Talk:Continued fraction

"Complete convergent"?

The paper patz5.pdf cited under Pell's equation says on page 2:

... If $x_{n}=(P_{n}+{\sqrt {D}})/Q_{n}$ is the n-th complete convergent of the simple continued fraction for $\omega =(P_{0}+{\sqrt {D}})/Q_{0}$, ...

but I can't find any definition of complete convergent. Is it just another name for a complete quotient? If it is, perhaps someone sufficiently knowledgeable could mention it in that page. Hv (talk) 13:46, 22 May 2021 (UTC)

Requires citations and clarifications

This is a long article; and I jumped in to the "Examples" section. The table completely mystified me, in terms of "how do I get from the 'stuff' in the table to a continued fraction?" What I'm getting at is that the column and row headings are neither descriptive nor described -- at least a quick definition of terms or a "For example, using the table able, we can generate pi with the following continued fraction. Notice how the values <bla bla bla> correspond to the table entries <bla bla bla>."

The citation that I'd like is one in the "History" section, where it states, "300 BCE Euclid's Elements contains an algorithm for the greatest common divisor which generates a continued fraction as a by-product" -- unfortunately, neither hotlink "Euclid's Elements" nor "greatest common divisor" leads me to text that deals with the "greatest common divisor which generates a continued fraction as a by-product" goal that I was looking for. Each leads me to its respective "narrow" definition, but nothing leads me to the overall concept about how GCD relates to continued fractions. 198.84.205.118 (talk) 00:03, 29 May 2021 (UTC)

I have edited the first item of section "History", and added in section "Motivation and notation": "The sequence of the integers that occur in this representation [of a rational numbers by a continued fraction] is the sequence of the successive quotients that are computed by the Euclidean algorithm". D.Lazard (talk) 08:32, 29 May 2021 (UTC)

Too hard formula represented on article

Correct formula for programming conversion $a/b$ to continued fraction is very simple and like this within loop:

       an1 := an0 / bn0; // integer div (you need this $a_{n}$)
       bn1 := an0 % bn0; // integer mod
       an0 := bn0;
       bn0 := bn1;

Mvitaminus (talk) 23:28, 2 June 2021 (UTC)

Division by zero

The article currently indicates "where a_i and b_i can be any complex numbers." Before we required b_i to be 1 -- because we were defining a simple continued fraction before -- and we also required that a_i be a positive integer. As a consequence, this prevented division by zero in every convergent. The current definition has no such safeguard. Should it? —Quantling (talk | contribs) 17:21, 2 May 2022 (UTC)

Reciprocal of 1

The article suggest that the reciprocal of a number is given by adding/removing a zero at the begining. It is not true for 1 or -1 since their reciprocals are themselves. 2601:648:8601:93A0:AC29:71C5:7EE0:6ABA (talk) 06:55, 13 December 2022 (UTC)

Did you work out the value of the continued fraction obtained when you add a zero at the beginning for these numbers? Remember that continued fraction representations are not always unique. —David Eppstein (talk) 07:01, 13 December 2022 (UTC)

The golden ratio is not the most irrational number.

If you use square roots, instead of fractions, you can get something more irrational than the golden ratio, because many square roots already become irrational with 1 iteration, which does not apply with fractions.

example: 3+sqrt(7+sqrt(15+sqrt(1+sqrt(292+sqrt(... 84.151.244.169 (talk) 15:07, 8 May 2023 (UTC)

I don't think that there is a precise meaning for "most irrational number". The point in the article is to highlight that the continued fraction expansion of the golden ratio φ is all ones. It turns out that this means that regardless of the fraction p/q, it is the case that the value (|φ − p/q|) × q² is large in comparison to what can be achieved with rational approximations for other irrational numbers. —Quantling (talk | contribs) 16:20, 8 May 2023 (UTC)

Revamping the page

Hello!

I just wanted to check in with anyone that might care whether or not they'd take issue with me trying to reorganise the page. I've become quite interested in continued fractions not just as a weird means of 'calculating numbers', but as an alternate means of representing numbers between the integers to that of the usual decimal 'negative power series', and I'd really like to try to do justice to them through updating the page.

Some notes I've taken thus far as as follows:

- Motivation and Notation section spends a lot of time explaining how to calculate the continued fraction form from usual decimal negative power series form, not enough time talking about actual motivations/history/desirability, and the notation. Either I can change the name and collate related information or remove from this section and write a clearer explanation of the method elsewhere, preserving Pier4r's request above.

- Whole separate Notation section exists, which actually discusses 'alternative notations' to the ones presented in the 'motivation and notations section'; I think this heading should be changed and it should be subsumed by a broader section on notation.

- Repeated references throughout to the effect of "sqrt(2) actually equals 1.41421..., so you can calculate this from its continued fraction form [1;2,2,2,..] by doing so and so." Seems to be a neglect for the consideration of a continued fraction representation of a number as equally 'valid' as the power series representation, probably due to unfamiliarity and the somewhat cumbersome but necessary notation. To be clear, I think there's little reason to not switch the notation such that for example pi = 3.7(15)1(292)111213... (in continued fraction form) = [3;1,4,1,5,9,2,6,..] (in power series form) -- now imagine analogously saying that " pi actually = 3.7(15)1(292)111213..., so you can calculate this from its power series form via... ". I personally think it's reductive and unnecessary, so I wonder what you guys might think of this point in particular.

