# Talk:Regular number

This is a new, sourced, write-up of the reference to this term I happen to know. Eric Weisstein has inserted a a note in MathWorld about a (slightly different) generalization of the term, and the article on that was deleted, perhaps rightly. Septentrionalis 21:10, 13 July 2006 (UTC)

This seems fine, except that I'd wager there are other uses of the term out in the literature somewhere. I'm not quite bored enough to go looking for them right now, but if someone else is, maybe he'd like to make a disambiguation page. --Trovatore 18:53, 30 July 2006 (UTC)
I have better sourced and significantly expanded this article, and merged from Hamming number some additional content. I didn't include a link to MathWorld because that describes a different concept ({2,5}-smooth as opposed to {2,3,5}-smooth numbers) on which there seems less scholarly work. —David Eppstein 06:51, 13 March 2007 (UTC)

I don't think that Regular number is a good location for this article.  I haven't ever seen that term used to mean a 5-smooth integer, although I have seen it used once or twice with other meanings.  Further, the term Hamming number is used fairly widely.  Although they will always be "5-smooth" to me (or perhaps {2, 3, 5}-smooth), I think that Hamming number is the right place for this article.  Thoughts?
CRGreathouse (t | c) 21:09, 2 March 2008 (UTC)

"Regular number" is the term used in the context of Babylonian studies; see the titles of several of the references. Similarly, Hamming number is the term used in the context of functional programming. Both names seem somewhat specialized, as does the awkward "{2,3,5}-smooth number". But as long as we don't go for the "ugly number" name, I don't really have a strong opinion on which should be primary. I suppose "Hamming number" would fit with the longstanding mathematical tradition of naming things after latecomers to the study of those things (Hamming wasn't even the first to talk about algorithms for computing these numbers, he was merely the first to talk about generating them in order). —David Eppstein (talk) 21:13, 2 March 2008 (UTC)
Of course if we were to find the name of the Babylonian who first classified these numbers I'd be happy to name the page after her/him. :) CRGreathouse (t | c) 22:52, 2 March 2008 (UTC)
http://mathworld.wolfram.com/RegularNumber.html says: "A regular number, also called a finite decimal (Havil 2003, p. 25), is a positive number that has a finite decimal expansion." http://www.britannica.com/eb/topic-496213/regular-number says: "Regular numbers are those whose prime factors divide the base; the reciprocals of such numbers thus have only a finite number of places (by contrast, the reciprocals of nonregular numbers produce an infinitely repeating numeral). In base 10, for example, only numbers with factors of 2 and 5". The Babylonians used base 60 so a Babylonian regular number is consistent with the Britannica definition. MathWorld is about the reciprocal and assumes base 10, so no factor 3 allowed. How about splitting the article? Regular number could use the Britannica definition but also mention other definitions. Hamming number would only be about 5-smooth numbers. Babylonian mathematics and everything base-related could stay in Regular number, and base-independent 5-smooth things could move to Hamming number. PrimeHunter (talk) 02:12, 3 March 2008 (UTC)
I think much of the value of the article has to do with bringing together the different names people have used in very different subject areas for the same concept. Splitting it would lose that. The MathWorld decimal definition already went through an AfD and was deleted. And much of the content here looks specific to base 60. But if you can find sufficient sources to support an article on the general concept of regularity in any base, I don't see any reason to object to having the "regular number" title point to that article and having this article under a different title, with cross-links between the two. —David Eppstein (talk) 02:22, 3 March 2008 (UTC)
I didn't know Wikipedia:Articles for deletion/Regular number. It's probably best to keep the article in one place, maybe with a brief mention that other definitions exist. PrimeHunter (talk) 03:14, 3 March 2008 (UTC)
Why is this article not renamed yet?  It seems it has also been voted to be deleted once: https://en.wikipedia.org/wiki/Wikipedia:Articles_for_deletion/Regular_number
How about calling it "Babylonian regular numbers"?  "Regular numbers" is clearly not a right name (show me a non-Babylonian who believes that these numbers are more regular than the others). --Cokaban (talk) 14:16, 13 October 2013 (UTC)

