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Reference the "Binomial series" and remove examples when n is not a positive integer.

I suggest that following the first sentence: "In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial." that a sentence is added stating "For binomial expansions where the exponent is not a positive integer, see the article regarding the Binomial series."

Then, I suggest that the examples (pasted below) regarding square roots (n=1/2) and negative exponents (n=-1) be removed, because they are clearly out of scope of this article.

"Another useful example is that of the expansion of the following square roots:

${\displaystyle {\sqrt {1+x}}=\textstyle 1+{\frac {1}{2}}x-{\frac {1}{8}}x^{2}+{\frac {1}{16}}x^{3}-{\frac {5}{128}}x^{4}+{\frac {7}{256}}x^{5}-\cdots }$

${\displaystyle {\frac {1}{\sqrt {1+x}}}=\textstyle 1-{\frac {1}{2}}x+{\frac {3}{8}}x^{2}-{\frac {5}{16}}x^{3}+{\frac {35}{128}}x^{4}-{\frac {63}{256}}x^{5}+\cdots }$

Sometimes it may be useful to expand negative exponents when ${\displaystyle |x|<1}$:

${\displaystyle (1+x)^{-1}={\frac {1}{1+x}}=1-x+x^{2}-x^{3}+x^{4}-x^{5}+\cdots }$"

See from http://www.haverford.edu/physics/MathAppendices/Binomial_Expansions.pdf, which apparently are public domain and in my opinion has a better "Statement of the theorem" than this article. For example, it also states that "The series in eq. (1) can be used for any value of n, integer or not, but when n is an integer the series terminates or ends after n+1 terms." 207.107.66.194 (talk) 15:00, 6 July 2015 (UTC)

I agree with both proposed edits (removal of out of scope examples and addition of a sentence pointing to the binomial series, possibly to the lead of the article where it can easily be located). Sławomir Biały (talk) 17:38, 6 July 2015 (UTC)

References

1. Notes on Binomial Expansions and Approximations

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Examples

In the examples section there are some with fractional or negative exponent, but the generalized binomial theorem that deals with those cases is only mentioned later. Up to this point only non-negative integer values of $n$ have been considered and the summation is always from 0 to ${\displaystyle n}$. — Preceding unsigned comment added by Pacosantosleal (talk • contribs) 12:24, 25 November 2015 (UTC)

Meaning of n choose k

${\displaystyle Does{n \choose k}}$ mean n_k ? — Preceding unsigned comment added by Skk146 (talk • contribs) 15:49, 27 November 2016 (UTC)

History

Joel B. Lewis I think it's better to cite Rashed here, because the source used is St Andrews, but while O'Connor and Robertson quote Rashed's statement, and apparently agree with it, the statement itself is Rashed's alone, and would be best cited directly to the source where he makes it. Not doing so when the source is readily available online is very poor scholarly practice. Best regards.---Wikaviani (talk) 00:37, 5 May 2018 (UTC)

recent edits

To help settle this recent conflict, I'm starting this talk page section. To the IP who keeps edit warring their preferred version in, there are a lot of inappropriate spacing changes you're making, as well as things like changing the capitalization of piped links (the link part, not the visible part); this should not be done. There are also a lot of invisible wikicode changes you're making. These are fine in and of themselves, but edits should not be made just to do that. See WP:COSMETICBOT for further info about these. WP:BRD is a pretty good idea. If someone reverts your changes from the status quo, it's really up to you to try to justify the changes, not make snarky retorts that don't address the disagreement. So again, please discuss here rather than trying to re-add. –Deacon Vorbis (carbon • videos) 21:05, 28 April 2019 (UTC)

The only reply to your only point, which is to assert "inappropriate", is to assert "appropriate". I am not going to further engage in pointless rhetoric with people want to claim that some standard should be met, never stating that standard, and moving it when it pleases them.80.65.247.112 (talk) 22:57, 28 April 2019 (UTC)