- The continued fraction notation version of a bunch of mathematical constants in the Motivation and Notation section seems really helpful to me for familiarising the reader with this perspective on these numbers, and I would like to preserve something similar, but when you look for its context you see that all this space is actually serving to elucidate infinite continued fractions, which is off-topic from the heading. I'd like to flesh out some of these kinds of examples with more numbers that aren't just infinite cfs, and reserve discussion of infinite cfs for maybe the section 'Infinite continued fractions and convergents'.

- Notice that there's no mention nor use of the 'repeating' notation usually seem with 'decimal' notation of numbers like 1/3 or 1/7, only ellipsis like sqrt(3) = [1;1,2,1,2,1, 2,...]. I'd like to explicitly incorporate that.

- Having read the page quite a few times, I'm confused as to whether it's about cfs in the canonical form or the generalised form. Given that a page exists solely for the generalised form, I'd be inclined to dedicate this to the canonical form, but I also feel that that would be too specific and might mislead people given the name. The diagram in the introduction shows it in the canonical form, mention of definition as 'the reciprocal of another number' somewhat suggests the understanding that it's about the canonical form, the intro makes the delineation between the two and suggests a prioritisation of the canonical form, yet the section on Basic Formula immediately jumps into the generalised form, despite that formula being mirrored on the generalised cf's page. That formula is then repeated in the later section titled 'generalized continued fraction(s?)', which I feel again is redundant and I'd like to remove and possibly move over any interesting information to its respective article if it's not there already.

There's more I'd like to add, including some interesting patterns I've found myself, some restructuring to be done, and more I need to study in order to be able to really speak on some topics. I'd like to ensure I'm factoring in the suggestions that others have already made too, and in particular I'd love to be able to address Manoguru's concerns about the natural operations of numbers in continued fraction form, but that's all I can speak on for now.

Please do let me know your thoughts on my potential changes, thank you for reading if you made it this far! CallumMScott (talk) 14:17, 14 August 2023 (UTC)

how "larger term" better approximation

I don't like the sentence

The larger a term is in the continued fraction, the closer the corresponding convergent is to the irrational number being approximated.

It goes on to explain that the golden ratio is the hardest to approximate because all terms of its continued fraction are "1".

I think what the sentence should say is something like "the larger a term is, the more that one term improves the approximation." But then I want to natter on about percentage reduction in absolute error.

I would like to hear from someone who understands the article before I try to "improve" it.

Jmichael ll (talk) 20:41, 8 November 2023 (UTC)

Which is closer to 4: 41⁄7 or 41⁄3?

The greater a partial quotient is, the less effect it and its successors have on the number; in other words, the more accurate the fraction already is. —Tamfang (talk) 05:32, 15 November 2023 (UTC)

Perhaps I'm mistaken in this concern, but isn't there a problem about what we mean by "the corresponding convergent"? Would it be an improvement to add a "next", as in

The larger the next term in the continued fraction is, the closer a convergent is to the irrational number being approximated.

?

I think both are true and mean slightly different things. Dhrm77 (talk) 15:40, 15 November 2023 (UTC)

Constructive observation

A real number $x$ has an infinite continued fraction expansion iff it is apart from all rationals: $\forall q:\mathbb {Q} .\exists m:\mathbb {N} +.|x-q|\geq (1/m)$. This is constructively stronger than being irrational (not rational).

46.33.143.125 (talk) 15:52, 28 January 2024 (UTC)

What does "apart from all rationals" mean, if not "irrational"? —Tamfang (talk) 17:28, 27 March 2024 (UTC)

Constructively, "apart" has a stronger meaning than "not equal". Two numbers are "apart" if they differ by at least some $1/n$. 46.33.143.125 (talk) 19:04, 13 April 2024 (UTC)

Can you give an example of an irrational number which is not "apart" from the rationals? –jacobolus (t) 19:37, 13 April 2024 (UTC)

What is an example of an irrational that has a finite neighborhood containing no rationals?? —Tamfang (talk) 21:07, 13 April 2024 (UTC)

The definition in the top comment here lets you pick a different neighborhood excluding each rational number. But I don't understand what's different about it than the concept of "irrational" per se. –jacobolus (t) 22:22, 13 April 2024 (UTC)

References

Proposal: move this to "Simple continued fraction", so that the current lemma can be used for generalized continued fraction

As already noted in 2006 (Talk:Generalized_continued_fraction#Not_"generalized"_enough?), the current lemmata continued fraction and generalized continued fraction are at variance from much of the mathematical literature. I therefore offer my help to move

• this article continued fraction to simple continued fraction, and
• generalized continued fraction to continued fraction,

in agreement with

• DLMF https://dlmf.nist.gov/1.12
• Weisstein https://mathworld.wolfram.com/ContinuedFraction.html
• this most pertinent book https://www.google.de/books/edition/History_of_Continued_Fractions_and_Pad%C3%A9/rxzsCAAAQBAJ?hl=de&gbpv=1&dq=brezinski+history+of+continued&printsec=frontcover

Dyspophyr (talk) 15:23, 23 October 2024 (UTC)

Hi @Dyspophyr, I think your suggestion makes sense; although, in my opinion, it might make even more sense to merge the two articles...

