Talk:Continued fraction/Archive 2

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Archive 1

Archive 2

"Exact" continued fraction

If Wikipedia will permit, I think you should start a brand new article called "Exact continued fraction" (abbreviation ECF). Why?

I found a couple of tables in The On-Line Encyclopedia of Integer Sequences™ (OEIS™) by typing

"exact continued fraction" (including the quotes)

in the blank space provided. The tables are:

"Exact" continued fraction of e

which I frankly admit is an actual improvement on the simple continued fraction for e. To compare them:

Simple : e = [2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, ..., 4n, 1, 1, 4n+2, 1, 1, ...];

"Exact": e = [3, -4, 2, 5, -2, -7, 2, 9, -2, -11, 2, ..., 4n+1, -2, -4n-3, 2, ...].

I intentionally showed the patterns where each method arrives at the same convergent. The latter converges one-third more quickly.

I conjecture that the convergence speed improvement could be -38,196601125010515179541316563436... % less terms on average (i.e. one minus the golden ratio), or equivalently that to get the same approximation as an ECF of order n, you'll need to compute phi×n terms in the CF. And that the ratio between the number of terms in the CF of any irrational of large enough order n and the number of terms equal to 1 in the CF is also the golden ratio on average: the ECF us saves with all terms equal to 1 (but it still compresses many others).

There may be some theorem about this, but this is visible in the expansions of pi and e (and may be this could help demonstrate which one of pi+e and pi×e is transcendantal (and then the other one would be algebric, i.e. a zero of a finite polynomial with integer coefficients !), and the same thing between pi^e and e^pi or iⁱ, simplifying a lot of trigonometric problems and topologic problems within complexes (however it will be clearly difficult to domonstrate that, given that the ECF does not create a bridge with GCF where pi is defined with a period. 09:31, 16 August 2010 (UTC)

"Exact" continued fraction for Pi

The continued fraction pattern for pi is random for both simple and "exact". Again the latter converges close to one-third more quickly:

Simple : pi ≈ [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, 15, ...];

"Exact": pi ≈ [3, 7, 16, -294, 3, -4, 5, -15, -3, 2, 2, 2, 2, 3, -85, -3, 2, 15, ...].

Here I found the COMMENT from the editor (Serhat Sevki Dincer) illuminating, with patterns for other popular "exact" continued fractions:

If we use "closest integer function" instead of the common practice of using Floor(x) when calculating continued fractions, we obtain a sequence of (not just positive but also occasionally negative) integers which approximate the original number better "per term" in the sequence. I call such continued fractions as "exact".

For instance 3+1/(7+1/16)=3.14159292, 3+1/(7+1/15)=3.141509434;

3+1/(7+1/(16+1/(-294+1/3)))=3.141592653619, 3+1/(7+1/(15+1/(1+1/292)))=3.141592653012;

It is easy to see that as long as the fractional part of x(n) is in [0, 0.5) usual continued fraction and exact continued fraction agree in terms, but whenever fractional part of x(n) gets to be in (0.5, 1) then exact continued fraction gives better approximations more and more at each term.

In fact you can reduce the case of convergence only to the open intervals (0, 0.5) and (0.5, 1), because the tie-breaking rule only applies to the continued fractions for rationals, not those for irrationals. (If ever you get the exact fractional part 0.5, the only difference you'll see is in the last two terms of the finite continued fraction but this will not change the number of terms with a different sign for the last term equal to 2 or -2, depending only on the direction of rounding of the term before it. What is really import is (in that case only, that the rounding function remains symetric around 0).

Why did I choose the round half towards zero tie-breaking rule ? Only because the conventional continued fraction for 1/2 is [0; 2] but not [1; -2], and the conventional continued fraction for -1/2 is also preferably the symetric [0; -2], and not [-1; 2]. verdy_p (talk) 08:31, 16 August 2010 (UTC)

Another example is that, exact continued fraction of golden ratio is 2,-3,3,-3,3,... which gives better approximations for any same amount of initial terms when compared to the usual 1,1,1,...

For |x|>2, ECF(1/x) = [0, ECF(x)].

ECF(sqrt(3))=2,-4,4,-4,4,...

ECF(1/sqrt(3))=1,-2,-3,4,-4,4,-4, ...

ECF(-x) is just ECF(x) with signs reversed.

x(n)-a(n) is in [ -0.5, 0.5 ], hence for n>0, |a(n)| >= 2.

[End of editor's quote.]

I would have no objections whatsoever to your incorporating the contributions you wanted to introduce here on that page. − Glenn L (talk) 09:37, 13 August 2010 (UTC)

Tentative support for "exact" continued fractions, in a separate article, or at least in a self-contained section within this article. While these "exact" continued fractions have some nice features, it is also the case that some of the features for "simple" continued fractions are lost. For instance, with simple continued fractions, the truncated prefixes alternate giving upper and lower bounds for the number they are approximating. So, yes, if you have a citation that can back you up, then there is room for "exact" continued fractions somewhere. But, please be very careful to keep them separate enough that it is easy to see which properties apply only to "simple" continued fractions and which properties apply only to "exact" continued fractions. Thanks! Quantling (talk) 13:48, 13 August 2010 (UTC)

And this is not a discovery. You cite authors that published their article in 2007 or 2008, but I personnally learned to use the nearest integer method more than 20 years ago, and used it for creating faster convergents when implementing transcendantal functions since long, simply because they are numerically more stable (especially for computing exponentials with numbers in floatting-point notation, or for sine and cosine): this gives correct monotony in the expected intervals with several additional derivation/differentiation of the result, something you don't get with only positive terms in "simple" continued fractions, or that requires computing at least 6 or 7 additional terms for the IEEE-754 double precision, and many more (with significantly reduced performance) when you are building high-precision libraries, and even in that case, the result is often approximated with 2 additional ulps of error when working at the same maximum precision (and there's no way to correct it).

All good math library are using the division to the nearest integer to compute the terms of of the continued fraction, instead of the floor.

The fact that when using the nearest integer division reduces number of the terms by about 1 third for pi or e is even better than what I learned initially (it was just assumed that it would require 25% less terms (but at that time, the convergents were still not computed with many terms like they are today.

Note 1: I never learned the term "exact" continued fractions, because they cannot be exact for irrationals (as they still have an infinite number of terms), and they are as equally exact as with the other "simple" method for rationals. The nearest integer method just generates a contined fraction that converges faster.