If you're unwilling to discuss, then you have no basis to continue to insist on your changes. If there's objection to those changes, you need to pause and resolve that objection rather than just plowing ahead. I've made some specific points here which should be addressed. Since you're unhappy with my use of "inappropriate", I'll elaborate on the spacing. You're using little-used manual spacing templates that aren't well known, aren't needed, produce inconsistent results, run contrary to MOS:MATH, and thus make maintenance more difficult. You've also added delimiter sizing commands that aren't needed – like ${\displaystyle (x^{n})}$ vs. ${\displaystyle \left(x^{n}\right).}$ This produces no difference in rendered output. A small handful of the changes you made were reasonable, mainly display mode vs. text mode for binomial coefficients. I reinstated those (which you took back out in your rush to revert me), but the vast majority were either unnecessary or actively harmful. I don't know why you feel such a need to see these edits stand, but there are good reasons why they shouldn't. –Deacon Vorbis (carbon • videos) 23:18, 28 April 2019 (UTC)

I notice that 80.65.247.112 doesn't want to say much on this Talk page, and doesn't leave edit summaries to give some hint as to his motivation. Neither of these habits is compatible with the way we do things on Misplaced Pages. Conversely, Deacon Vorbis not only shows a willingness to use the Talk page constructively, he shows an enthusiasm for doing so. Therefore I endorse the standard advocated by Deacon Vorbis. The editor at 80.65.247.112 should either adopt the Misplaced Pages way of doing things, or get used to the idea that all his hard work will one day amount to nothing. It should be an easy decision. Dolphin (t) 13:16, 29 April 2019 (UTC)

Multiple issues

Currently only a history is provided with enough citations and the rest is poorly sourced. Would be nice of someone adds a couple of links out there. DAVRONOVA.A. ✉ ⚑ 08:27, 17 June 2019 (UTC)

That's only one issue :p. Abstractly you are certainly right; concretely, are there any particular places you think are suspect (either, might be wrong, or with inappropriate weight, or original research)? Because "please make the article better" alone is not very constructive. --JBL (talk) 14:28, 17 June 2019 (UTC)

Original Research in substitution of of e^ax and e^bx in general Leibniz formula section of generalizations

Hi! The substitution of e^ax and e^bx into the general Leibniz formula seems to have no sources and thus seems to be original research. Although I understand that this substitution is correct mathematically, but there are no published sources having the same idea. Please take a moment to review my edit and verify my changes. The two changes I made are:

• Added tag to the e^ax, e^bx substitution claim
• Added to the source "Calculus in One and Several Variables" by Robert Seeley since the book does not contain the Leibniz formula at all. The "Calculus for One Variable" can be found here, available for digital borrowing. This is single-variable calculus part of the combined book. Therefore, the Leibniz formula is expected to be in here. However, after extensive searching, no record of the Leibniz formula was found here.

I understand that a similar issue was addressed under the "Multiple issues" section, but I wanted to keep a concrete discussion just for this change. If a citation is found, please feel free to enter a citation. Otherwise I will be forced to remove the content according to the Misplaced Pages: Original research policy. Thank you. --Dh*Phoenix (talk) 17:40, 20 June 2020 (UTC)

This isn't OR; I'm having trouble finding a good ref for it, but a quick google search finds a lot of message-board type expositions of this fact. While a source would be ideal, the ease with which one finds non-RS discussion + WP:CALC probably lets this one stand. As for the Leibniz rule, there's another source at the main article that you could probably use if you think it's better. You could always just cite Abramowitz & Stegun too, or probably hundreds of other Calculus textbooks. –Deacon Vorbis (carbon • videos) 18:54, 20 June 2020 (UTC)

Ref added for the binomial theorem as a corollary of the Leibniz rule. –Deacon Vorbis (carbon • videos) 19:38, 20 June 2020 (UTC)

Yes, I'm sorry: this is not original research. Thank you for the source! And yes, I find the main article Leibniz rule citation better: I'll correct that.--Dh*Phoenix (talk) 05:47, 21 June 2020 (UTC)