Should Wikipedia record (and mention in such a prominent place as the opening paragraph of the article) a nonce term apparently made up and only ever used in the context of some forgotten computing contest? And may I ask what reason was given for such a label by whoever coined it? The term "ugly number" strikes me as terribly POV, and moreover its strong pejorative character seems completely ludicrous and unwarranted given the high notability and usefulness of these numbers for all kinds of purposes (starting by their utmost importance in geometry and music). It seems as if whoever thought such a dismissive label as "ugly" was somehow befitting for these numbers, must have thought that "number beauty" is measured by arcanity and lack of pragmatic value, so that the quantities that govern myriad aspects of our daily lives would appear "ugly" because they are to be found all around. 213.37.6.23 (talk) 09:57, 8 May 2008 (UTC)

Agreed. Removed. SethTisue (talk) 11:28, 9 May 2008 (UTC)
However, now "ugly number" redirects to this page without being mentioned anywhere in the article.

I expanded a previous footnote into a paragraph about computing sexagesimals in order, after discovering the Knuth and Bruins references on the subject as well as the Gingerich citation we already had. I'd also like to add something like "Eppstein (2007) describes an algorithm for computing tables of this type in linear time for arbitrary values of k.", with a later bibliography entry Eppstein, David (2007), The range-restricted Hamming problem. But as you can see, that would be a little self-serving. If someone else thinks this would be an appropriate addition, please go ahead and add it. —David Eppstein (talk) 18:22, 27 May 2008 (UTC)
The formula i use for calculating regulars in base 60 less than 60^N, is KN^2/2 - AN + B,  where K=(ln60)^3/ln(2)ln(3)ln(5) = 56.002707, A = 4.7, and B = 2.  For new entries, the formula is KN - C, where C = 32.  This is for N places.  In base 120, these are K=89.5324377, A=12.95, B=4, C=47.  The error is less than 1 over a range of N=30. Wendy.krieger (talk) 09:33, 19 February 2017 (UTC)

Is there a reason that this article keeps mentioning that these numbers can divide powers of 60? It seems strange, because that's just another way of saying that their prime factors are one of 2,3, or 5, isn't it? If there's anything more than that (that I'm missing), it probably should be clearly stated. Luminifer (talk) 05:18, 1 June 2009 (UTC)

60 is more important than 30 in the connections with the Babylonian number system. —David Eppstein (talk) 06:19, 1 June 2009 (UTC)
Sure, I can see that it's important historically, but shouldn't it just be stated that it was important historically? Mentioning it over and over again makes it pretty confusing in my opinion, when it could be stated more simply. Luminifer (talk) 16:13, 1 June 2009 (UTC)

In an article about the Hamming problem it is nice to have the solution. Since the article discusses how non-trivial the solution of the problem in a non-lazy language is, it is a good idea to include the non-lazy solution itself, especially, since it is only 10-lines long. The language of implementation is irrelevant, Python was just an example. I propose to undo the removal of the solution. GrGBL (talk) 11:58, 30 May 2010 (UTC)

I have to say that I support Arthur Rubin's removal of your Pyhton code. I don't think it adds anything to the article, and, if it is code you wrote yourself, then it is original research. Gandalf61 (talk) 14:02, 30 May 2010 (UTC)
It is not 10 lines long, it is 22. There is, however, a very similar 10-line non-functional Python implementation in the description of this article's figure (function A051037). I agree with Rubin and Gandalf: in general, detailed implementations of algorithms do not belong in Wikipedia articles and in this particular case it's too far off-topic to be helpful. In addition, it causes WP:NPOV problems by unbalancing the article too far in the computer-science direction, when that is only one of several aspects of the subject. —David Eppstein (talk) 16:22, 30 May 2010 (UTC)
I suspect the root of misunderstanding is the "merge" of this page with "Hamming problem" (it redirects here). You are absolutely right: in an article about regular numbers, code is useless; OTOH, in an article about "Hamming problem", its solution in a lazy and non-lazy languages is a must. I propose to make a separate article about Hamming problem.
Btw, the real code is 10-lines long and the rest is min_element function that is a part of C++ STL but seems to be absent from Python.
The fact that an interesting part of an article is hidden in the code that generates illustration is not obvious for a reader (btw, that code is not an example of traditional approach since it uses "yield").GrGBL (talk) 11:18, 31 May 2010 (UTC)
It's not intended as part of the article. And if you removed the "yield" statements or replaced them by prints you'd get a similar-length program that did the same thing. I still think that Wikipedia is not a code repository: pseudocode should be preferred to code and code should only be included to the extent that it provides a clearer description of an algorithm than other forms of text. So I'm wondering what the point of a separate article on the Hamming problem is: what aspect of the problem is not already adequately represented here? Why is a non-functional version important? Can you find reliable sources for that version? —David Eppstein (talk) 16:50, 31 May 2010 (UTC)
The point of a separate article is to discuss the Hamming problem itself: why the problem is significant, what are the approaches to its solution, solutions themselves (code or pseudo-code is not important, but code is more precise and is easier to verify). This can be done in this article, but apparently it has nothing to do with Regular numbers and thus a separate article (intended for a different auditory) is a better approach. GrGBL (talk) 11:18, 2 June 2010 (UTC)
You didn't answer the question: what is there to discuss about the Hamming problem that is not already in the article and would not belong in this article? Your non-functional Python code is not a good answer, because you have not yet convinced me that anything like it belongs in Wikiedia at all: it's not a clear way of describing an algorithm and it's not supported by sources. —David Eppstein (talk) 15:25, 2 June 2010 (UTC)