Best, Malparti (talk) 16:11, 23 October 2024 (UTC)

"Continued fraction" is by far the WP:COMMONNAME for the topic currently at continued fraction. That is where anyone intersted in the topic would expect to find that article. It should not be moved. —David Eppstein (talk) 17:33, 23 October 2024 (UTC)

@David, you may be right that texts about `b0+1/(b1+1/(...` commonly call them "continued fraction". However, it is also true that texts about `b0+a1/(b1+a1/(...` commonly call these "continued fraction" as well. If under the header "continued fraction" we only treat the case `a1=a2=..=1`, then we are out of sync with much of the mathematical literature. If we start under this header with the generic case, then we do nothing wrong. Of course at some point we have to mention the special case `a1=a2=..=1`. -- Dyspophyr (talk) 17:49, 23 October 2024 (UTC)

@David Eppstein "Continued fraction" is by far the WP:COMMONNAME for the topic currently at continued fraction." → I agree, but I was under the impression — and I might be wrong about this! — that "continued fraction" is also the WP:COMMONNAME for the topic currently at generalized continued fraction. Hence my suggestion that maybe the two articles could be merged.

At any rate: I think we agree that what matters is that people who search for "continued fraction" should be told fairly early in the article that "continued" fraction is a generic term that some people use in a strict sense to refer to what are also known as simple continued fractions; and some other people use in a loose sense to refer to what are also known as generalized continued fractions. Then, whether there should be {one article} vs {a main article about simple continued fractions and a specialized one on generalized continued fractions} vs {a main article about generalized continued fractions and a specialized one about simple continued fractions} is not completely clear: to me, these three options seem to make sense...

One argument that goes in the direction suggested by @Dyspophyr is that several of the resources linked in the article continued fractions define "continued fractions" as generalized continued fractions (Britannica, Encyclopedia of mathematics, to some extent Wolfram MathWorld, etc).

Best, Malparti (talk) 20:09, 23 October 2024 (UTC)

@Malparti, there are two pragmatic reasons against merging: The present article is very long, and uses notation that is in conflict with the generic article. Therefore I rather suggests that the generic article, moved here, be given a section on the `a1=a2=..=1` case, which then refers to "simple continued fraction" for deeper information. -- Dyspophyr (talk) 17:49, 23 October 2024 (UTC)

I oppose this move. This encyclopedia is a general resource, not devoted solely to mathematicians. Non math readers who want to learn about "continued fractions" should get this article, not a more general one.

I think the issues you cite about the relationship between the articles could be reduced by adding a WP:Hatnote to this article. Among all readers who enter, those who are looking for the mathematicians view would then quickly find the other article. Johnjbarton (talk) 18:18, 23 October 2024 (UTC)

I don't think a hatnote is a good solution. But generalized continued fraction should be linked from within the lead section. –jacobolus (t) 19:02, 23 October 2024 (UTC)

This seems like reasonable evidence that the present wiki-naming is against the common naming convention. It also seems like the Press–Teukolsky and Jones–Thron sources at generalized continued fraction also use "continued fraction" for what wiki now calls "generalized continued fraction." Is there good counterevidence, that "generalized continued fraction" is actually common terminology for this? Gumshoe2 (talk) 18:25, 23 October 2024 (UTC)

There are many sources calling this "simple continued fraction" and many sources calling the other one "generalized continued fraction" (though more sources just use the name "continued fraction" with the specific meaning clear from context), but I think the "simple" variant is overall a better article for this title. It seems fine to use the title Generalized continued fraction for that article, I don't think readers will be confused.

What we could do, however, is enlarge the section at this article about generalized continued fractions to provide a somewhat more detailed summary. –jacobolus (t) 19:12, 23 October 2024 (UTC)

Edit: after doing some more skimming in the literature, I'm somewhat leaning towards mostly merging these articles under the name Continued fraction, and then possibly splitting out any excessively detailed sub-sections into more specific articles. –jacobolus (t) 20:56, 23 October 2024 (UTC)

If it's possible to do without making the article unwieldy, this seems to me like a very satisfactory solution. (But I have no opinion on whether it's possible!) Gumshoe2 (talk) 21:03, 23 October 2024 (UTC)

The current article is mostly unwieldy because it has a poor structure: there are too many small top-level sections, limited narrative flow, and not much high-level vision. Even without changing the scope it would be improved significantly by someone with knowledge about and care for the subject doing a some significant housecleaning (probably with some nontrivial shortening / cutting of material here now). I don't feel qualified or motivated to take on a job like that though. –jacobolus (t) 22:24, 23 October 2024 (UTC)

@Johnjbarton »Non math readers who want to learn about "continued fractions" should get this article, not a more general one« - Why? Non-specialist readers come here because they somehow encountered the notion "continued fractions" out in the wild, often applied to beasts of the non-simple type. They will be very confused by our current narrow definition, as I was when I first came here. A hatnote would help (@jacobolus why not even a hatnote??). However, using standard terminology from the onset would help much more. -- Dyspophyr (talk) 19:41, 23 October 2024 (UTC)