Note 2: I learned the method by rounding 0.5 to zero instead of 1, and rounding -0.5 to zero as well (i.e. round half towards zero, exactly like in my table above, instead of towards infinity, or instead of round half down or round half up), because it preserves the symetry of terms in the finite continued fractions for rationals. For irrationals, the tie-breaking rule of the rounding mode does not make any difference because the exact half is never reached for irrationals (otherwise their continued fraction would be finite and the number would be rational). The reason for this choice was about a theorem about rational series, but I can't remember what it was exactly, but it was related to sign properties. And they were definitely not called "generalized continued fractions" (because generalized continued fractions may contain arbitrary numerators, when the "exact" continued fractions still use contant numerators always equal to 1).

Note 3: all theorems about the bounding conditions for the convergence of "simple" continued fractions (using only positive terms) are equally valid with continued fraction whose terms are computed with a "nearest()" function instead of "floor()". Notably, the "exact" continued fractions are also providing convergents with upper and lower bounds at any time: if you currently have only lower bounds, just compute a single additional term using rounding down or up (in the direction indicated by the parity of the number of negative terms already computed).

In theory the "simple" continued fractions are only useful for demonstrating the theorem that all conctinued fractions can provide an upper and a lower bound approximant at any time; once you know that, you can abandon them and use only "convegents using terms computed with the "nearest" method. verdy_p (talk) 07:59, 16 August 2010 (UTC)

Lead cleanup

Hey, all, I edited the lead/intro, to (hopefully) improve clarity and conciseness. I give a prose definition, which we really should have. Thanks to Encyclopedia Britanica for giving a concise prose explanation; it is elegant but surprisingly rare in the search results I saw. I wikilink basic concepts like "integer" and "fraction". I get the terminology and distinction between generalized vs regular out-of-the-way upfront, as incoming links may not disambiguate that stuff. I also moved the defining formulae to a section all their own. While the formulae are very useful, they also make the intro/lead hard to read from a human-language standpoint, and (I feel) make the article less accessible to people unfamiliar with the subject.

Now, two caveats:

The generic formula for a finite continued fraction has a bit of a kludge in it. The terminating denominator should perhaps be fully subordinate to the diagonal dots, rather than stuck in with the dots as part of the same denominator. But I don't know the markup (I was copy-and-paste coding here), so I just stuck it all together. Or maybe that's okay.
Everything I know about continued fractions I leaned from this article. :) So my edits should be checked for accuracy by a subject matter expert.

Hope this helps. Cheers! —DragonHawk (talk|hist) 17:47, 29 November 2010 (UTC)

There is no such thing as an infinite expression

Alas, an expression (mathematics) is "a finite combination of symbols that are well-formed according to the rules applicable in the context at hand." While I appreciate the attempts at clarity, unfortunately we cannot lean on the concept of infinite expression to get there. The infinite continued fraction is defined only as the limit of a sequence of finite continued fractions, as described in the text that was removed.

The new recursive definition also has the problem that it cannot terminate; to me, it appears to prohibit finite continued fractions. If you are to rescue that sentence, perhaps something like the following would be better:

In mathematics, a continued fraction is an expression of a number as either an integer or the sum of an integer and the reciprocal of a continued fraction.

That way, at least, it can terminate after finitely many steps. However, this revised sentence still inappropriately permits infinitely many steps, because it does not force one to eventually use the integer-only alternative.

I am going to revert for now, but I urge you to give it another go, addressing these concerns. — $Q$ uantling (talk | contribs) 19:37, 30 November 2010 (UTC)

There is absolutely no mathematical problem defining infinite expressions, any more than there is in defining infinite sequences or infinite graphs or other infinite combinatorial objects, and there is no shortage of reliable sources that use the exact phrase "infinite expression" in a mathematical context [1]. Mathematically, an infinite expression is just an infinite tree in which the interior nodes are labeled by mathematical operations and the leaf nodes are labeled by mathematical values, just like a finite expression. And even if one uses a definition of "expression" that restricts it to a finite thing, an "infinite expression" can still be well defined (as a generalization rather than a specialization of the case of a finite expression). Also note that this is a general issue that has little to do with continued fractions, so we shouldn't need an extended discussion of this in the article itself. One has to be careful in defining the meaning of an infinite expression: what does it evaluate to? But that doesn't seem to be what you're bothered by, so I'm left at a bit of a loss. What do you think a notation like

a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{\cfrac {1}{a_{4}+\ddots }}}}}}}}.

is, if you don't think infinite expressions exist? —David Eppstein (talk) 20:56, 30 November 2010 (UTC)

Thank you for your response. The notation itself is fine, and I am willing to come to terms with "infinite expression." However, I am hoping you will address the correctness of the first sentence, "In mathematics, a continued fraction is an expression of a number as the sum of an integer and a fraction, the numerator of which is 1 (one), and the denominator of which is the sum of an integer and another such fraction." Do you agree that this prohibits finite continued fractions? Also, what do you think about changing "sum" to "formal sum"? —

Q

uantling (talk | contribs) 21:32, 30 November 2010 (UTC)

Yes, I am bothered by the numerical meaning of the infinite continued fraction as currently defined in the text. If what we mean by it is a limit of implied finite continued fractions, then why not say so? —

Q

uantling (talk | contribs) 21:32, 30 November 2010 (UTC)

The numerical meaning is covered briefly in the lead section, towards the end, where it says that truncating the expression leads to a sequence of numbers that is guaranteed to converge to a limit. So the answer to "why not say so" is that we already do. —David Eppstein (talk) 21:47, 30 November 2010 (UTC)

Thank you for making that edit to address my concerns. —

Q

uantling (talk | contribs) 14:43, 1 December 2010 (UTC)

"Quantling" wrote: "The infinite continued fraction is defined only as the limit of a sequence of finite continued fractions,". But two different infinite continued fractions can have the same limit (although with "simple" continued fractions this can't happen). Moreover, although a number, (e.g. e) may be the limit of a sequence of finite continued fractions, nonetheless one would not say therefore that e is a continued fraction. Michael Hardy (talk) 21:37, 30 November 2010 (UTC)

I have reverted to the previous definition, which is simpler and clearer. The new definition was simply wrong, because it did not restrict a_i to be a positive integer for i > 0; patching it up to correct this would have made an already overly convoluted definition even worse. Gandalf61 (talk) 22:12, 30 November 2010 (UTC)

I think it is very important to have a prose definition, for reasons of accessibility. That is both accessibility to those who don't "think" in mathematical formulae, as well as accessibility to people with reading/vision deficiencies. I, for one, do not find the current presentation at all clear. I will attempt to improve the prose. Please do me the favor of working with me and not simply reverting. —DragonHawk (talk|hist) 03:36, 1 December 2010 (UTC)

Take two

Okay, I've reworded things to try and improve correctness: (1) Added remark that integers (other than the first) are positive (2) Explicitly mentioned termination. The thing I really like about Britannica's definition is its simple elegance: It expresses the concept perfectly (at least for the way I think). I didn't really "get" what the magic of a continued fraction was until I read that. The recursive nature seems fundamental. (Perhaps there should be discussion of that somewhere in the article? Or at least a mention/link to recursion.) The fact that there's a terminating condition is critical, of course, but I think it improves comprehension to articulate the core concept, and then define the edges.