Hello fellow Wikipedians,
I have just modified one external link on Regular number. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:

Added archive https://web.archive.org/web/20110721032832/http://11011110.livejournal.com/95519.html to http://11011110.livejournal.com/95519.html
When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
This message was posted before February 2018. After February 2018, "External links modified" talk page sections are no longer generated or monitored by InternetArchiveBot. No special action is required regarding these talk page notices, other than regular verification using the archive tool instructions below. Editors have permission to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the RfC before doing mass systematic removals. This message is updated dynamically through the template {{source check}} (last update: 5 June 2024).

If you have discovered URLs which were erroneously considered dead by the bot, you can report them with this tool.
If you found an error with any archives or the URLs themselves, you can fix them with this tool.
Cheers.—InternetArchiveBot (Report bug) 21:45, 5 April 2017 (UTC)

This review is transcluded from Talk:Regular number/GA1. The edit link for this section can be used to add comments to the review.
Reviewer: Shushugah (talk · contribs) 21:51, 18 December 2021 (UTC)

@Shushugah: This does NOT use a deprecated style for referencing. It uses footnotes for referencing. The use of parenthetical author (year) short references in footnotes, pointing to more complete references in a later section, is not deprecated, neither is the use of parenthetical author (year) forms in the text of the article, as part of sentences of the article rather than as references. The only thing that was deprecated was extra-textual references formatted parenthetically in article text rather than in other ways. There used to be some of these in the article; I removed what was I believe the last one in preparing for this nomination, in Special:Diff/1059406410. The only remaining parenthetically-formatted reference, to Stormer 1897, occurs within another footnote, so is not covered by that deprecation. —David Eppstein (talk) 22:18, 18 December 2021 (UTC)