Hatnotes are not intended to be used for disambiguating closely related topics like this. See WP:RELATED. –jacobolus (t) 22:19, 23 October 2024 (UTC)

Ok, seems like you have a lot of input here. I was mainly pushing against the idea that the decision should be based on what mathematicians want. Johnjbarton (talk) 22:21, 23 October 2024 (UTC)

@jacobolus »many sources calling the other one "generalized continued fraction"« - can you show us some of these sources? -- Dyspophyr (talk) 19:45, 23 October 2024 (UTC)

Some evidence: Google Scholar search for "the continued fraction" (a phrasing that makes sense only for the version described in this article): about 31200 results. Google Scholar search for "simple continued fraction": about 2730 results. Or, if you prefer phrasing where this distinction is even less ambiguous: "the continued fraction expansion": about 11700 results; "simple continued fraction expansion": about 1040 results. So avoiding simple and using a definite article to indicate the uniqueness of the expansion (something that would not be true for generalized continued fractions) is about 10x more of a WP:COMMONNAME than using simple. —David Eppstein (talk) 20:01, 23 October 2024 (UTC)

@David Eppstein ""the continued fraction" (a phrasing that makes sense only for the version described in this article)" → I am nitpicking a bit here, but I disagree with this because I've used expressions such as "the continued fraction of the theorem" and "the continued fraction XXX" to refer to generalized continued fractions in work indexed by Google Scholar. Malparti (talk) 20:19, 23 October 2024 (UTC)

That's why I included the second variation, with "expansion". It didn't make much difference in the relative proportions. —David Eppstein (talk) 20:23, 23 October 2024 (UTC)

@David Eppstein Sorry, but I don't understand: I could see myself use something like "the continued fraction expansion below" to refer to a generalized continued fraction expansion (and in fact I've done that; and read it as well). So wouldn't those be false hits for your stats? Sorry if I am missing something. Anyway: I am not arguing that "continued fraction" isn't the common name for simple continued fraction — we agree on that. My "concern" (though the word is a bit excessive) is that it might also the common name for generalized continued fraction. Malparti (talk) 20:40, 23 October 2024 (UTC)

Any query is going to have false hits. I do not expect the number of false hits to be significant nor to change the proportions much for these queries. —David Eppstein (talk) 20:42, 23 October 2024 (UTC)

Maybe I've misunderstood your point, but this evidence seems perfectly compatible with the claim that "generalized continued fraction" is not a common name for the topic of wiki article generalized continued fraction and moreover that both are typically called "continued fraction." Gumshoe2 (talk) 20:41, 23 October 2024 (UTC)

Sure. But when two related but distinct topics share the same name we still need two articles on them, and in such cases when one of the two topics is by far the WP:COMMONNAME (that is, the topic usually meant by that name, not merely the name usually used for that topic) we let that topic have the unmodified name and modify the name of the less-common topic. Exactly as is the status quo for these two articles already. —David Eppstein (talk) 20:45, 23 October 2024 (UTC)

I find that rather problematic, any reasonable wiki-reader would clearly think that "generalized continued fraction" is the typically understood name for this concept. (In fact, until today I have been such a reader of these particular pages.)

However I can appreciate that the one special case is the most important and deserves the most central coverage. I don't see any easy solution; however, from what I can see, at minimum I think a note should be added somewhere near the top of generalized continued fraction to say that the concept is typically (or at least very often) simply called "continued fraction." Gumshoe2 (talk) 21:00, 23 October 2024 (UTC)

If it is really the case that "generalized continued fraction" is not used much for these things, so much so that it would be misleading to use that title, another alternative would be to use a disambiguator, like continued fraction (non-unit). —David Eppstein (talk) 21:09, 23 October 2024 (UTC)

I think something like that would be much more satisfactory than the present situation.

(However, just to emphasize, my only knowledge of this matter comes from this thread. For all I know, "generalized continued fraction" is actually common language for this – but so far I haven't seen any reason to think this.) Gumshoe2 (talk) 21:13, 23 October 2024 (UTC)

Other names include "non-simple continued fraction" and "irregular continued fraction". –jacobolus (t) 22:00, 23 October 2024 (UTC)

I like those better than my disambiguation above. —David Eppstein (talk) 22:08, 23 October 2024 (UTC)

Continued fraction (complex) would be a natural disambiguation. fgnievinski (talk) 02:49, 24 October 2024 (UTC)

"Complex" is potentially confusing, because it seems to more commonly indicate that the "integers" in the fraction are Gaussian integers rather than describing what kind of numerators are used. –jacobolus (t) 03:59, 24 October 2024 (UTC)

@David Eppstein (also @Jacobolus): "But when two related but distinct topics share the same name we still need two articles on them" → Yes, but that was my point from the start: are these topics distinct enough that we need two articles? From my perspective (i.e, from the perspective of someone not working on continued fractions, but who has used them in their research), I wasn't really convinced that this was the case... Hence my suggestion to merge the two articles (and possibly keep dedicated articles for more in depths discussions). Malparti (talk) 21:01, 23 October 2024 (UTC)