Applications

The section on the infinite series for e is alright up until "This indicates that e can be written as a series:", at which point rigor and accuracy is lost. See Rudin, Priniciples of Mathematical Analysis, Section 3.3.1, for a proof. The mistake in the page as it exists is that it is not sufficient to say that the in the limit kth term is 1/k! implies that the entire series of 1/k! terms is equal to the original series. — Preceding unsigned comment added by 192.80.55.86 (talk) 18:20, 29 November 2021 (UTC)

History section

@Jacobolus: Hi, I just came to see your recent edits. In my humble opinion, sources like Amulya Kumar Bag, Biggs ... cannot challenge the views of prominent experts of the history of mathematics like Roshdi Rashed, Robertson, O'Connor as per WP:WEIGHT. According to these sources, Al-Karaji discovered the binomial theorem. please let me know if you think I'm missing something. Best.---Wikaviani 21:18, 2 December 2024 (UTC)

You can read the historical Indian sources or their translations directly for yourself. It is not at all controversial to claim that what is described is binomial coefficients / "Pascal's triangle". I haven't looked at Rashed's book (paper?) but this sounds like a miscommunication.

It is entirely plausible though that these topics were developed independently in India, the Islamic world, and China. Who counts as "discoverer" or "inventor" seems like a pointless argument; we should just try to describe precisely what various historical scholars did, and let readers draw their own conclusions. –jacobolus (t) 21:46, 2 December 2024 (UTC)

I have checked your sources, honestly, nothing very impressive, some obscure mathematicians of Indian origin, mainly. Also, the historical section should give a fair representation of the history of this theorem, the discovery of it might be pointless for you, but not for our readers. The page you're looking for is 63 in Rashed's work. Here, another source repeating what Rashed says :"THE BINOMIAL THEOREM : A WIDESPREAD CONCEPT IN MEDIEVAL ISLAMIC MATHEMATICS" (PDF). core.ac.uk. p. 401. Retrieved 2019-01-08..---Wikaviani 22:00, 2 December 2024 (UTC)

The authors involved here are world class experts in the history of Indian mathematics. I find your dismissive tone pretty insulting to be honest, and elevating one or another religious or ethnic group for point-scoring purposes by erasing the contributions of other groups is one the things I find personally most unpleasant and unfortunate about discussions of math history both off and especially on Misplaced Pages. Mathematics is a great human achievement which should be celebrated rather than fought over. Rashed does good work, and is well worth citing as an authority on developments among medieval Islamic mathematicians, but it's also not like he's omniscient. –jacobolus (t) 23:44, 2 December 2024 (UTC)

It's about discussing the quality of the sources, not about making it personal, and that's a legit concern of mines. So according to you, "Amulya Kumar Bag" is a world class expert ? Wow, so how many influencial books has this guy published ? with what publisher ? how many awards has he ? how about his academic career ? The source number 3 that you use repeatedly in the section about Indian maths is a 60 years old paper with barely any information about its publisher.

"Mathematics is a great human achievement" : agreed, so what ?---Wikaviani 10:27, 3 December 2024 (UTC)

I'm going to stop engaging with this now, because I find your sarcasm very unpleasant. I recommend you adopt a more agreeable attitude before engaging in Misplaced Pages talk page discussion. Don't delete paragraphs of material from this page without consensus. –jacobolus (t) 16:23, 3 December 2024 (UTC)

So convenient, you refuse to achieve a consensus here, ok then, I'll revert back to the status quo version. This is not "sarcasm" but legit concerns about the quality of the sources you cite as I told you already above. Also, even Amyula kumar Bag says that some sources say that the so-called Pascal triangle found by Indian mathematicians has nothing to do with it. Best.---Wikaviani 18:26, 3 December 2024 (UTC)