David Eppstein the deprecated source reads plainly to me to discontinue using Harvard style (name, year) inline i.e with This discussion supports the deprecation only of parenthetical style citations directly inlined into articles. Within a footnote itself is completely fine of course, but the way Gingerich (1965) was linked, I'd have expected to open Wikipedia article Gingerich (1965). Leaving this aside, the double usage of inline parenthesis reference AND footnote at Heninger, Rains & Sloane (2006)..... If someone wants to change a reference, they need to change a reference link twice, and in a very non consistent way from other reference styles in this article. Admittedly I personally don't care so much about the citation style including the deprecation, but in this case, I found 3 different inline styles, and that was confusing regardless. The article is so close to a GAN, I hope this is something we can resolve together. ~ 🦝 Shushugah (he/him • talk) 22:40, 18 December 2021 (UTC)
The deprecation discussion was entirely about references, and not about article text. The linked guideline you point to is entirely about references and anything it says about how to write the text of an article would be entirely out of scope and off-topic. The text in article sentences like "...after Richard Hamming, who proposed..." "...sequences at the On-Line Encyclopedia of Integer Sequences..." "...given by Srinivasa Ramanujan in his first letter to G. H. Hardy" "...interpreted by Neugebauer..." "Honingh & Bod (2005) list ..." "...were popularized by Edsger Dijkstra. Dijkstra (1976, 1981) attributes..." is text, not a reference. You can tell because it forms a grammatical part of the sentence it is contained in, rather than sitting there as an extra-textual lump like a footnote marker pointing to something else. The fact that the text happens to use parentheses in some instances, or that it happens to use publication years in some instances, does not make it any less text, and therefore not subject to any deprecation rules about referencing style. If I wrote "In 2005, Honingh & Bod" instead of "Honingh & Bod (2005)", it would have the same meaning as text, merely being unnecessarily un-concise. I argued strenuously at the deprecation RFC that this deprecation would lead to exactly this misunderstanding, and it appears that in doing so I was correct. If we're going to get pissy about referencing style, I might further point out that the WP:GACR do not even require references to have a consistent style, only that they have an appropriate layout into sections. —David Eppstein (talk) 23:02, 18 December 2021 (UTC)
I am not being pissy and would like for you to WP:AGF. I referenced 1b earlier, when I meant to reference 2b which refers to citation styles specifically, not the layout. I have corrected the criteria checklist and apologise for the confusion. I'll make a follow up edit for clarity.I am sympathetic to the fact that [inevitable] confusion was created with that RFC, but that's how consensus works. I still don't understand in this example 5-limit musical scales other than the familiar diatonic scale of Western music have also been used, both in traditional musics of other cultures and in modern experimental music: Honingh & Bod (2005) list 31 different 5-limit scales, drawn from a larger database of musical scales. is either an unreferenced statement completely, or does Honingh & Bod (2005) function as both text and a reference? In other cases you've made a footnote at end of the sentence, so could you do that for all these examples? I don't mind the text style that references the year, but once it's hyperlinked, it's not pure text for me, but is a reference/external link, and an inconsistent one at that. I'm new to GAN, so am happy to seek a 2nd opinion/review from an experienced GAN reviewer. ~ 🦝 Shushugah (he/him • talk) 01:36, 19 December 2021 (UTC)
There already were a couple of footnotes like that (in-text callout to author followed by the same callout in a footnote). I added more, I think catching all of them. It may seem silly to do it that way, but it has the advantage that the footnote marks the end of the text that is being referenced by that author. For instance, the footnote to Knuth is somewhat further down than the end of the sentence with the in-text callout. If you think of a footnote as something used to verify the content of an article, and the in-text callouts as intended to tell readers the history of who did what when rather than where to go to find out more, it might make the intent clearer. As another example, the in-text callout to Bruins (1970) is not used in a footnote (instead, the footnote goes to Knuth, who cited Bruins). —David Eppstein (talk) 08:50, 20 December 2021 (UTC)
@Shushugah: {{harvtxt}} here is used as an in-text attribution as opposed to a citation, which appears to be allowed. This also makes sense because the original reason for deprecating parenthetical citations was because of added clutter, but in the case of an in-text attribution the author would already have to be in the text anyway. So, in my view, that deprecation does not apply to this use of {{harvtxt}}, but I do think it is valuable to see what others think. (For the record, there are 150 GAs that use {{harvtxt}} in a prose context (but many of these uses are in footnotes themselves.)) eviolite (talk) 04:07, 22 December 2021 (UTC)
Happy to accept this GAN as successful! It's remarkable that 15 years ago this article was successfully deleted and today is of GA status. Thank you both for bearing with me regarding citation styles. I also found Wikipedia:What the Good article criteria are not which explicitly specifies that consistent citation is not required, so my apologies for insisting on that! Happy holidays everyone! ~ 🦝 Shushugah (he/him • talk) 10:53, 22 December 2021 (UTC)

I tried to edit the article, pointing out that sometimes the term "regular number" is used for non-integers, and adding something in the section on Babylonian mathematics about reciprocal tables and the remarkable one containing all the six-place regular numbers. David Eppstein systematically reverts whatever I do, claiming that my reference doesn't say what it says and that it doesn't matter if it does, and that I don't do the reference in the right way. I claim that those are not sufficient reasons to revert what I do. Eric Kvaalen (talk) 18:22, 14 January 2023 (UTC)

The fact that your reference doesn't say what it says is in fact a sufficient reason for reverting. Also, see WP:NOTDICTIONARY. This article is about a topic, the integers whose prime factors belong to {2,3,5}. It is not about a word or phrase, the phrase "regular number". Encyclopedia articles are about topics; dictionary entries are about words.  So if that word or phrase is sometimes used with a different meaning, that is not a fact about the topic of the article. —David Eppstein (talk) 18:39, 14 January 2023 (UTC)
But the article does say what I said it says. I even quoted the sentence to you in one of my edit comments. And your other arguments are not sufficient reasons for reverting everything I do. I don't see why you make such a big issue out of my pointing out that some people use the term, in the context of Babylonian mathematics, for certain fractions as well. Why should you revert an edit like that? And everything else I wrote. Eric Kvaalen (talk) 09:14, 15 January 2023 (UTC)