Also: as I said, I'm not an expert on the topic so my opinion isn't really informed; as a result, I think I've contributed what I had to contribute to this conversation. What I could do — if that's useful — is set up a slightly more robust method to try to search the literature and try to determine "what is being called what, in which field" and "what is the most common topic, in which field" [my guess is that simple continued fractions overwhelmingly used in topics related to number theory; but that generalized continued fractions might be more common in other areas]. As long as it doesn't take me more than, say, one hour, I'm happy to do that if that's helpful. Cheers, Malparti (talk) 21:08, 23 October 2024 (UTC)

Looking some more, I find plenty of examples but also a similar number where "generalized continued fraction" instead means a higher-dimensional analog of a continued fraction, which I'm not sure we have any articles about. –jacobolus (t) 20:47, 23 October 2024 (UTC)

@David Eppstein Let me look into your argument based on a Google Scholar search. You found "continued fraction" is 10 times more frequently used than "simple continued fraction". That sounds perfectly credible. Even a higher ratio would not have surprised me. But it does not support your conclusion.

You argue: In the scholarly literature, term X ("continued fraction") is more frequently used than term Y ("simple continued fraction"), hence X is the WP:COMMONNAME for Wikipedia article X. Funnily, your argument would work exactly the same way after we had done the proposed renaming. Why? Because you nowhere refer to the contents of the articles.

In drawing your conclusion, you implicitly assume that the scholarly works about X and Y deal with the thing defined in our article X. This, however, is not the case. Quite many texts (reference works, textbooks, research reports) that contain the term X assume a definition that is less restrictive than in our present article X. They all, however, are compatible with the definition given in another WP article, currently named Z ("generalized continued fraction"), which defines Z to be a strict superset of X.

In such a situation the obvious solution, advocated by @Malparti, is to merge articles X and Z under lemma X. However, this would lead to a loss of version history. For this only reason I propose to move X to Y ("simple continued fraction") before porting a digest of it to the new common article X'.

Does this help to elucidate the situation? -- Dyspophyr (talk) 05:59, 24 October 2024 (UTC)

No. Try fewer words.

A hint: I didn't refer to the content of the articles because I think we can all agree on what they are: one is about nested fractions of a specific form with 1 in the numerators, and the other is on a similar form without the restriction on the numerators.

So your theoretical maunderings about how maybe these articles could be about some entirely different topic than what they are about seem to be entirely based on counterfactuals. —David Eppstein (talk) 06:32, 24 October 2024 (UTC)

In few words: This WP article has numerators = 1, many of the scholarly works you counted have not. Therefore your Google Scholar count proves strictly nothing. -- Dyspophyr (talk) 08:09, 24 October 2024 (UTC)

@Dyspophyr "However, this would lead to a loss of version history" → I do not think this is a good argument against merging: that argument would prevent any merge of any two articles. A better argument would be that implementing the merge properly would be a lot of work, and that there may be no one willing/qualified to do this.

Leaving these considerations aside, I still think that merging under the name continued fraction (with the option of keeping specialized articles under the names simple continued fraction and generalized continued fraction, if that's really necessary) would be the best solution. @Jacobolus and @Gumshoe2 seem to partly support this; @Dyspophyr seems to be against; @David Eppstein hasn't explicitly voiced an opinion on this (or I missed it). Malparti (talk) 08:10, 24 October 2024 (UTC)

I agree that the final outcome should very much look like both articles are merged. With possibly some more technical sections in a specialized article or two. But how to get there? As the two articles have conflicting notation, a straightforward merge with subsequent refinement seems not viable. Thence my proposal to start with the moves, then migrate and consolidate contents, proceeding section by section. -- Dyspophyr (talk) 08:21, 24 October 2024 (UTC)

Sorry, I haven't read all the above (but some of it).

It seems to me the title Continued fraction should be given to a shorter and not too technical article, dealing with both simple and generalized continued fractions. You could call it a merge, but only of core parts of the two present articles. And then, Simple continued fraction could be largely identical to what is presently at Continued fraction (but improved, if anyone can do that! -- and without the bits on generalized continued fractions), while Generalized continued fraction is largely unchanged - the new shorter article linking those two as main articles.

We could start by renaming Continued fraction as Simple continued fraction, followed by a recreation of Continued fraction as a disambiguation page, subsequently expanding that to cover main points from both types.

This is perhaps a less ambitious but also more realistic plan than a full merge. Nø (talk) 08:58, 24 October 2024 (UTC)

@Nø, unfortunately your proposal doesn't address my main concern: that our current article Generalized continued fraction is misnamed. It is misnamed because it describes mathematical objects that are commonly called "continued fraction", not "generalized continued fraction". Conversely, if the latter term appears in the mathematical literature, then typically one of the following holds:

• They use the term interchangeably, and inconsistently, to mean the same as "continued fraction";
• They write "generalized continued fraction something", and actually mean "generalized continued-fraction something", i.e. they generalize "something", not "continued fraction";
• Or they really generalize the concept of "continued fraction" in ways that have nothing to do with the standard continued fractions described in our articles.

