1 has been listed as one of the Mathematics good articles under the good article criteria. If you can improve it further, please do so. If it no longer meets these criteria, you can reassess it. Review: September 29, 2024. (Reviewed version).

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Numbers Top‑importance This article is within the scope of WikiProject Numbers, a collaborative effort to improve the coverage of Numbers on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.NumbersWikipedia:WikiProject NumbersTemplate:WikiProject NumbersNumbers articles Top This article has been rated as Top-importance on the project's importance scale. Mathematics Top‑priority This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.MathematicsWikipedia:WikiProject MathematicsTemplate:WikiProject Mathematicsmathematics articles Top This article has been rated as Top-priority on the project's priority scale.

A fact from 1 appeared on Wikipedia's Main Page in the Did you know column on 17 October 2024 (check views). The text of the entry was as follows: Did you know... that the number 1 is its own square, square root, and factorial? A record of the entry may be seen at Wikipedia:Recent additions/2024/October. The nomination discussion and review may be seen at Template:Did you know nominations/1.

This is a section to discuss various facts about the number one, whether (and how) they should be included, and hopefully, seek consensus on the issue. Allan Nonymous (talk) 14:40, 8 August 2024 (UTC)[reply]

I am happy to go through the article line by line if need be. Polyamorph (talk) 18:45, 8 August 2024 (UTC)[reply]

We should do this for all the articles in question, I believe. I will be removing my own SYNTH for all articles I was involved in, which should conclude after a couple of days. For this article, I think most points are valid frankly, they are very tacit and precise, as well as sourced. Radlrb (talk) 21:08, 8 August 2024 (UTC)[reply]

"Mathematically, the number 1 has unique properties and significance."

Redundant sentence, yes, we know 1 is special, its in the lede. We can remove this here

"In normal arithmetic (algebra), the number 1 is the first natural number after 0 (zero) and can be used to make up all other integers (e.g... etc.)." While correct, the way this is phrased is slightly incorrect, this is a fact about the successor function and the succession of numbers. I have rewritten this to that effect. Allan Nonymous (talk) 14:12, 9 August 2024 (UTC)[reply]

Why are you not waiting for a response first? I thought you said you were going to stop editing until there was a consensus reached, in WP:WP Numbers for guidelines to be established, and here for each point needing discussion. Polyamorph asked you that he can go line by line. Radlrb (talk) 14:41, 9 August 2024 (UTC)[reply]

The lead is meant to summarise the article contents, so one would expect content present in the article to be present in the lead. That said, I don't have an issue removing that first sentence. I have modified your revised sentence relating to the succession of numbers so that it fits better as the first sentence of the paragraph. For this I feel it is important to mention that it succeeds 0. Polyamorph (talk) 15:12, 9 August 2024 (UTC)[reply]

The latter part, that it follows 0, is indeed fundamental. Thank you for returning that point, it felt a bit vacuous without it. 0 and 1 are adjacent, and with remarkably dissimilar and similar properties, so it is important to mention their juxtaposition in the set of natural numbers. Radlrb (talk) 16:18, 9 August 2024 (UTC)[reply]

A number following another number isn't really "fundamental" in fact, almost no fact about a number is fundamental (which is an important part of group theory). Is it due? Maybe, I would say no, per the emerging guideline at NROUTINE (as 2 follows 1, etc.) Allan Nonymous (talk) 17:53, 9 August 2024 (UTC)[reply]

It is fundamental in that it is first, the very beginning of countability, which needs to be mentioned so that an understanding of the distinction between N_0 and N^+1 can be introduced, and the "disambiguity" that comes from this, where different authors make use of either different definition in different ways depending on contexts. Secondly, because their properties are soo powerful individually, the proximity between them makes it more remarkable than that from between for instance, 100001 and 100000. Radlrb (talk) 20:29, 9 August 2024 (UTC)[reply]

1 being the first or 2nd element of the naturals is already discussed everywhere else, however, this fact might actually qualify on being a basic fact. Allan Nonymous (talk) 20:35, 9 August 2024 (UTC)[reply]

"1 is the only natural number that is neither composite (a number with more than two distinct positive divisors) nor prime (a number with exactly two distinct positive divisors) with respect to division." is redundant with prior discussions of the primality of one, which cover this topic and are mentioned here.

