-\int _{1}^{\infty }{\frac {t-\lfloor t\rfloor }{t^{2}}}dt$ for all positive integers $r$.

Proof: This result is a special case / easy consequence of Theorem 1 in the paper:

Saffari, B. Sur quelques applications de la "methode de l'hyperbole" de Dirichlet a la theorie des nombres premiers. (1970) Enseign. Math. (2) 14, (p.205-224)
Link to publisher: L’Enseignement Mathématique
(I do have a copy of this document, which I am willing to share upon request).

Why am I interested in this? Quite frankly, there are not all that many number-theoretic questions one can ask about the factorial, so I am a little bit surprised that this is not mentioned in the article. The reason I am aware of this result is that I wrote my bachelor thesis on this exact result. I did try to add the above formula to the Wikipedia page, but I had a hard time figuring out how to cite and write up it up. The reason for this is that: (1.) B. Saffari does not state the above special case of his formula, and I am not aware of any place in published literature where anyone has written this exact formula. (2.) The only exception to (1.) is ironically my own Bachelor thesis: Number of prime factors of the factorial , but this has not been published in any Journal, so I refuse to cite it on the page. If you look in the thesis (Theorem 5.4, p.34), you see how one obtains the above formula from Saffari's Theorem, in addition to some information about the constants in the formula.

Is there anyone who can recommend a course of action? Or who would be willing to create this entry?
Thanks. : )

Amusing numbers (talk) 19:36, 16 June 2021 (UTC)

old notation

The history section should mention the old notation consisting of a line to the left and below, see here for examples. This was quite common well into the first half of the 20th century; I have some old books that use it. I'm not sure how to write it though (except as an image). Probably ⁿP_n should be mentioned too. McKay (talk) 05:32, 4 October 2019 (UTC)

This notation is still commonly used in India. So I added it, but it got undone. MaharajaAD (talk) 06:19, 24 June 2021 (UTC)

Why is the alternative factorial sign being undone?

Why is the alternative factorial notation being undone? Not only has this notation been used in a very popular textbook in India, but also many teachers use this notation. But I did not found this notation in Wikipedia. Then on searching about the notation of factorial in Google, I was surprised to see that this notation, which such a popular book uses, is not found anywhere except a very uncommon website (didn't try to remember the name of that website). So I thought of adding that here. MaharajaAD (talk) 06:17, 24 June 2021 (UTC)

It doesn't belong in the lead. If there is a reliable source for this notation it can go elsewhere in the article (maybe the history section), with a description of the context in which this alternative notation is used. —David Eppstein (talk) 14:28, 24 June 2021 (UTC)

So, shall I create a new heading about notation and there put both the notations there? MaharajaAD (talk) 15:42, 24 June 2021 (UTC)

I have created a new subheading about notation and put it there. MaharajaAD (talk) 16:39, 24 June 2021 (UTC)

Noting a change

So, an editor changed

The volume of an n-dimensional hypersphere of radius R is...

to

The volume of an n-dimensional ball of radius R (in other words, the volume of the portion of n-dimensional Euclidean space lying inside an n-dimensional sphere of radius R) is...

I have no idea what any of that means, but it's a new anon editor and looks like a big change so I just thought I'd point it out. Herostratus (talk) 01:10, 24 October 2021 (UTC)

I think the text before is better. If a person wants to find out what a hypersphere is they can go to it's article. In light of this, I change the sentence to "The volume of a hypershere of radius..." as the "N-sphere" part is part of what it means to be a hypersphere.War (talk) 01:17, 24 October 2021 (UTC)

OK all to the good then. Herostratus (talk) 11:26, 24 October 2021 (UTC)

About 0! and the edits I made and reverted.

After inserting yet another motivation for setting 0!=1, I looked over your list one more time and realized that your remark about extending the recurrence relation to 0 duplicated my edit. FYI, here is what I wrote and deleted:

• Substitute, n = 2, into the dispersion relation, n! = n·(n-1)!, which yields 2!=2·1! and 1! = 1 (as one would intuitively expect.) Now substitute, n = 1, into the same dispersion relation to obtain 1 = 1/0! and 0! = 1.

As a Wikiversity editor I much prefer the more pedantic explanation...but Wikipedia is not Wikiversity--Guy vandegrift (talk) 04:34, 8 November 2021 (UTC)

Then after reading Help:Footnotes I realized that it is OK to put the explanation in a footnote.--Guy vandegrift (talk) 04:40, 8 November 2021 (UTC)

It would be OK, if it was also properly sourced. Now it's just wp:original research. I have removed it. Feel free to put it back with a proper source. - DVdm (talk) 09:45, 8 November 2021 (UTC)