I agree with you that the resulting article Continued fraction should be not too long nor too technical. Ideally it should be as readable and polished as Integral. With more technical material relegated to a number of specialized articles.

-- Dyspophyr (talk) 09:34, 24 October 2024 (UTC)

@Dyspophyr I don't think it is misnamed: "generalized continued fraction" is a correct, standard name for these objects; it's just that in practice they are often just referred to as "continued fractions" (most likely because people can be bothered to use the long name every time). Maybe some people would insist that these are the "true" continued fractions and should get the short name; but I don't think that's the debate here: I think for many of us the problem is that if you pick an occurrence of the phrase "∅ continued fraction" uniformly at random in the literature, it could refer to a simple or to a generalized continued fraction.

If the probability that it refers to one of the variants is much higher than the other, then this variant should get the article with the short name. If they are comparable, we have to think about whether we want a disambiguation, a merge, etc.

So to avoid unnecessary discussions, the first step should probably be to see which term is used for what in what context. David Eppstein suggested that "continued fraction" overwhelmingly referred to simple continued fractions, and he might be right about this; however (1) I was not convinced by his Google searches and (2) that goes against my limited experience of the topic (I've worked a bit with continued fractions — as tools rather than as objects of studies — in topics related to combinatorics / complex analysis, and in doing so I had the impression that I saw "∅ continued fraction" used a lot to refer to the generalized variant).

In case someone is interested: in combinatorics [resp. probability theory], the generating functions of some combinatorial classes [resp. random variables] can be expressed as generalized continued fractions. There is a substantial literature on the topic and a whole zoo of relevant classes of fractions (S-fractions, J-fractions, M-fractions, etc). After skimming that literature a few years ago, I got the impression — again, I might be wrong! — that in that literature, people typically say "generalized continued fraction" at most a few times [and sometimes never] when they introduce the concept; and then go with continued fraction. Whence the problem that stumble upon one occurrence of the phrase "∅ continued fraction", it could likely refer to a generalized continued fraction.

Something that I have already mentioned is that I think that whether "∅ continued fraction" will typically refer to simple vs generalized one will depend on the field. I guess that in number theory, where people are interested in continued fraction expansions of irrational numbers, etc, simple continued fractions are the norm; however, in other fields (e.g, related to combinatorics / probability theory / statistical mechanics) this might be different. Malparti (talk) 10:27, 24 October 2024 (UTC)

I appreciate your efforts, @Malparti. Yet all this remains speculative as long as we haven't seen a few references to authorative texts that indeed use the term "generalized continued fraction" to refer to the thing with numerators ≠ 1. I can easily provide a few more references that do completely without "generalized" -- Dyspophyr (talk) 10:36, 24 October 2024 (UTC)

But what are autoritative sources? Specialist literature is NOT what we consider the best sources in wikipedia, if quality secondary or tertiary sources are available. -- Mathworld may not be the best source for wikipedia purposes either, but none the less: It clearly supports the view that "Ø continued fractions" refers to objects within the class they specifically call "generalized continued fractions", but that it in some context may refer to a subset only, viz. "simple continued fractions". Linguistically, I think it in this situation would make more sense to talk about "simple vs. general" rather than "simple vs. generalized", but "general" seems to have less traction than "generalized". Nø (talk) 10:53, 24 October 2024 (UTC)

@Dyspophyr I looked at the first 10 hits for "generalized continued fraction" on Google Books (one book is listed twice, hence the 9 results). In what follows, I used "simple continued fraction" and "generalized continued fraction" as on the current version of Wikipedia:

• [1]: refers to a simple continued fraction $[a_{n}]$ where $a_{n}$ is a vector.
• [2]: refers to a generalized continued fraction $[a_{n},b_{n}]$ where $a_{n}$ and $b_{n}$ are variables / functions, as opposed to numbers.
• [3]: mostly used to refers to the generalized continued fraction $[a_{n},b_{n}]$ where $a_{n}$ and $b_{n}$ are numbers, but also to the same notion as [2].
• [4]: same as [3].
• [5]: not clear.
• [6]: used only for the generalized continued fraction $[a_{n},b_{n}]$.
• [7]: same as [3].
• [8]: not clear.
• [9]: I think same as [2].

I did that fairly quickly, so there might be some mistakes.

Anyway, on this small sample, it seems that you are correct in saying that "generalized continued fraction" often refers to something a bit more general than what is described on Wikipedia (namely, [2]); but that some authors also use in the way described on Wikipedia.

If you want to do things properly, we should agree on a methodology before-hand (e.g, what queries, on which platforms, how many results, how do we classify them) and then do it. It is a bit of work though, it would probably take me one hour or two; so unless several people here agree that this would be helpful, I'm not going to do it. Best, Malparti (talk) 12:22, 24 October 2024 (UTC)

Google doesn't show me any of the pages you linked. Exceeded number of views, wrong jurisdiction, or whatsoever.

All but one seem to be younger than 2006, so may have been influenced by the terminology here in WP.

Some of them are from number theory. I guess we all agree that current WP terminology is fine for number theory.

Do the others prove more than some fringe usage? -- Dyspophyr (talk) 12:56, 24 October 2024 (UTC)

@Dyspophyr did you use Google Books? Do you get similar results with the first ten results it displays? Malparti (talk) 17:01, 24 October 2024 (UTC)

Please let me attempt to summarize the factual base of this discussion.