@JayBeeEll Do you want to take this one on? I'm not in the mood to deal with persistent rudeness today. @Wikaviani, please read Misplaced Pages:Civility, Misplaced Pages:What Misplaced Pages is not § Misplaced Pages is not a battleground and Misplaced Pages:Here to build an encyclopedia before making further edits or comments. –jacobolus (t) 18:38, 3 December 2024 (UTC)

Well, I'm quite baffled to see an editor who has been editing here for 20 years blatantly throwing baseless accusations of rudeness while faced with an editor who tries to discuss the quality of the sources ... How about complying with WP:CONSENSUS ? WP:UNDUE ? WP:BURDEN ? WP:RS ? WP:ASPERSIONS ? and so on ? You have made many edits to the article without any consensus and while some of them are cosmetic, others aren't. @JayBeeEll: I would appreciate your input about the reliability of Amyula kumar Bag for this topic. Thanks. Best.---Wikaviani 22:05, 3 December 2024 (UTC)

"Tries to discuss" would be a more convincing summary if you had skipped the unjustified content removal, edit warring, repeated insults, sarcasm, air quotes, and unexplained purity tests about which historians are "true" enough.

From a content perspective, please just be straight forward about your goals. My speculation is that you don't think the history of combinations (n choose k) and Pascal's triangle should be discussed in a page about the binomial theorem. You can try to build consensus for this position but you should explain yourself clearly and give some clear rationale for your position, instead of just blanking content and revert warring about it.

The sources are clearly fine: Bag's paper in the Indian Journal of History of Science was written by a professional historian in a respectable peer reviewed journal, has been widely cited since, and unquestionably meets WP:RS. Bibhutibhushan Datta, Radha Charan Gupta, Kripa Shankar Shukla were all celebrated and decorated mathematical historians. I'm sure if you put in some effort could find yet further secondary and tertiary sources discussing this topic, including by European/American authors if you don't like Indian historians. –jacobolus (t) 23:50, 3 December 2024 (UTC)

• "would be a more convincing summary if you had skipped the unjustified content removal, edit warring, repeated insults, sarcasm, air quotes, and unexplained purity tests about which historians are "true" enough." : Uh ? are you aware of WP:PA ? you must know that baseless accusations qualify as personal attacks ? where are my so-called insults ? please clarify or this will end up at ANI.
• "The sources are clearly fine: Bag's paper in the Indian Journal of History of Science was written by a professional historian in a respectable peer reviewed journal, has been widely cited since, and unquestionably meets WP:RS. Bibhutibhushan Datta, Radha Charan Gupta, Kripa Shankar Shukla were all celebrated and decorated mathematical historians. I'm sure if you put in some effort could find yet further secondary and tertiary sources discussing this topic, including by European/American authors if you don't like Indian historians" : I never said I don't like Indian historians, I just said that Bag has not the expertise to challenge prominent historians of maths like Robertson, Rashed etc baseless accusations of yours, again ... Interestingly, you have not included Bag in your blue links lists above by the way ... Also, please read WP:ONUS, I don't need to achieve consensus since you are the one who make repeated changes to the article, not me, thus you need to achieve consensus, not me. I will step out this article for now as you are not in the mood to have a constructive discussion. I will wait for more input from other editors.

---Wikaviani 08:55, 4 December 2024 (UTC)

We got into this because you blanked a whole relevant and RS-sourced paragraph, based on the complaint that "none of the cited authors is a prominent expert of this topic", because Jean-Claude Martzloff  is a mathematical historian primarily focused on China rather than India and Norman L. Biggs is a professional mathematician (though in this case writing a peer-reviewed paper in a top mathematical history journal).

That blanking was inappropriate, so was quickly reverted by @JayBeeEll, but you edit-warred to re-blank the paragraph, this time saying "sorry, but I insist" because you don't consider Martzloff or Biggs to be a "true historian of maths". In my opinion these sarcastic comments about both Martzloff and Biggs were insulting and out of line.