Thank you Firefangledfeathers. I certainly do object to his removal of what I wrote about the reciprocal table. And the fact that he systematically reverts everything I do. He doesn't like something about the format of the reference I gave (I don't understand what), so he uses that as one of his reasons for reverting the whole edit. As for the question of whether we should say that the term is sometimes used to mean certain fractions, I don't see what's so bad about mentioning this, perhaps with revised wording, like "can be used" or "has been used". That's a fact, and I don't see why we must prevent readers from being exposed to it. Eric Kvaalen (talk) 06:02, 22 January 2023 (UTC)

EK, in my work as third opinion volunteer, I try as much as possible to avoid sharing an opinion on conduct disputes, per the instructions given here. I encourage you to follow the steps of WP:CONDUCTDISPUTE. On the reciprocal table, do you feel adding a mention of it helps the reader understand something about regular numbers? Could you explain what, exactly? "Has been used" would be an improvement, in my view, but not enough of one to garner my support. There's still the issue of how formally or analogously Fowler and Robson are using the term, and it's very unclear how due it would be to mention this use of the term based on just the one source. I think both issues could be solved by finding more sources that use "regular number" to refer to non-integers. Absent such sources, I'm likely to continue to oppose including the content. Firefangledfeathers (talk / contribs) 06:21, 22 January 2023 (UTC)
I don't wish to start an official "conduct dispute". But I will address your question. Our article should give interesting information about regular numbers, including their history. The fact that someone thousands of years ago found all the regular numbers with six sexagesimal "digits" is amazing. (I did this in a spreadsheet and found 980!) And he found their inverses, which of course are fractions if we consider the starting numbers to be integers. So yes, this should be in the article, in the section on Babylonian mathematics. I still think we should mention the fact that fractions can also be called regular. We have one person, David Eppstein, who says "no they can't", and we have two authors, David Fowler and Eleanor Robson, who say "they can". Why should David Eppstein's opinion rule? Eric Kvaalen (talk) 18:20, 24 January 2023 (UTC)
It is false to say that David Fowler and Eleanor Robson have weighed in on whether a Wikipedia article on a certain class of integers should include a terminological variation involving non-integers and have taken your side on it. Don't make up falsehoods to bolster your narrative. —David Eppstein (talk) 19:10, 24 January 2023 (UTC)
That's not what I said. I said that they say that certain fractions can be called regular numbers. To be more precise, they didn't say it, they just do it. Eric Kvaalen (talk) 17:04, 25 January 2023 (UTC)
@Firefangledfeathers: Are you still watching this Talk page? What do you think of my argument that we have two experts in Babylonian mathematics who use the term "regular number" for certain fractions, and only one person who is saying we mustn't say that? And what about my contention that I should be allowed to mention the table of regular numbers and their reciprocals in the section on Babylonian mathematics? Eric Kvaalen (talk) 10:21, 31 January 2023 (UTC)
Hi Eric Kvaalen. I prefer to let local discussion go stale again, which it has, so thanks for the ping. I remain unconvinced on fractional regular numbers. The "two authors who say they can" argument just doesn't fly. Any published view could be promoted with this rationale, violating WP:DUE. Of the hundreds of sources (I'm estimating) that are relevant to this topic, is the only support coming from one page of one paper? I still think the best avenue to inclusion of this is to find more sources.
I think a shorter mention of the six-place table is worth mentioning. I have some questions/comments. Is it true that Inaqibıt-Anu's tablet "contains all the six-place reciprocals of regular numbers"? Fowler and Robson do not make this claim, and Knuth references a tablet that has all the six-place (or fewer) reciprocals of regular numbers that start with 1 or 2. He says that other tablets continuing the list were made but have since been lost. Am I misinterpreting? I also see Knuth's quote in context as referring to the sorting of the large list of numbers (more than on just the one surviving tablet), which is a bit outside the scope of this article. Unfortunately, while I can access Knuth's original paper here, I haven't been able to find an accessible version of his correction, which Fowler and Robson emphasize we must read. Firefangledfeathers (talk / contribs) 18:51, 31 January 2023 (UTC)
@Firefangledfeathers: I am not claiming that the term "regular number" MEANS integers and fractions with the given form, I'm just claiming that it is sometimes used for fractions. The single reference (Fowler and Robson) suffices to prove this. I don't see why David Eppstein finds it so important to prohibit this remark!
You're right about the table of inverses. It has been a month or two since I made the edit, so I couldn't remember exactly what my source of information was, but I have used my Firefox history to relook at all the articles I looked at back then, and my conclusion is that that particular fact came from this same Wikipedia article, lower down, in the section "Algorithms": "A related problem, discussed by Knuth (1972), is to list all k k-digit sexagesimal numbers in ascending order, as was done for k = 6 by Inakibit-Anu, the Seleucid-era scribe of tablet AO6456." That sentence was added a few years ago by David Eppstein, but it is apparently incorrect, as you have discovered. I have also tried to find the correction to Knuth's article. The on-line archive of the relevant issue of the Communications of the ACM (here) does not contain it. But I have found a version by Knuth from after the correction (here). Just don't know what the correction was! (Maybe you can tell us, since you have already looked at the uncorrected version.) Anyway, I propose to add a correct mention of this table in the section on Babylonian mathematics.
Eric Kvaalen (talk) 20:25, 3 February 2023 (UTC)
 On "I am not claiming ...", I didn't say you were! If something I said made it seem like that, it wasn't intentional. You've reiterated your argument, but it hasn't changed my mind. Thanks for linking CotACM, which does include Knuth's correction. It's in the "Letters" segment, and here's a PDF link. It makes it clear that Knuth's opinion that Inaqibıt-Anu had collected all the up-to-six place regular reciprocals is just that: an opinion. Knuth owns up to misreading a source and quotes another researcher who thinks there was only one tablet. Sadly, it looks like the surviving tablet doesn't even include all the reciprocals starting with 1 or 2, only 136 of the 231. The line in §Algorithms needs to be updated, at the very least, to state that the extent of Inaqibıt-Anu's accomplishment is only hypothesized by Knuth, and that even he doesn't think the scribe got them all. Firefangledfeathers (talk / contribs) 02:46, 4 February 2023 (UTC)