Continued fractions are finite or infinite expressions of the form b0+a1/(b1+a2/(b2+... They are used in various fields of mathematics. In certain fields, principally number theory, the canonical form has numerators a1=a2=...=1, and is usually called "Ø continued fraction". From the perspective of this field, it feels natural to designate continued fraction with numerators ≠ 1 as "generalized continued fraction". This term, however, is rarely used in the mathematical literature because continued fraction with numerators ≠ 1 are rarely used in number theory, whereas in other fields they are just called "Ø continued fraction". From the perspective of those fields, e.g. numerical analysis, the name "generalized continued fraction" feels wrong and violates the principle of minimal surprise.

Presenting both perspectives in one article faces the additional challenge that number theory writes continuous fractions as a0+1/(a1+1/(..., i.e. uses variable `a` in place of `b`.

Can we agree this far? -- Dyspophyr (talk) 11:13, 24 October 2024 (UTC)

I would recommend using the letter a for the denominators and b for the numerators, which seems quite common in sources I've seen. –jacobolus (t) 16:45, 24 October 2024 (UTC)

Note also that the current situation causes quite many inconsistencies in our analysis-related articles:

• Several articles have a link to generalized continued fraction overlayed with text "continued fraction":
  • Beta_function
  • Complex_plane
  • Convergence problem
  • Generating_function
  • Lentz's_algorithm
  • Stieltjes_transformation
• Other articles link to continued fraction but have numerators ≠ 1
  • Padé_table
  • Exponential_function
• Chain sequence has an "infinite continued fraction" with numerators z
• Complex_plane presents an "infinite periodic continued fraction" with numerators z
• Gauss's continued fraction says in its very first sentence "Gauss's continued fraction is a particular class of continued fractions ", then has numerators ≠ 1
• Pochhammer_k-symbol#Continued_Fractions,_Congruences,_and_Finite_Difference_Equations has "Continued Fractions" in the section header, and has numerators ≠ 1
• Solving_quadratic_equations_with_continued_fractions#General_quadratic_equation talks of "the continued fraction solution" and has numerators c

-- Dyspophyr (talk) 12:37, 24 October 2024 (UTC)

• Oppose move. By far the common name for a (simple) continued fraction is continued fraction. For instance, Hardy and Wright, Khinchine, Brezinski, Rockett and Szüsz, Hensley, etc. Generalized continued fractions are much more rarely used in mathematics. Indeed, many standard textbooks on continuedfractionsdo not even treat the generalization! Tito Omburo (talk) 13:40, 24 October 2024 (UTC)

@Tito OmburoI fully agree with your statement "By far the common name for a (simple) continued fraction is continued fraction." But this not the point. I am proposing certain moves not because the current lemma article title is inadequate for the continued fractions of number theory but because we need the lemma very same title to also cover the continued fractions of numerical analysis. Please read the above.

I happen to hold Brezinski in my hands. The first numerator ≠ 1 is on page 16, and many more follow.

Your last two sentences are neither right nor wrong; there is no way to weigh the relative importance of number theory and its continued fractions against numerical analysis and its continued fractions. We need to make an effort to be precise in our arguments. -- Dyspophyr (talk) 14:12, 24 October 2024 (UTC)

No idea what you're talking about: "current lemma". This is an article about continued fractions. You seem confused. Textbooks on continued fractions are pretty clear on this point, so I defer to sources here. 14:59, 24 October 2024 (UTC)Tito Omburo (talk)′

"Lemma" is a standard term in lexicography, see Lemma_(morphology). Of course this was potentially confusing in a mathematical context, I apologize. -- Dyspophyr (talk) 15:15, 24 October 2024 (UTC)

Yes, that was very confusing. XOR'easter (talk) 16:15, 24 October 2024 (UTC)

As for textbooks:

• https://www.google.de/books/edition/Analytic_Theory_of_Continued_Fractions/oN1TDwAAQBAJ has the first numerator ≠ 1 on page 2.
• https://www.google.de/books/edition/Nonlinear_Methods_in_Numerical_Analysis/elIyqgYk6RsC has the first numerator ≠ 1 on page 3, in the very first equation.

-- Dyspophyr (talk) 15:25, 24 October 2024 (UTC)

• Oppose move on the grounds articulated above by Tito Omburo and the fact that it would be a distraction from fixing the organizational issues identified by Jacobolus. The current article is mostly unwieldy because it has a poor structure: there are too many small top-level sections, limited narrative flow, and not much high-level vision. Shuffling around the article titles is not a fix for the real problems. XOR'easter (talk) 16:20, 24 October 2024 (UTC)

Move, modify, or restart?

In the above discussion, it became clear that this article currently covers continued fractions as they are usually defined in number theory, namely with numerators all being unity. This, however, is inadequate in numerical analysis, where continued fractions are usually defined as b0+a1/(b1+a2/(..., i.e. with numerators generally ≠ 1. Currently, this latter case is treated in the article generalized continued fraction. That article title is unfortunate because it conflicts standard use in numerical analysis, violates WP:COMMONNAME and the principle of minimal surprise, and also causes inconsistencies within this wikipedia, as listed above.