But whatever, fair enough. Since you didn't like these quasi-tertiary sources, I expanded the section adding closer secondary sources by subject experts. I don't know much about Bag – he is a professional mathematical historian specializing on the history of Indian mathematics who wrote a good number of widely cited papers in the 60s–70s which are still being cited today, and remains active. The other cited authors included Bibhutibhushan Datta and Radha Charan Gupta. In an edit summary you sarcastically and insultingly called Bag's paper a "'source'" with gratuitous air quotes, and then in your comment here you insultingly summarized all of these sources as "nothing very impressive, some obscure mathematicians of Indian origin". Neither Datta nor Gupta is obscure: these are two of the most famous, celebrated, and prolific historians of Indian mathematics.

I was at that point quite unhappy with your repeated rude comments, but what put you over the line was "So according to you, 'Amulya Kumar Bag' is a world class expert? Wow, so how many influencial books has this guy published? with what publisher? how many awards has he? how about his academic career? The source number 3 that you use repeatedly in the section about Indian maths is a 60 years old paper with barely any information about its publisher." – Here you put more sarcastic air quotes, plus this time a string of sarcastic and insulting rhetorical questions. (To answer them though: Yes Amulya Kumar Bag is a world-class expert on the history of Indian mathematics and science. He doesn't have a Misplaced Pages article about him yet (feel free to write one), but here's a Google Scholar page. According to a quick web search he's a fellow of the Indian Academy of Sciences and was the editor of the Indian Journal of History of Science from 2002–2018. In particular his 1979 book Mathematics in ancient and medieval India has been very widely cited. I'm not sure which awards he has won, I'll leave you to research that. Both Datta and Gupta won multiple awards for their mathematical history work (e.g. Gupta won the Kenneth O. May Prize in 2009; Rashed won the same prize in 2017), and Datta's book History of Hindu Mathematics (with Singh) is one of the most influential ever written about the subject. As for that "its publisher", we are talking about the Indian Journal of History of Science published by the Indian National Science Academy.) I found your tone here to be completely unacceptable, insulting both to me personally and to these professional scholars. Since you are now making threats, I demand that you retract your insults.

After that, we have more sarcastic and exaggerated language from you which I found to be insulting: "So convenient," "baffled blatantly", "Uh?" " Interestingly,". Please cut that out now.

––jacobolus (t) 17:38, 4 December 2024 (UTC)

I guess you and me don't have the same definition of the word insult. Let me be clear about that, if my comments were insults, go ahead and report me, otherwise, stop casting aspersion by labelling as "insults" legit concerns of another editor about that 60 years old source. Also, for your information, a mathematician is not a historian of maths, Biggs and Bag may be respectable mathematicians, but they are not expert in the field of history of maths. You seem to consider that since a publisher is reliable, that's enough to make the source reliable, this is wrong and you probably know that. Let me remind you what is a reliable source here, for Misplaced Pages (WP:SOURCEDEF) :

"When editors talk about sources that are being cited on Misplaced Pages, they might be referring to any one of these three concepts:

• The piece of work itself (the article, book)
• The creator of the work (the writer, journalist)
• The publisher of the work (for example, Random House or Cambridge University Press)

Any of the three can affect reliability. Reliable sources may be published materials with a reliable publication process, authors who are regarded as authoritative in relation to the subject, or both. These qualifications should be demonstrable to other people."

This is cristal clear, being published by a good publisher is not enough to make a source reliable, thus, while the Indian journal of history of science might be a great publisher, the author (Bag) is all but a world class expert and quite outdated since the sources that come after him, like Rashed or Robertson don't share his views (WP:AGEMATTERS). Besides, even if Bag was a world class expert like you claim, his views should not be given such an undue weight since they are not supported by the mainstream of reliable sources that are cited just after, especially when that source makes such an extraordinary claim about about the discovery of something equivalent to the triangle of Pascal in the 10th century. By the way, you keep editing the article while there is an ongoing dispute here, which is all but a constructive behaviour.---Wikaviani 08:59, 5 December 2024 (UTC)

I don't intend to "report" you, which seems like a waste of time and energy. Let's assume (per WP:AGF) that your repeated inaccurate, sarcastic, condescending (and in my opinion quite insulting both to me personally and to the scholars cited here) comments were an honest mistake. I am merely asking you to please stop now. Specifically: I expect no more air quotes; no more sarcasm or condescension (along the lines of "so convenient", "interestingly", or most recently "for your information"); no more comments like "nothing very impressive"; no more sarcastic rhetorical questions. If you can cut the attitude and engage respectfully, we have no problem.