"The product of 0 numbers (the empty product) is 1 and the factorial 0! evaluates to 1, as a special case of the empty product." This is about 0!, and should probably be paired with facts about 1!. In general, this, and facts about 1 being the first of many sequences should be rewritten as "By convention, 1 is the start of several sequences."

"Any number multiplied or divided by 1 remains unchanged. This makes it a mathematical unit, and for this reason, 1 is often called unity." The etymology is unsourced, and it looks like this sentence means to say a different thing. What is meant to be said here is that 1 is the identity under multiplication in the real numbers.

"Consequently, if is a multiplicative function, then must be equal to 1." this is NOFFTOPIC. Removing

"This distinctive feature leads to 1 being is its own factorial, its own square and square root, its own cube and cube root, and so forth." Given there is nos source, this is WP:OR. Removing.

"By definition, 1 is the magnitude, absolute value, or norm of a unit complex number, unit vector, and a unit matrix (more usually called an identity matrix)." This is clunky phrasing of the concept, given that magnitude, absolute value and norm are all norms.

"1 is the multiplicative identity of the integers, real numbers, and complex numbers." This is correct, but true because each of these sets of numbers extends the integers, since we have already discussed the multiplicative identity above, it is redundant. If someone wants to describe why 1 is also the multiplicative identity of these sets (as well as quotient fields, among others), feel free to do so but that should probably belong somewhere further down in the article. Allan Nonymous (talk) 18:14, 9 August 2024 (UTC)[reply]

I haven't had a chance to review these comments yet (and I'm about to hit the sack) but I've reverted your changes because you did not have agreement to remove the content. You have to give us a chance to actually read your comments and respond before removing content in mainspace. By continuing to delete content before getting consensus is simply a continuation of the same disruptive pattern that you have been warned for. I am expecting this page to be edit protected shortly. (@Daniel Case and Johnuniq: pinging two admins familiar with this case for their independent assessment.) Polyamorph (talk) 21:18, 9 August 2024 (UTC)[reply]

I really wish you would engage with the edits made rather than reverting them, but, to be perfectly honest, it is your choice as to how you wish to respond. Either way, now that you've reverted, I hope you can address the rationale behind the edits. Allan Nonymous (talk) 22:09, 9 August 2024 (UTC)[reply]

Just stop deleting content. You don't have consensus and have been warned many times to stop. When I have engaged with your edits directly in mainspace, you have reverted my changes anyway. So stick to your agreement to stop editing and discuss. This means wait for a response before going ahead and implementating your deletions. Polyamorph (talk) 07:01, 10 August 2024 (UTC)[reply]

@Beland: as an admin who is actively helping mediate the current disputes across numbers articles. Your perspective would be welcome. Polyamorph (talk) 07:15, 10 August 2024 (UTC)[reply]

Short version is a blunt but bedrock cliche: It's probably best to comment on content, and not the behavior of other editors.

Long version is a smattering of responses to specific events and general advice which might be useful to anyone in this discussion:

The edit history will remember everything that is removed, and you can always look at a previous version. It doesn't have to be put back in the article in order to be reviewed. It seems unhelpful to revert a removal without giving an actual objection on the merits, especially if you are in fact intending to look through the removed content in detail and ponder the reasons given. After a dispute has been identified, edits, including reverts, should generally be trying to make progress. Instead of putting everything back blindly, it would be more helpful to only put back what you actually have a content-based reason to keep, and explain that reason as much as possible. It's also helpful, instead of just restoring the same text again, to put up a new version that responds to the concerns expressed by the remover, for example with better sourcing if the objection was "original research" or "NOTOEIS".