Therefore, for the sake of good coverage of numerical analysis, this article must change. Sorry, number theorists, however happy you are with the current state of affairs, this article must and will change. The remaining question is how.

Proposal 1, move:

• Move continued fraction to regular continued fraction.
• Move generalized continued fraction to continued fraction.
• Migrate enough contents from old to new continued fraction so that the numerators-all-1 case is also decently covered.

Proposal 2, modify:

• Modify the current article so that both number theory and numerical analysis are adequately addressed.
• Start from the more generic definition b0+a1/(b1+a2/(..., but move on soon to introduce also the case a1=a2=...=1.
• Migrate more technical material out to specialized articles.

Proposal 3 (by User:Nø), restart:

• Move continued fraction to regular continued fraction.
• Make continued fraction a disambiguation page.
• Gradually expand continued fraction into a self-contained overview article.

-- Dyspophyr (talk) 15:47, 24 October 2024 (UTC)

Pro 1:

• Preserves history
• Easy to get started

Con 1:

• Number theory will be ill served unless step 3 is brought to a good result.

Pro 2:

• Can be done gradually

Con 2:

• Can lead to inconsistencies, especially because variable name a is used differently in number theory vs numerical analysis.

Pro 3:

• Fair to both communities
• Can be done gradually
• Preserves histories

Con 3:

• Both communities will suffer a bit, until the disambuation page has grown into a decent article that feels like a good landing page for links from both fields

-- Dyspophyr (talk) 16:02, 24 October 2024 (UTC)

I think I'd (weakly) support proposal 2 (with the second-place alternative being leaving the articles mostly as-is), after having looked at several elementary books about the number theory variant written decades apart which lead with something along the lines of:

A continued fraction is an expression of the form $$a_{0}+{\cfrac {b_{1}}{a_{1}+{\cfrac {b_{2}}{a_{2}+{\cfrac {b_{3}}{a_{3}+\ddots }}}}}},$$ where $\{a_{i}\}$ and $\{b_{i}\}$ are integers. In this book we will concern ourselves with the special case where $b_{i}=1,$ called a simple continued fraction. ... (and then go on to mostly just use the name continued fraction for the "simple" case.)

But I think it would only work if someone is willing to do a significant amount of work improving the merged article's content, not just copy/paste paragraphs around. I don't personally feel motivated to do it. I'd recommend trying to at least sketch out a draft of a proposed merged article in user namespace or maybe at Talk:Continued fraction/Draft merge or the like, so participants here can better evaluate the concrete intention. –jacobolus (t) 16:56, 24 October 2024 (UTC)

I agree with what @Jacobolus said: I think proposal 2 would be the best, although (1) I don't think it's high-priority and (2) I think it can only work if someone qualified is willing to put the time and effort into it. Malparti (talk) 17:06, 24 October 2024 (UTC)

I oppose all three. The elementary version of the topic (with unit numerators) needs to be what readers see first, under the name "continued fraction", per WP:TECHNICAL, because that is the simplest and most accessible version of the topic. Re Malparti's earlier query: I have not formulated an opinion on whether a merge is ok, as a general thing, because I have not looked carefully at whether it is possible to do that and not overload the merged article. But if it is to be merged, the basics need to be front and center. In particular, what I oppose in proposal 2 is not the idea of a merge overall, but the specific "Start from the more generic definition". In mathematical topics, starting from a position of generality is usually a very good way to make material abstract and inaccessible. —David Eppstein (talk) 19:41, 24 October 2024 (UTC)

@David Eppstein I worked hard to understand somehow your standpoint, and I conceded that the focus on unit numerators is standard in number theory and that changes shall respect the wishes of editors from that background. Now I would appreciate if you could also try to understand the standpoint of numerical analysis where all kinds of numerators occur, and the restriction to unit numerators does not feel natural at all.

For me, the request to start with numerator = 1 feels a bit like Division_(mathematics)#Notation starting with a/2 instead of a/b. -- Dyspophyr (talk) 20:23, 24 October 2024 (UTC)

I'm pretty sure that, in fact, elementary school students learn about halving before they learn about more advanced forms of division. See Dyadic rational § In mathematics education. The difference is that we can reasonably expect most of our readers to be past that point already, but I don't think we can reasonably expect readers of this article to come into it with an advanced understanding of the topic. —David Eppstein (talk) 20:30, 24 October 2024 (UTC)

@David Eppstein One issue is that the current lead seems kind of vague and cagey about what "continued fraction" means, and then the rest of the article implies a very restrictive definition. This ends up being confusing/misleading to readers who encounter other kinds of continued fractions and come here looking to find out about them. This same concern probably motivates initial explanations along the lines of what I wrote above, which is found in a variety of books focused on "simple" continued fractions. Even if we want to keep this article's subject the same, I think it's helpful to lead with a definition giving an explicit expression with arbitrary integer numerators and then include text along the lines of that found in Olds (1963), "From now on, unless the contrary is stated, the words continued fraction will imply that we are dealing with a finite simple continued fraction." –jacobolus (t) 20:33, 24 October 2024 (UTC)