I've also stuck the basic equation into the lead, the same way we normally do page images. We normally like to have a "defining illustration" as part of the lead, along with a prose description. This is the same thing, really. It's just that because the subject matter is a mathematical concept, the defining illustration is a formula. The math markup gets rendered as an image anyway. Good? Bad?

Now, one of the concerns raised is that properly, there is no such thing as an "infinite expression". Fine. Cool. But simply saying that doesn't really do anything to advance the article. We need to address:

What's the right term, then? I could write "A continued fraction is a thingy consisting of the sum of an integer and a fraction, where ...". What's the term to use instead of thingy?
The math markup for an infinite continued fraction looks like an expression to me. (And it was like that when I got here. <grin>) Should it be tweaked somehow? Perhaps by explicitly giving it as a limit? Or this just one of those cases where something which is technically slightly improper is still well-understood in common usage?

If there are still problems, by all means, identify them here, or fix the article if you can. I'm sure we can work together to come up with something which makes the article better for everyone. (I'm actually really excited by this prospect (sad as that may sound). I've learning some neat stuff in the process of trying to improve this article, and I hope that I can continue my enlightenment *and* help make it more accessible to other less math-savvy folks (like myself).)

Clear skies! —DragonHawk (talk|hist) 05:08, 1 December 2010 (UTC)

Regarding this edit by Gandalf61. While I'm pleased to see people collaborating on the prose, I think this particular edit is redundant. Specifically:

"Optional". The fact that the fractional component is "optional" is explained in the sentence after the next. Human language isn't like a formula; it's okay for a later sentence to refine a concept from an earlier sentence. Now, if people think adding "optional" to the first sentence significantly helps understanding (more than it costs in wordiness), by all means, let's have it. But if the objection is that the lack of "optional" makes the first sentence less "mathematically pure", I think we need to have further discussion.
"numerator of the fraction component". I think that's completely redundant. "Numerator" implies fraction, by definition. It doesn't need to be explicitly qualified. Given the concerns about the sentence being cumbersome, I really think this should be omitted.

Gandalf61, I'd like to hear your take; and if others have opinions, please post. —DragonHawk (talk|hist) 10:17, 1 December 2010 (UTC)

DragonHawk - I think you are displaying a high degree of ownership behaviour here. You seem to be completely opposed to any modifications to your own text, no matter how small. Personally, I think your new recursive definiton is needlessly baroque and opaque. Without the word "optional" it is not only opaque, it is also wrong. But no matter - we can put this straight again after you have got bored and moved on. Gandalf61 (talk) 12:55, 1 December 2010 (UTC)

Gandalf61:

I'm not opposed to modifications (e.g., I really like Quantling's latest idea), but I do think you and I disagree over how to best approach a technical topic. You seem to favor strict mathematical correctness at every moment in the text (i.e., your "wrong" comment). I think it's not at all wrong to introduce a concept and refine it in immediately following sentences. Is this something you're willing to discuss further, or is it an involute principle for you? I can work with the later, although I would prefer to strive towards mutual understanding.
Regarding "baroque"; I'm confused. I usually think of "baroque text" as having superfluous words or overly complicated sentence structure. That's exactly what I was trying to avoid by removing the "fraction" bit, for example.
As far as "after you have got bored and moved on" goes, I don't find that attitude helpful at all.

Kind regards, —DragonHawk (talk|hist) 16:40, 1 December 2010 (UTC)

I have made the first sentence define only the infinite continued fraction, so that we can be both accurate and avoid the need for the "optional" which is being debated. —

Q

uantling (talk | contribs) 14:43, 1 December 2010 (UTC)

Quantling - well done ! I think your version is an improvement. Gandalf61 (talk) 15:10, 1 December 2010 (UTC)

I, too, think Quantling's idea of starting with the definition of an infinite continued fraction is great. It avoids the need to spend separate verbiage making the distinction of infinite vs finite later on. Nice! —DragonHawk (talk|hist) 16:20, 1 December 2010 (UTC)

Sorry to come a bit late to this discussion, the damage seems to have settled already. I totally agree with Quantlings original remarks that an expression is by definition finite. It is not that one cannot envisage infinite mathematical objects that are somewhat like expressions, but the things one does with expressions require them to be finite. The main application is giving meaning to an expression by defining the meaning of the operations in the expression: for this to work, it must be possible to give meaning to the atomic subexpressions first then work up to larger and larger subexpressions, each time by applying an operation to subexpressions with already determined meaning. If one should wish to contemplate infinite expressions, then one must first think very deep what one wants to allow. There need not be any atomic subexpressions, nor any root; for instance the infinite tree with nodes indexed by Z×Z, each representing a subtraction where the left operand of the node (i,j) is given by the node (i+1,2j) and the right operand by the node (i+1,2j+1), perfectly well matches the definition by David Eppstein; there is no way one can even start giving a meaning to such an "expression" (nor would there be any place to end). There is an analogy with proofs, which must also by definition be finite, because only in that way can one deduce from the validity of individual steps the validity of the conclusion. Logicians or computer scientists know more about this; it would be good to compare what is written in the corresponding articles. I know that while mathematics involves many notions where an infinite number of operations is potentially involved, any meaning given to a "result" of that is defined by a limiting process of things involving only finitely many operations. One can define (if one likes) polynomials as (equivalence classes of) expressions, but not formal power series; there the sum notation is just a notation for an object defined by an infinite sequence of coefficients; no infinitely many additions are involved (even though one can consider the limit of finite sums, which converge in the appropriate topology).