Amulya Bag is a professional historian of mathematics and science who was for an extended time the editor of a respected journal of mathematical history.

"extraordinary claim" Well let's just look at the translated 10th century text directly, shall we:

"After drawing a square on the top, two squares are drawn below (side by side) so that half of each is extended on either side. Below it three squares, below it (again) four squares are drawn and the process is repeated till the desired pyramid is attained. In the (topmost) first square the symbol for one is to be marked.. Then in each of the two squares of the second line figure one is to be placed. Then in the third line figure one is to be placed on each of the two extreme squares. In the middle square (of the third line) the sum of the figures in the two squares immediately above is to be placed; this is the meaning of the term pūrṇa. In the fourth line one is to be placed in each of the two extreme squares. In each of the two middle squares, the sum of the figures in the two squares immediately above, that is, three, is placed. Subsequent squares are filled in this way.

What is described here is precisely Pascal's triangle – indeed, even in a more "modern" form than Pascal himself used.

"You seem to consider that ..." – You are putting words in my mouth, but you are also setting up a standard that has nothing to do with WP:RS. Scholarly papers about mathematical history written in reputable mathematical history journals clearly meet the wikipedia reliable source standard.

I think you are somewhat mixing up what these various sources are claiming (and in particular the claims we are repeating here); to be specific, I think you are interpreting claims made about the history of combinations (which are numerically the same as binomial coefficients) as claims about the binomial theorem per se. Binomial coefficients, as numbers, occur in multiple contexts in mathematics. One of the context where they appear is in combinatorics, in counting the number of subsets of size ⁠${\displaystyle k}$⁠ of a set of size ⁠${\displaystyle n}$⁠. The paragraph about Indian examples is mainly about this combinatorial occurrence of these numbers. A distinct context is the algebra of polynomials, where these numbers appear as the coefficients of a binomial multiplied by itself some number of times.

There's really no question that there are multiple Indian sources ranging over a wide time period discussing binomial coefficients as numbers, especially in a combinatorial context. The Bhagavati Sutra describes combinations up through ⁠${\displaystyle n=4}$⁠. (Here's O'Connor & Robertson on this since you consider them an acceptable source: "Permutations and combinations are used in the Sthananga Sutra. In the Bhagabati Sutra rules are given ... in the cases where n = 2, 3, and 4. The author then says that one can compute the numbers in the same way for larger n."). Pingala's description of making verses with various metres is in my understanding somewhat cryptic, but the elaboration given by his 10th century commentator describes exactly Pascal's triangle with the standard method of generation. (O'Connor and Robertson's summary: "In a commentary on this third century work in the tenth century, Pascal's triangle appears in order to give the coefficients of the binomial expansion." ) The fraction-multiplication rule for finding n choose k is very clear and explicit in multiple sources from the 8th–9th century (here's a page by Ian G Pearce on O'Connor and Robertson's website discussing Mahavira's version).