Personally, I try not to unilaterally repeatedly undo the changes someone else has made (whether adding or removing), especially if they are exactly the same material, unless there's a clear policy violation like unsourced material in a BLP or something. Even if it's an NPOV violation, I can usually live with the "wrong" version for a few days while it's under discussion by adding a problem tag to the article to highlight the dispute for readers.

People do need to work with others to resolve conflicts and build consensus, but I don't find procedural objections to be particularly helpful. Often if I do A people will complain that I need to do B first. Then other times I'll do B, and people will complain I need to do A first. This isn't a bureaucracy, it's a community of volunteers. I think instead of trying to force other people into our own way of doing things, it's better just to be happy that someone is actually paying attention to articles and trying to build a better encyclopedia and try to make progress from whatever most recent starting point they've given you. For reasonable requests, though, I do often indulge people in whatever procedure they are demanding, because this tends to make them less unhappy and more cooperative, and removes a lot of excuses for saying "no". It does sloooow things down, though, and generally I'm an impatient person, so it takes a bit of effort and is quite frustrating when we end up getting slowly to the same place I thought I should have been allowed to get to quickly. There are also times when an in-depth back-and-forth makes for better-sourced, more neutral, clearer, more informative, or just generally better content, and that's the benefit of having other people question my work and make me defend it.

If there's not an obvious way to make progress by refining content (for example if the dispute is whether or not a fact is important enough to include) then the way to make progress is discussion. I think the time to go back to making edits is when the discussion has either concluded, or reached a point where redrafting to address others' concerns is useful. If the discussion is stalled, I usually hold off on jamming my preferred version back in until I can get another editor to sanity-check my arguments and make sure I'm not in the wrong or at least not in the minority of editors. (Wikipedia:Third opinion is very helpful for that on low-traffic articles, but in this case there are plenty of watchers who will no doubt chime in or could be summoned from the WikiProject discussion.)

Not sure why everyone has gotten so emotionally involved, but this is just a list of facts about numbers. Yes, our articles should be clutter-free, but readers will be able to cope for a day or two by skimming past some boring parts if needed. Having a few facts gone for a day or two that we later decide to restore isn't going to ruin anyone's day. If they are looking for a specific fact, kind of by the definition of no original research, if it's something we're supposed to have here, they should be able to look it up somewhere else. If I need to get some sleep before slogging through some dense content and saying something intelligent about it, I'd consider just leaving things a mess overnight and going to do that. I know I'm more thoughtful and less prickly after a good night's rest, and much better at handling a complicated negotiation, and my brain will work on it overnight anyway and I might have some better ideas in the morning. -- Beland (talk) 08:59, 10 August 2024 (UTC)[reply]

I have created a sandbox at Talk:1/Draft 1. I suggest we can edit this collaboratively, justifying each edit with detailed edit summaries or here if necessary. Once we arrive at a version we can all agree on then we can request a history merge into mainspace. I suggest we focus on the In mathematics section since this is the most contested. Polyamorph (talk) 14:36, 10 August 2024 (UTC)[reply]

Answering some of the points noted in the section above:

• "1 is the only natural number that is neither composite (a number with more than two distinct positive divisors) nor prime (a number with exactly two distinct positive divisors) with respect to division." is redundant with prior discussions of the primality of one, which cover this topic and are mentioned here. - agree that if this is covered in the subsection "Primality". Polyamorph (talk) 17:03, 10 August 2024 (UTC)[reply]
• "Any number multiplied or divided by 1 remains unchanged. This makes it a mathematical unit, and for this reason, 1 is often called unity." The etymology is unsourced, and it looks like this sentence means to say a different thing. What is meant to be said here is that 1 is the identity under multiplication in the real numbers. - since unity is an often used synonym of 1, it should be explained. But I agree this etymology does not seem correct and needs revising.
• "Consequently, if is a multiplicative function, then must be equal to 1." this is NOFFTOPIC. Removing - not off topic, but it is duplicating detail already provided on the multiplicative identity.
• "This distinctive feature leads to 1 being is its own factorial, its own square and square root, its own cube and cube root, and so forth." Given there is nos source, this is WP:OR. Removing. - I have added a source.
• "1 is the multiplicative identity of the integers, real numbers, and complex numbers." This is correct, but true because each of these sets of numbers extends the integers, since we have already discussed the multiplicative identity above, it is redundant. If someone wants to describe why 1 is also the multiplicative identity of these sets (as well as quotient fields, among others), feel free to do so but that should probably belong somewhere further down in the article. - I don't see where the redundancy is/was (it's mentioned in the lead but since the lead is a summary of the article contents then it is expected to reproduce some of the main text. I've merged this into the top of the section.