Note that recursive/inductive definitions such as using Backus–Naur Form always implicitly assume finite expressions and therefore termination of the recursive process; allowing circular definitions that do not obey such a restriction immediately gives occasion for all kinds of paradoxes. I think it is especially important that the current article should avoid any talk of infinite expressions, precisely because continued fractions are often introduced with an enormous naiveness with respect to the meaning of writing expressions with ellipses in them. If there weren't any finite continued fractions, there would be no way to give a meaning to infinite ones, and the current lead suggests exactly the opposite (even though the following paragraphs make clear that it cannot be like that). Marc van Leeuwen (talk) 14:05, 5 December 2010 (UTC)

I don't think anyone is trying to argue that a proper expression can be infinite. But nobody's yet given the correct usage. To reiterate: What's the correct term? I could write "A continued fraction is a thingy consisting of the sum of an integer and a fraction, where ...". What's the term to use instead of thingy? —DragonHawk (talk|hist) 14:07, 6 December 2010 (UTC)

In this case, a "thingy" is an "infinite sequence of finite continued fractions," (where the first finite continued fraction of the sequence is an integer, and each subsequent finite continued fraction of the sequence is built from the previous continued fraction's finite sequence of integers plus one additional positive integer). Noting that a finite continued fraction is an expression we could say that a "thingy" is an "infinite sequence of expressions" (with restrictions that the expressions be finite continued fractions with the aforementioned property). The reasons that "infinite expression" isn't available for this purpose is that it is ill defined. In too many cases, infinite self-referential definitions are, at best, ambiguous and, at worst, self-contradictory (e.g., set of all sets). I liked the old article lead, that started with finite continued fractions, defined them well, and then used a limiting sequence of them to define an infinite continued fraction. Perhaps we can revert to there and then try to tweak for improvements. — $Q$ uantling (talk | contribs) 17:16, 6 December 2010 (UTC)

I believe you are mistaken when you say that an infinite continued fraction is "an infinite sequence of finite continued fractions". At the very least, I would like to see a cite from a reliable source that identifies continued fractions with sequences before considering such a bizarre and non-standard notion for inclusion in the article. It is true that one normally defines the value of the infinite continued fraction as the limit of a sequence of finite continued fractions, but that is not what you said and it is not the question that DragonHawk asked.

As was indicated above by other editors, such as David Eppstein, there is really no problem with "infinite expression". An expression is a tree structure; mathematicians deal with infinite tree structure all the time. For example, Khinchin says:

An expression of the form $a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+\cdots }}}}$ is called a regular or simple continued fraction. … The number of elements may be finite or infinite. … In the second case we shall … call it an infinite continued fraction.

(Khinchin, A. Ya. (1964). Continued Fractions. The University of Chicago Press. p. 1.)

As you see, Khinchin has no problem with defining an infinite continued fraction as an infinite expression. The only difficulty occurs when one tries to say what the infinite expression means. Khinchin continues: "We cannot immediately assign any numerical values to an infinite continued fraction. Until we adopt some convention, it is only a formal notation, similar to that for an infinite series whose convergence or divergence is not brought into question."

I do not want to get involved in the details of the rewrite, but in my judgment, it is unwise to avoid terms like "infinite expression" merely out of a misguided fear of unfamiliar infinite objects, and doing so is likely to complicate the article to the point where the general reader will have trouble understanding it. For example, if one were afraid of infinite expressions, one could define continued fractions the way Jones and Thron do, as a (possibly infinite) sequence of pairs of numbers representing the partial numerators and denominators, and then associate these with a third sequence of the partial quotients. Infinite sequences may be more familiar to some people than infinite expressions, but I do not think they are intuitively clearer or that they improve the presentation.

But I suggest that the other contributors to this thread should look at the definition in Rockett and Szüsz, which might make everyone happy: it makes clear the continued fractions are to be understood as expressions of a certain type, but the actual definition is in terms of the sequence of partial denominators, and then there is no trouble in slipping in the idea that the sequence might be infinite. —Mark Dominus (talk) 15:22, 7 December 2010 (UTC)

What do you mean by an "infinite expression" on integers? For instance, if I have a directed graph where each vertex with out-degree of zero is assigned an integer value and every other vetex is assigned an operator (for combining one or more integers) is that an "infinite expression"? Does the out-degree of each vertex have to be finite? Does the graph have to be acyclic? Does the graph have to be connected? Does the graph have to have a root vertex from which all other vertices can be reached? Or, if we're in search of an alternate for "infinite expression", which attributes (finite out-degree, connectedness, etc.) would you want "thingy" to have? — $Q$ uantling (talk | contribs) 17:28, 7 December 2010 (UTC)

Why are you so skeptical about infinite expressions, when you seem to admit without difficulty the concept of an infinite graph or an infinite sequence? There is little conceptual difference. And in any case this is quite far off topic for this article since the expressions that are actually used for continued fractions are so highly constrained that it hardly matters what level of additional generality we allow. —David Eppstein (talk) 17:37, 7 December 2010 (UTC)

(ec) All but one of your objections apply just as well to finite expressions as they do to infinite expressions. I do not understand the point of bringing up a whole lot of ill-defined cases that are just as meaningless when finite as when infinite, unless you are trying to argue that the notion of an "expression" is ill-defined in general. Are you really approaching this discussion in good faith? If so, I suggest that you learn what a tree is before participating further; the definition always requires that the graph be acyclic and connected. —Mark Dominus (talk) 17:44, 7 December 2010 (UTC)

What does "(ec)" mean? There are very good definitions for infinite graph and infinite sequence. Likewise, a finite expression is well defined in pretty much any book on software compilers, and can be defined in terms of a finite rooted tree. For "infinite expression" it seems natural to stick with rooted trees. But we still have some decisions that were moot in the finite tree case, yes? For an infinite series of real values with a sum that is convergent, but not absolutely convergent, the order of adding the terms together is important; so for our infinite expression, I think we're going to have to stick with finite degree vertices to avoid having to somehow infer the desired birth order for infinitely many children of a given vertex. The finite degree will necessarily limit us to countably many vertices, I think, so maybe we don't have to consider whether to permit uncountable numbers of vertices. Are there any other ambiguities or gotchas out there that we should watch out for? Since it is not our duty to do original research, is there any source that rigorously defines an infinite expression to this level of detail?