Where there's some amount of ambiguity is the question of what counts as the "binomial theorem" per se. Amulya Bag makes the claim that Pingala's rules for verses should be considered as binomial expansion, because they involve forming metres as arrangements of two types of syllables, so for instance with length three we get all of the syllable patterns (aaa, baa, aba, aab, bba, bab, bba, bbb), which can be grouped by how many a's and b's they have and then counted, yielding the counts (1, 3, 3, 1), the 3rd row of Pascal's triangle. For Bag, this is a form of binomial expansion. However, there's an argument that could be made that these aren't binomials in the sense of sums of variable numerical quantities ⁠${\displaystyle x+y}$⁠ to be expontiated like ${\displaystyle (x+y)^{3}={}\!\!}$${\displaystyle xxx+yxx+xyx+xxy+{}\!\!\!}$${\displaystyle yyx+yxy+xyy+yyy={}\!\!}$${\displaystyle x^{3}+3x^{2}y+3xy^{2}+y^{3}}$. (Partly for this reason, I didn't repeat Bag's more opinionated claim in the article, but only his clearly factual claims. I think discussion in detail of this point is too in the weeds, and should be relegated to a more detailed history of binomial coefficients article if it is to be discussed at all.)

Per Rashed, the oldest known description of the the binomial theorem per se, i.e. taking powers of a binomial sum of numbers, at least for exponents greater than 3, can be found in al-Samawʾal's book from the 12th century, credited by al-Samawʾal to al-Karajī.

These different claims are not contradictory, and we don't need to reject the claim that Indians were working on combinatorics or expressed Pascal's triangle in order to accept Rashed's claim that the earliest version of something pretty close to the binomial theorem as we think of it today is al-Samawʾal/al-Karajī. Frankly we don't even need to reject Bag's claim that Pingala's verses are a kind of binomial expansion to also accept Rashed's claim. These claims are just not in any kind of conflict.

As for your concern about sources though: These sources are clearly "reliable" by Misplaced Pages standards because they are written by reputable mathematical historians in reputable peer-reviewed history journals or published in scholarly books, have been widely cited by other scholars, and are based directly on primary historical sources whose translations we can easily read directly and understand. We don't need to rely on anyone's subjective interpretation: the texts are right there in black and white. –jacobolus (t) 19:10, 5 December 2024 (UTC)

This wall of text is mainly your own interpretation of this topic, what I see on my end, is an editor who is not capable to provide serious sources for the claim "The Chandaḥśāstra by the Indian lyricist Piṅgala (3rd or 2nd century BC) somewhat crypically describes a method of arranging two types of syllables to form metres of various lengths and counting them; as interpreted and elaborated by Piṅgala's 10th-century commentator Halāyudha his "method of pyramidal expansion" (meru-prastāra) for counting metres is Pascal's triangle.".

Bag is not an expert historian of maths and is a bit outdated while Jayant shah's field of expertise is "computer vision" (Jayant shah is source number 12). I will remove this sentence but leave in the rest of your work since it is quite well-sourced. Best.---Wikaviani 08:27, 9 December 2024 (UTC)

Bag is not an expert historian of maths – This is a falsehood which you now know to be false because we have been over this several times. Bag is a professional historian of mathematics who spent his career in the field and was the editor of a respected history of mathematics journal.

If you remove this perfectly fine sentence you will be reverted. Your behavior and comments here continue to well outside Misplaced Pages policy and norms. –jacobolus (t) 15:54, 9 December 2024 (UTC)

Ping also @Slawekb, who somewhat expanded this section in July 2015 and may be interested to weigh in / may have other recommended sources. –jacobolus (t) 01:23, 4 December 2024 (UTC)

It might be useful to have an article History of binomial coefficients to consolidate Binomial coefficient § History and notation, Pascal's triangle § History, and Binomial theorem § History. That would leave more room to describe specific historical versions and their context in more detail and list more historical examples without unduly burdening these articles which otherwise are in a hurry to get to the mathematical content. This article could then make do with a more concise summary of 2–3 paragraphs. I don't have the motivation to write such a thing in the near future, but if someone feels inspired I'd be supportive of that effort. –jacobolus (t) 17:46, 3 December 2024 (UTC)

• C-Class level-5 vital articles
• Misplaced Pages level-5 vital articles in Mathematics
• C-Class vital articles in Mathematics
• C-Class mathematics articles
• Mid-priority mathematics articles