I was complaining that David Eppstein wouldn't let me mention how Fowler and Robson use the term because he disagrees with them. Anyway, I have now edited the article to include a correct description of the tablet of Inaqibıt-Anu. Thanks for the link to the correction of Knuth. Actually, the old version contains an interesting appendix which is not in the newer version. You know, he has offered a check to anyone who would find an error in one of his papers, and I wrote to him once about an error in a paper quoting him, but apparently it wasn't in his original! He wrote me a nice reply. Eric Kvaalen (talk) 11:00, 8 February 2023 (UTC)

The version you added still has some of the issues mentioned above, and some new ones:  Firefangledfeathers (talk / contribs) 18:28, 8 February 2023 (UTC)
An additional minor issue: the new reference violates WP:CITEVAR by using Citation Style 1 ({{cite journal}}) rather than Citation Style 2 ({{citation}}).
FFF's point about the Knuth quote being about sorting rather than calculation is important.
I also have the feeling that some of this new material is heading into material that does not "stay focused on the topic without going into unnecessary detail", one of the Good Article Criteria (#3b) so something that we need to pay attention to for continued Good Article status. The topic of this article is the integers of the form 
  
    
      
        
          2
          
            i
          
        
        
          3
          
            j
          
        
        
          5
          
            k
          
        
      
    
    {\displaystyle 2^{i}3^{j}5^{k}}
  
, not really other stuff the Babylonians might have done (like carrying out sorting tasks). —David Eppstein (talk) 18:40, 8 February 2023 (UTC)

I made a couple of those changes. I changed the citation from "cite journal" to "citation" but it made absolutely no difference in the result! I don't think Knuth was referring just to the sorting. After all, it's not that hard to sort a couple hundred numbers (although I admit it might be more difficult with clay tablets in place of index cards!), whereas calculating those inverses is a LOT of work! Eric Kvaalen (talk) 10:38, 9 February 2023 (UTC)
In fact, "cite journal" and "citation" do produce different results. Most obviously. Cite journal. Separates. The citation. Into lots of. Annoying. Sentences. With periods. Citation does not. —David Eppstein (talk) 12:22, 9 February 2023 (UTC)