In articles on mathematics it is okay to use terms without defining each and every one of them, but, IMHO, it is not okay to use terms that do not have accepted definitions. I believe the latter is a disservice to the reader. — $Q$ uantling (talk | contribs) 19:12, 7 December 2010 (UTC)

Ec means edit conflict. And you are still unnecessarily conflating two very different things: the form of the expression and its meaning (since you seem to prefer computer science terminology: the syntax and the semantics). Evaluation order is a semantic notion, it has nothing to do with the syntax. And since it is our duty to avoid original research, we should stick to definitions of continued fractions that can be sourced instead of trying to synthesize definitions from sources that are about expressions but not about continued fractions per se. —David Eppstein (talk) 19:16, 7 December 2010 (UTC)

There is no accepted syntactic definition for an "infinite expression" that I know of. If you know of one that is rigorous enough to take sides on the subtle issues (e.g., whether vertex degree should be finite), please let me know. Lacking such a definition, I continue to believe that it is a disservice to the reader to use "infinite expression" as if it were a mathematical concept. — $Q$ uantling (talk | contribs) 19:39, 7 December 2010 (UTC)

The subtle issues have nothing to do with continued fractions, since in the case of continued fractions the form of the fraction necessarily restricts the degree to be finite, and since there is a standard accepted semantics (the limit of the convergents) for these expressions. If you are interested in a nice language for looking at syntactic issues mathematically, see symbolic combinatorics, but your failure of imagination and your limited experience with mathematical constructs are not a reason to similarly blind the other readers of this article. —David Eppstein (talk) 19:42, 7 December 2010 (UTC)

I agree that "infinite continued fraction" does not depend on the subtleties. However, if, in a mathematical article, we say that an A (e.g., an infinite continued fraction) is a B (e.g., an "infinite expression") then, IMHO, B had better be something. In this case B is not well-defined, and thus, IMHO, a disservice to the reader. (The parts about "failure of imagination" and "limited experience with mathematical constructs" and "blinding" other readers are not helpful. Please consider removing them; and you can remove this parenthetical note along with them.) — $Q$ uantling (talk | contribs) 19:56, 7 December 2010 (UTC)

Multiple book sources just use "infinite expression" without getting distracted by the formalism: [2] (Euler!), [3], [4], [5], [6], etc. If you can find a source which insists that the precise class of things that can be called "infinite expressions" must be defined before the term is used, in the context of continued fractions, then we can add a footnote about that or something. Otherwise, we risk committing original research. —David Eppstein (talk) 21:10, 7 December 2010 (UTC)

Good work with the sources! From them and a little more poking around, it appears to me that "infinite expression" is used to mean the union of objects of type infinite series (including, e.g., a decimal expansion), infinite product, and infinite continued fraction. I think this resolves the dispute. — $Q$ uantling (talk | contribs) 21:44, 7 December 2010 (UTC)

Take three

Discussion here has petered out (though there continues some higher level discussion at Wikipedia talk:WikiProject Mathematics#Infinite expressions). I am making a WP:BRD edit in the article to restart things. — $Q$ uantling (talk | contribs) 15:49, 9 December 2010 (UTC)

Now that the wording "infinite expression" appears in clear in the lead, let me just reiterate my objection to this type of description briefly. While it is possible to make some sensible definition of infinite expressions, mentioning them here (or in the series article) is a very bad idea, because it suggests that these notions are based on a general notion/theory of infinite expressions that is a straightforward generalisation of finite expressions. In reality there is no such theory, and the meaningfulness of individual instances of infinite expression derives from a case-by-case approach, showing that the potential pitfalls of infinity are avoided. The infinite expression article does not address this clearly, and its existence can reinforce a naive belief that everything is well founded (I personally prefer that it were removed altogether). This aspect is more pernicious for continued fractions than for series (where divergent examples abound) as it superficially appears that everything always converges. But this is too tiring for me to keep discussing endlessly. Marc van Leeuwen (talk) 12:44, 18 February 2011 (UTC)

I share your belief that "infinite expression" is neither so well established nor even so well defined as "(finite) expression." I have added the article on the former, and the link to it from continued fractions because, IMHO, this is better than what was before, which was text and a link to the more rigorous (but less appropriate) expression (mathematics). That is, I put in the word "infinite" precisely because I agree with you that likening continued fractions to (finite) expressions is inappropriate.

I also agree that the infinite expression article does not well address the many pitfalls associated with trying to define a general concept of infinite expression. Despite that we are wearing you down, I hope that you will make an attempt to rectify this. I think it would be quite valuable for the article. Thanks! — $Q$ uantling (talk | contribs) 14:33, 18 February 2011 (UTC)

I took a stab at adding "not well-defined" and "self-inconsistent" to the infinite expressions page. Please take a look, and suggest changes or modify. — $Q$ uantling (talk | contribs) 14:55, 18 February 2011 (UTC)

circular definition

The new definition contained in the opening sentence is a bit circular. It might be better to say "the sum of the integer part of the number and the reciprocal of another number, then replacing that other number by the sum of its integer part and the reciprocal, etcetera". Tkuvho (talk) 03:12, 12 December 2010 (UTC)

It's not circular, it's recursive. —David Eppstein (talk) 00:05, 13 December 2010 (UTC)

Yes, but it does not help the reader who does not yet know what a continued fraction is. In this sense it is circular. One should not confuse a definition and an introduction. Tkuvho (talk) 01:54, 13 December 2010 (UTC)

Indeed one should not confuse a definition with an introduction. The recursive description of continued fractions is an elegant way to introduce the subject; people can then scroll down to see a more detailed definition. The iterative definition (current version), although it's logically correct, is longer and feels a little clumsy. If someone doesn't already know what a continued fraction is, then they'll need to read some of the article and study some examples, no matter how the opening sentence is phrased. I'd be in favour of reverting to the recursive version. Jowa fan (talk) 05:38, 13 December 2010 (UTC)

I agree that the recursive presentation is more elegant. I am just wondering whether it is actually mathematically coherent. Can one make it into a precise recursive statement? Note that one chooses an arbitrary integer at each stage, so this some kind of "free-choice" recursion. Has this sort of thing been formalized? Tkuvho (talk) 06:11, 13 December 2010 (UTC)

You could try looking at Conway's "Mathematicians' Liberation Movement" (an appendix to his book On Numbers and Games). Conway argues for much stronger (transfinite) recursive constructions. —David Eppstein (talk) 12:27, 13 December 2010 (UTC)

Interesting. What are we being liberated from? Axiom of foundation? Tkuvho (talk) 13:08, 13 December 2010 (UTC)