Polyamorph (talk) 17:03, 10 August 2024 (UTC)[reply]

On unity:

This makes it a mathematical unit, and for this reason, 1 is often called unity.

This is unsourced and it is not clear that this is correct, the unity synonym is important to mention, but is it simply from Anglo-Norman word for one? Really that belongs in the Entymology section. I've removed from the draft section. Polyamorph (talk) 12:19, 12 August 2024 (UTC)[reply]

I think linking to unit is ok. "for this reason,..." is dubious for the reason you state. I think that one is called unity has to be mentioned somewhere. Tito Omburo (talk) 12:23, 12 August 2024 (UTC)[reply]

The SOED gives 1570 as a citation for "unity" meaning one, which I think predates any notions of "unit" being an identity multiplier. Imaginatorium (talk) 13:14, 12 August 2024 (UTC)[reply]

• Note given the page protection has expired, I asked an admin to history merge the draft back to mainspace, so we can continue editing there. Polyamorph (talk) 15:17, 14 August 2024 (UTC)[reply]

@Dedhert.Jr: Do you think this is improved sufficiently to increase the rating back to class B? Also, any opinion on what is needed to push this to GA? Polyamorph (talk) 08:54, 21 August 2024 (UTC)[reply]

@Polyamorph See WP:QUALITY for looking the criteria of B-class: they are suitably referenced with reliable sources in terms of inline citations, covers on topics and avoid inaccuracy, defined structure, reasonably well-written, and approriate the wording in understandable way (see WP:TECHNICAL).

See WP:QUALITY as well for GA. But if someone would like to engage the nomination of GA, I have a discussion from a user whose make good qualities. This discussion is only if someone already has done it once but would like to pursue the understanding of how to have a better preparation, but of course, it is fine to find any tips from another user. You can also check the six criteria.

For a small summary from me, although I am not actually expert at this field. This leads me to a quick question: Is the article primarily focused on mathematics? If so, should we have put first the "In mathematics" section? Also, what is the point of adding sign language video in the infobox, clock, and typewriter as well? Is there any mention somewhere, unless if these both are important or explained in the examples phrasing? Try to use prose for a better influence of reading instead of bullets. You might need to look for more examples in our mathematics GA. Dedhert.Jr (talk) 10:04, 21 August 2024 (UTC)[reply]

(edit conflict) Possibly agree with "In mathematics" moving up, although usually entymology and history sections come before others. The typewriter is an illustration of the missing numeral 1, as referenced in the main text "As such, many older typewriters do not have dedicated key for the numeral 1 might be absent, requiring the use of the lowercase letter l or uppercase I as substitutes.[16]". The clock is an example of the use of J instead of I or 1, as referenced in the main text "It is also possible to find historic examples of the use of j or J as a substitute for the Arabic numeral 1.[17][18][19][20]" I'm not sure about the sign language, but I guess if the infobox includes all representations of the number then for completeness this should include the sign language. There seemed to be agreement at WP:NUM that the infobox can contain items not introduced in the article per MOS:INFOBOXEXCEPTIONS. Polyamorph (talk) 10:25, 21 August 2024 (UTC)[reply]