I believe the intent is to liberate us from gratuitous pedantry — why reduce everything to a set in an unnatural way when it can perfectly well be defined as an object of its own type? —David Eppstein (talk) 23:17, 13 December 2010 (UTC)

To prevent a set of all sets and other oddities, and their associated self contradictions, mathematicians have been very careful to codify what infinite constructions are permissible. While not every case of an infinite recursion or a circular definition necessarily gets us into "set of all sets" territory, it is not ours to do original research establishing that a given circular definition is permissible. We should stick to the well established constructions. One such well established construction is that of an infinite sequence; which is why I would define an "infinite continued fraction" as a sequence of "finite continued fractions" (as formed from the finite prefixes of the sequence of integers for the infinite continued fraction). From there we can define the value of the infinite continued fraction as the limit of the sequence of values for the finite continued fraction, etc. If such a formal definition is too stilted for our lead paragraph, I would alert the reader to our "sloppiness" by starting with "Roughly speaking, " or similar. —

Q

uantling (talk | contribs) 14:59, 13 December 2010 (UTC)

I sympathize with your concern that "infinite expression" is an ill-defined concept. However, one can't always usefully replace an infinite object by a sequence. A real number is "really" a sequence of rationals, but it would not be helpful to our readers to insist on this point of view. The "infinity" in "infinite continued fraction" is not that different from the infinity in the definition of a generic real. Besides, reals can be defined this way; I don't remember if this is mentioned at our page for constructions of R. Tkuvho (talk) 15:14, 13 December 2010 (UTC)

[3; 7, 15, 1] 100x more accurate representation of π than 3.1416?

The Motivation section states, "As an approximation to π, [3; 7, 15, 1] is more than one hundred times more accurate than 3.1416." By my math, [3; 7, 15, 1] is 355/113, which is about 2.7 * 10^-7 greater than π; 3.1416 is about 7.3 * 10^-6 greater than π. So unless I'm mistaken, this continued fraction is only ~28 times more accurate. Rather than correct the number I'd prefer removing this sentence, as the previous sentence shows another example with a similar scale difference in accuracy. --dzhim (talk) 20:09, 25 September 2011 (UTC)

The statement has been fixed, see [7]. (talk)

Not so: it still says "is more than one hundred times more accurate", which as pointed out appears to be incorrect. Quondum^talk_contr 07:54, 25 November 2011 (UTC)

The section presently compares the approximations 355/113 and 333/106. The error in the latter is 8.3 * 10^-5, which is more than hundred times greater than 2.7 * 10^-7. Isheden (talk) 10:35, 25 November 2011 (UTC)

My bad – I should read more carefully. But the point being made is a little "pointless", if I may say so. In any efficient representation system, the step increase in accuracy is in line with the amount of information encoded. In a base-1000 positional system, every digit added increases the accuracy by roughly 1000 times. I think the the entire paragraph should be scrapped as being non-encyclopedic. Quondum^talk_contr 11:20, 25 November 2011 (UTC)

I guess the point with this is to illustrate Theorem 5, Corollary 2 later in the article. I agree though that the point being made is not very clear. Isheden (talk) 13:48, 25 November 2011 (UTC)

I think part of the problem is that the section is really trying to motivate rational approximation rather than continued fractions, with continued fraction a means to find rational approximations. It comes down to whether you'd rather say a km is 5/8 of a mile or .62 of a mile. Both use two digits and are about the same accuracy, but if I'm driving through Canada and see a sign saying Windsor is 158 km away, I'd rather use the 5/8 when trying to figure out in my head if I need to stop to get gas. I have my doubts about the "more accuracy with less information" argument so perhaps the section should be scrutinized a bit more skeptically, but I do think there should be some kind of motivation given in the article.--RDBury (talk) 02:32, 26 November 2011 (UTC)

The point about the "best possible" rational approximation (under the given constraint) is probably one worth making. This example most certainly does not make it, it only points to an arbitrary example of a big increase in accuracy in one step. My point about information (I think I could prove a statement like "no representation can be more efficient than a positional system with respect to accuracy", and you seem not to disagree) is merely to say that the article should not create the impression that in a general sense continued fractions are "more accurate" than others for uniformly distributed real numbers (which seems like what the example was trying to do, before being corrected, and now I don't know what it succeeds in illustrating). If an illustration/motivation of in a certain sense the "best possible" rational approximation given the size of the numbers in numerator and denominator is what is intended, then a fresh approach is needed. It seems that (my guess) the length of the representation being even or odd distinguishes between whether it is the numerator or denominator that applies in the phrase I quoted here; the accuracy can actually diminish with the sequence being extended by 1. So simply taking a rational approximation from the sequence as a "best" approximation for your example (e.g. miles/km) is not generally a good idea; it must be done with insight. And, if such a laborious example of "best rational approximation" is needed, it should not be under "Motivation", but under the section titled "Best rational approximations", where some rules are at least given for locating best rational approximations. Quondum^talk_contr 05:46, 26 November 2011 (UTC)

I did a restructure of section, dropping about three paragraphs and rearranging and rephrasing the rest with a smattering if new material. Hopefully that will address the concerns raised here. To me, the biggest take-away-point of the section is not the comparison with decimals, but the parallel with the Euclidean algorithm, and this was not mentioned in the previous version.--RDBury (talk) 09:34, 28 November 2011 (UTC)

I agree with the general clean-up. I just would like to mention the "point of the pointless statement" that has now disappeared (I think). The point was that among the convergents obtained from a continued fraction expansion, those which truncate just before a large coefficient do a particularly good job at approximating. This is not intuitively obvious from an argument "the step increase in accuracy is in line with the amount of information encoded", whatever that is supposed to mean. On the contrary, that would suggest that one gets a large increase after including the large coefficient, which after all represents more information added than a small coefficient. But in fact the large coefficient can be interpreted as a comment on the previous approximation, namely that it proved to be particularly hard to improve upon. Marc van Leeuwen (talk) 11:02, 28 November 2011 (UTC)

The point made by Marc is quite interesting and well put; I had missed this pre-big-number accuracy. Such observations may belong in a general "Did you know?" section rather than as a motivation. Opening the motivation with an opaque example still strikes me as odd. Starting with Continued fractions are, in some ways, more "mathematically natural" representations... may be more "motivating". Quondum^talk_contr 13:15, 28 November 2011 (UTC)

Calculating continued fraction representations

In the section with the same name, it is claimed that the described method is suitable for all real numbers. Is this really true in practice? How would you proceed with e.g. an algebraic number such as ${\sqrt {3}}$ or a transcendental number such as e? Isheden (talk) 22:25, 23 November 2011 (UTC)

The only difference is that for irrational numbers the process does not terminate. So yes, it does work in practice.--RDBury (talk) 06:35, 25 November 2011 (UTC)

But according to the article, "the limited precision of floating point numbers will often lead to small errors, skewing the final result. Instead, floating point numbers should be converted to rational numbers." How should a number such as e be represented to ensure that the process does not terminate? Isheden (talk) 10:42, 25 November 2011 (UTC)

The accuracy of the result is always limited by the accuracy of your method of computation, even if you're just doing addition. Just as with computing the decimal expansion, the continued fraction expansion will only be accurate for a finite number of terms unless you know the exact value of the number and are doing the computations exactly. If you want more terms in the expansion then you have to start with a more accurate value for the number and do the computations more precisely. The phrase you quoted is somewhat inaccurate though and I'll delete it.--RDBury (talk) 01:58, 26 November 2011 (UTC)

Mathematics rating

The rating of this article should be updated. Clearly it is not Start class any more. Isheden (talk) 22:26, 23 November 2011 (UTC)

This has been done, though I might have given it a C.--RDBury (talk) 07:04, 25 November 2011 (UTC)

If zero is allowed

The article on e (mathematical constant) says, as this article says, that the CF is

e=[[2;1,{\textbf {2}},1,1,{\textbf {4}},1,1,{\textbf {6}},1,1,{\textbf {8}},1,1,\ldots ,{\textbf {2n}},1,1,\ldots ]],\,

But it adds that it can be written more harmoniously by allowing zero: (Hofstadter, D. R. "Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought" Basic Books, 1995)

e=[[1,{\textbf {0}},1,1,{\textbf {2}},1,1,{\textbf {4}},1,1,{\textbf {6}},1,1,{\textbf {8}},1,1,\ldots ]].\,

A scan of this article shows me no explanation to help me interpret that second expression. — Cpiral Cpiral 00:50, 30 November 2011 (UTC)

It seems that they used a comma rather than the normal initial semicolon to make the pattern more visually obvious. They are saying that if the rules on continued fractions are relaxed to allow zero, then the two expressions are equivalent (the second being more "hamonious" because with the zero there is an obvious pattern from the start). The expression with the zero is simply

1+{\cfrac {1}{0+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{4+\ddots }}}}}}}}}}}}}}

The Continued fraction article seems to give enough to interpret the expression. It does state: "Note that it is customary to replace only the first comma by a semicolon", which should be enough of a hint. Quondum^talk_contr 04:11, 30 November 2011 (UTC)

So you're saying that there they did two steps to get from one version to the other

[1;0,1...] is eq to [2;...] and
"The semicolon in the square and angle bracket notations is sometimes replaced by a comma"

And that the "harmony" is thereby two things

the pattern of numbers
the consistent use of commas

Now is there mention here of a rule (perhaps buried in the convergents formulas or the theorems) that a zero can relax, for any reason, as any but the leftmost quotent? It says "first integer may be zero", but that's it. The articles should harmonize. — Cpiral Cpiral 08:57, 30 November 2011 (UTC)

It does say "... where

a

₀ is an integer, any other

a_{i}

members are positive integers" under the section Basic formulae. I've edited the article to make it clear that this also applies to the infinite case. So the rule about all but the first value being positive (and hence non-zero) is there. The relaxation to zeros falls under the section Generalized continued fraction (perhaps not very obvious). The article unfortunately does not have a clearcut Definition section. Quondum^talk_contr 09:40, 30 November 2011 (UTC)

On the contrary, it seems that, since the section Generalized continued fraction follows the section Some useful theorems, which say "positive integers", that a generalized fraction must not allow zero.
Does the generalized fraction have a bracket notation? The main article has none.

Join the discussion at Talk:E (mathematical constant)#Continued fraction notation? — Cpiral Cpiral 10:23, 7 December 2011 (UTC)

For clarity, no restriction applies to the values used in a "generalized continued fraction"; they may be zero. The concise notation is not adequate for the purpose; aside from that I don't know. Quondum^talk_contr 12:21, 7 December 2011 (UTC)

Theorem 5

I think there is some mistake in theorem 5 - what I mean is if $\alpha =[a_{0},a_{1},\cdots ,a_{n},a_{n+1},\cdots ]$ , then $\alpha ={\frac {\alpha _{n+1}h_{n}+h_{n-1}}{\alpha _{n+1}k_{n}+k_{n-1}}}$ , where $\alpha _{n+1}=[a_{n+1},\cdots ]$ , so:
$\left|\alpha -{\frac {h_{n}}{k_{n}}}\right|={\frac {1}{(\alpha _{n+1}k_{n}+k_{n-1})k_{n}}}<{\frac {1}{(k_{n}+k_{n-1})k_{n}}}$

since $\alpha _{n+1}>1$ strictly for infinitive fractions. What do you think? — Preceding unsigned comment added by 82.41.56.99 (talk) 00:56, 4 January 2012 (UTC)

Try α=√2. The convergents are 3/2, 7/5, 17/12, ... . |3/2-√2|=1/11.65 which is between 1/10 and 1/14, |7/5-√2|=1/70.35 which is between 1/60 and 1/85, and |17/12-√2|=1/407.64 which is between 1/348 and 1/492, so while I'm not sure where your logic went wrong the theorem appears to hold.--RDBury (talk) 14:30, 4 January 2012 (UTC)

Introduction

The introduction starts off by taking about the continued fraction associated to a number. Would it be better to define a continued fraction intrinsically, say that every CF represents (evaluates to/converges to) a real number and then sy that every real number has a CF? Thue Siegel Roth (talk) 21:11, 8 February 2012 (UTC)

It's probably a matter of taste but I like it better the way it is. The Generalized continued fraction article would probably be the best place to cover convergence issues. The way I thinking about it is that, for example, a decimal expansion is really a infinite series, but we don't (I hope) start our article on the decimal system with material on convergence tests.--RDBury (talk) 06:30, 9 February 2012 (UTC)

I don't copmment much on Wikipedia, but wanted to say that I thought the example was great. It would probably be more encyclopedic to present the general case, but using a specific example, at least for me, was much more edifying. Good job.Zipperfish (talk) 16:12, 15 March 2013 (UTC)