@Polyamorph I know, but what does it have to do with the sign language in this article? Do we also have to put more videos about sign language in many articles on numbers: 2, 0, etc.? Is it relevant? Is it helpful? Is it an encyclopedia? Our article has too many problems to deal with, because of the infobox exceptions. Even so, I have to point out not all information can be included in the info box. The infobox is basically a summary. Dedhert.Jr (talk) 10:32, 21 August 2024 (UTC)[reply]

OK, the sign language is gone. What else should go from the infobox? Polyamorph (talk) 12:06, 21 August 2024 (UTC)[reply]

I've rearranged the sections so that "In Mathematics" is first, putting "etymology" and "modern usage" under a "As a word" heading. Polyamorph (talk) 12:16, 21 August 2024 (UTC)[reply]

@Polyamorph: I have watched you tidying up masses of these articles; thanks. But I disagree with removing 1 as the atomic number of nitrogen. You say it is in the disambiguation page, which is true. I think that atomic numbers are such a fundamental physical fact that they do belong in these articles, and hydrogen should be removed from the disambiguation list, because nobody, nobody, ever has looked up "1" to find hydrogen. 1 does not mean hydrogen, in any sense, but there is a neat mathematicophysical connection. (There were three snippets; I would be very open to a discussion as to whether they all belong.) Imaginatorium (talk) 09:02, 22 August 2024 (UTC)[reply]

OK. I was in two minds whether to keep it per WP:NUM/NOT. I'll put it back. Polyamorph (talk) 09:05, 22 August 2024 (UTC)[reply]

P.S. I've nominated the article for GA review, if someone who hasn't edited the article significantly wants to take a look. Polyamorph (talk) 09:07, 22 August 2024 (UTC)[reply]

P.P.S. I think being part of group 1 elements is more important than period 1, and so have restored the former and excluded the latter. Polyamorph (talk) 09:16, 22 August 2024 (UTC)[reply]

Thanks. That seems very reasonable. These articles do contain some amazing fluff, but a lot of it not susceptible to mass deletion. Imaginatorium (talk) 09:53, 22 August 2024 (UTC)[reply]

Not sure that any extended discussion of the definition of primality really belongs here at all. But it seems dubious at present (and even more in earlier versions). It feels we are being told that the truth is that primes are non-units, whereas of course it is only a matter of definition.

This is because 1 is the only positive integer divisible by exactly one positive integer, whereas prime numbers are divisible by exactly two positive integers and composite numbers by more than two positive integers.

Well, no, it is not because..., this is just a rewording to justify the convention. Is this really needed?

...exclude due to its impact upon the fundamental theorem of arithmetic and other theorems related to prime numbers.

Well, again it doesn't really affect the prime factorisation theorem, only its wording. You have to say "...unique factorisation up to order", but you could also say "up to order, sign, and units", meaning that as factorisations of 18, (3, 2, 3) and (1, -2, -3, -1, 3) are the same.

Anyway, I think this section should be minimalised, and mostly just point to the prime number article. Imaginatorium (talk) 10:51, 22 August 2024 (UTC)[reply]

Agree, it's just a matter of definition. It's convenient to not be prime, to avoid definitions getting cumbersome. It can be pruned. Polyamorph (talk) 11:34, 22 August 2024 (UTC)[reply]

 Done Polyamorph (talk) 12:30, 22 August 2024 (UTC)[reply]

@AirshipJungleman29, Launchballer, Dedhert.Jr, and Chaiten1: Seems like this was approved, promoted by AirshipJungleman29, and ready to go in prep area 6 until apparently rejected by Crisco 1492. What happens now? Polyamorph (talk) 15:55, 12 October 2024 (UTC)[reply]

I've asked for it to be restored here: Wikipedia_talk:Did_you_know#Template:Did_you_know_nominations/1 Polyamorph (talk) 17:05, 12 October 2024 (UTC)[reply]

It wasn't actually rejected, just pushed back forward! Phew Polyamorph (talk) 17:42, 12 October 2024 (UTC)[reply]