New Continued Fractions

New root approximating methods — Preceding unsigned comment added by 186.167.243.61 (talk) 03:41, 20 August 2013 (UTC)

"In lieu in the introduction, could be replaced by in place" — Preceding unsigned comment added by 181.47.65.156 (talk) 14:01, 5 July 2014 (UTC)

Contradicts itself in "Continued fraction expansions of pi"

It says: The third convergent of π is [...] 355/113. Then later on it says "The third convergent, therefore, is 333/106." I think the first statement should say "The fourth convergent ..." Olliehaffenden (talk) 10:49, 17 July 2014 (UTC)

Yes, you are right. 355/113 is the fourth convergent, not the third. I have changed the text in the article. Gandalf61 (talk) 14:51, 17 July 2014 (UTC)

Merge discussion

The article Convergent (continued fraction) does not discuss anything that would not be better discussed in context here. I propose merging it into this article. RockMagnetist (talk) 17:48, 14 January 2013 (UTC)

As a separate article it does not have any merit, since convergents are discussed here in all details. Also, I doubt that the dab page Convergent is a good solution, but still unsure whether it should be redirected to Limit (mathematics) #Convergence and fixed point. Incnis Mrsi (talk) 18:34, 14 January 2013 (UTC)

I think merging with this article is a good idea.--Hkleinnl (talk) 19:56, 15 January 2013 (UTC)

I also agree Joseph2302 (talk) 01:03, 24 January 2015 (UTC)

Well it looks already kind-of merged. Except that both articles are almost unwieldy in length, so some kind of multiple-article structuring is needed. There's a vast amount of material on continued fractions that neither of these articles touch on. e.g. bounds on the rate of convergence, the topology of Baire space, the relation of Pellian equations and characters and Gauss height of finite fields stuff, which is equivalent to the Riemann hypothesis ... none of this stuff is mentioned here. Not sure how this article got a B-class rating without even mentioning such critical info. 67.198.37.16 (talk) 19:28, 19 September 2015 (UTC)

Pell's equation statement

the section on Pell's equation states: Continued fractions play an essential role in the solution of Pell's equation. For example, for positive integers $p$ and $q$ , and non-square $n$ , it is true that $p 2 - nq 2 = \pm1$ if and only if $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}p/q$ is a convergent of √ $n$ .

But if I try this with convergents from Wolfram Alpha with Convergents[sqrt(119),8] I get {10, 11, 109/10, 120/11, 2509/230, 2629/241, 26170/2399, 28799/2640} and not all convergents satisfy Pell's equation as stated with if and only if above — Preceding unsigned comment added by 83.134.164.17 (talk) 08:22, 13 November 2015 (UTC)

Not a B-class article

There's a vast amount of material on continued fractions that this article does not even begin to touch on. e.g. bounds on the rate of convergence, the topology of Baire space, the relation of Pellian equations and characters and Gauss height of finite fields and classification os solution of the Pellian eqns, which is equivalent to the Riemann hypothesis ... and even more basic things, like the Stern-Brocot tree is the continued-fraction homeomorphism to Baire space... none of this stuff is mentioned here. Not sure how this article got a B-class rating without even mentioning such broad swaths of info.67.198.37.16 (talk) 19:34, 19 September 2015 (UTC)

The French and particularly the German versions of this article seem to be considerably more comprehensive. 67.198.37.16 (talk) 19:43, 19 September 2015 (UTC)

Also missing are simpler results: e.g. that the expansions of square-roots of integers contain repeated blocks of palindromes(!). There are also general identities (given e.g. in the wolfram website) that show that continued fractions can be reversed backwards-forwards. 84.15.191.139 (talk) 22:33, 15 November 2015 (UTC)

Convergent definition

I changed the name of the section "Infinite continued fractions" to "Infinite continued fractions and convergents" and changed the type of "convergent" from italics to bold in the text. "Convergent" is defined here, and used in many places later in the article, so it should be bold and its definition should appear in the table of contents.—Anita5192 (talk) 17:31, 17 January 2017 (UTC)

I also inserted two citations for the definition.—Anita5192 (talk) 17:33, 17 January 2017 (UTC)

Motivation section not so clear

While I understood how the method of the continued fraction works from the motivation and notation section, it does not explain why some operations happens (we do we need to get the reciprocal). So the following is what I would have liked to read, someone else can insert it in a better way I think but I also think it is ok for a talk discussion.

Consider, for example, the rational number 415/93, which is around 4.4624. As a first approximation, start with 4, which is the integer part; $415 / 93 = 4 + 43 / 93$ . Now we have the form $4 + a$ where $a$ is less than $1$ . Now we want to express $a$ with a fraction in the form 1/b. Therefore we note that $a = 1 / b$ that implies $b = 1 / a$ or that $b$ is the reciprocal of $a$ . So we continue until we get the form $4 + 1 / b$ .
43/93 is the reciprocal of 93/43 which is about 2.1628. Use the integer part, 2, as an approximation for the reciprocal to obtain a second approximation of $4 + 1 / 2 = 4.5$ . Now we can do the same process we did for 4.4624 but for 2.1628, that is the $b$ value mentioned earlier. Therefore we have $b = 2.1628 = 2 + 7 / 43$ . The remaining fractional part, 7/43, is the reciprocal of 43/7, and 43/7 is around 6.1429. Use 6 as an approximation for this to obtain $2 + 1 / 6$ as an approximation for 93/43 and $4 + 1 / 2 + 1 / 6$ , about 4.4615, as the third approximation; $43 / 7 = 6 + 1 / 7$ . Finally, the fractional part, 1/7, is the reciprocal of 7, so its approximation in this scheme, 7, is exact ( $7 / 1 = 7 + 0 / 1$ ) and produces the exact expression $4 + 1 / 2 + 1 / 6 + 1 / 7$ for 415/93.

Pier4r (talk) 11:25, 6 January 2018 (UTC)

Why the stop hand symbol on the table of "Calculating continued fraction representations"?

Why the stop hand symbol () on the table of "Calculating continued fraction representations"? Preceding comment signature by an IP address: 180.183.64.7 (talk) 04:29, 19 September 2019 (UTC)

This symbol was sneaked into the article inside an extensive edit on November 7, 2018 and is inappropriate in this context. I have now removed it. Thank you for pointing this out.

— Anita5192 (talk) 07:08, 19 September 2019 (UTC)