Talk:Continued fraction

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Mathematics grading:	Start Class	High Importance	Field: Analysis

According to the quite long tradition in number theory notation for a continued fraction is in between < and > brackets, so I do belive there's no need useing [ and ] brackets.
XJam [2002.04.02] 2 Tuesday (0)

I have seen brackets more often. I don't think it makes much of a difference though. AxelBoldt

I can barely follow this page. Firstly, the way it reads, it says that by choosing suitable values, a fraction can be made to equal or at least approach any number. So what? It is in no way clear as to how these numbers are chosen, and the demonstration of finding a series that converges on Pi makes no sense at all. It's like something is being assumed in the discussion that I'm not privy to. Also I wonder if anyone could add to it and say why the theorem was created in the first place - what problem did it solve?

As a minor contributor to this page, I decide to attempt a response to the comment above. A continued fraction is just another way to represent a number - on a par with the decimal system or Egyptian fractions. There is a clearly defined way to calculate the elements a_n in the continued fraction representation of any given number; this method is demonstrated in the π example, but it might help if the article explained it more directly. The continued fraction representation for some numbers is finite; for others it is infinite. The continued fraction representation for some numbers exhibits a pattern; for others it is apparently random. Continued fractions are of interest to mathematicians because they arise in the Euclidean algorithm for finding the greatest common denominator of two numbers, and they can be used to find the "best" rational approximations to a given number (this is mentioned in the π example, but again could perhaps be made clearer). Gandalf61 13:23, May 18, 2004 (UTC)

I agree that the page needs some better explanation for the general reader. A lot of wikipedia's math articles explain the what but not the why. I will try to add some better explanation this week. -- Dominus 14:16, 18 May 2004 (UTC)

I have added a section that tries to explain why continued fractions are considered to be interesting. -- Dominus 15:10, 18 May 2004 (UTC)

It looks like there's an error in the "Infinite continued fractions" section. The final step listed seems to be missing an a0 from the pn-1 generation. I think the real sequence should be the following:

$\frac{a_0}{1},\qquad \frac{a_0a_1+1}{a_1},\qquad \frac{ a_2(a_0a_1+1)+a_0}{a_2a_1+1},\qquad \frac{a_3(a_2(a_0a_1+1)+a_0)+(a_0a_1+1)}{a_3(a_2a_1+1)+a_1}$

Anyone else agree? -- Rckenned 15:41, 14 Jul 2004 (UTC)

I agree - and I have fixed it. Gandalf61 16:33, Jul 14, 2004 (UTC)

Seems that it is not totally fixed: the recursive formulae for h_n and k_n are wrong. The h_{n-1} and k_{n-1} need to be exchanged. Dscho 16:44, 8 Jun 2005 (UTC)

1 Looking a bit bare
2 Need for an expanded definition
3 Simple and not simple
4 Arithmetic sequences
5 Mistakes in article
6 General comments
7 History of continued fractions
8 Examples of periodic cont'd fractions
9 Simple continued fraction
10 Overly restrictive definition
11 Inaccuracy
12 Assessment
13 Clarification
14 Contradiction in article

[edit] Looking a bit bare

The page seems to be missing a lot of fundamental theorems to continued fractions and continued fraction expansions. I will try my best over the upcoming Easter holiday to get my notes out and starting putting some useful ones up (and some of the proofs if they are small enough). Also it is very much lacking examples and approaches to finding continued fractions, I will try my best to add some of these as well. Can anyone think of anything else that needs adding? Because I just finished a course in this so a lot of the material is fresh in my mind. zurtex

[edit] Need for an expanded definition

A few comments.

The definition of continued fractions here is too narrow. First of all, continued fraction theory is divided into analytic and arithmetic theory. The entry here is devoted entirely to arithmetic theory. In analytic theory, which is the focus of about half of the research, the numerators are not confined to being one. In fact, it is best to consider a continued fraction to be a formal expression of indeterminates (with numberator and denominator indeterminates.) For an infinite continue fraction, the indeterminates are taken in a complete normed field so that convergence can be defined. Continued fractions with elements from p-adic fields, formal Laurent series fields etc, have all been considered. These all fall outside of even the analytic and arithmetic categories I have mentioned above. With just formal indeterminates, continued fractions are even studied combinatorially. See the work of Flajolet, for example. BTW, for information purposes, I will mention that analytic theory refers to complex analysis, that is, calculus in the complex plane. So in this HUGE area of research, the elements of the continued fractions are assumed to be complex numbers. (Indeed, continued fractions are often viewed as products of fractional linear transformations, which are beautifully understood in the context of the complex plane.)

The associated entry on "generalized continued fractions" is even more naive. The expressions there are what are generally refered to as continued fractions, only the elements of the expression are not confined to being integers; indeed even in some of the examples given, for instance for exp(x), x is given as being complex, contraticting the earlier statement that the elements of the continued fraction being integers. I could go on about that entry, but I wont. Suffice it to say that today in mathematics, generalized continued fractions refer to several different things, among them: matrix continued fractions-- see the work of Levrie, and branched or branching continued fractions.

I wish I had time to write a complete entry on continued fractions, but I do not.

Here are some book suggestions for those enthusiastic about updating this entry. PLEASE have a look at the following books:

The best reference on the subject is:

1) "Continued Fractions with Applications", by Lisa Lorentzen and Haakon Waadeland, North Holland, 1992. (This covers analytic theory and a bit of arithmetic theory.)

On the arithmetic side, a great reference is:

2) "Continued Fractions" by Andrew M. Rockett and Peter Szusz. World Scientific, 1992. (This is a relatively up-to-date treatment of arithmetic theory.)

Older but good references are:

3) "Continued Fractions Analytic Theory and Applications", by William B. Jones and W.J. Thron. Addison-Wesley, 1980. (Analytic theory and history covered here.)

4) "Analytic Theory of Continued Fractions" by H.S. Wall. Chelsea, 1973. (Analytic theory).

5) "Die Lehre Von Den Kettenbruchen" Band I, II, by Oskar Perron, B.G. Teubner, 1954. (These two volumes cover both analytic and arithmetic theory and were written by the world's foremost expert on continued fractions of that day, also, I might add, a mathematician of great renown.)

Well, take care all. I am happy that people have taken the trouble to write an entry in wikipedia on continued fractions, a subject that has been near and dear to my heart for over 26 years.

Hi, yes,I agree the definition should be expanded. And several more quick remarks:

Please sign & date, don't post anonymously.
If this has really been near-n-dear for 26 years, surely you have a few hours to spare for your true love?
I'll copy the references over.

linas 03:22, 5 Jan 2005 (UTC)

Sorry for not signing or dating my comment.

I am q-analogue on Wikipedia. The date of my comment was (I believe Jan 4 2005).

To write up a full encyclopediac entry would take me, I believe, about 2 hours a day for a month or so. This is because the field is vast and certain parts technical. To work in the gradation from elementary description (usefull to most wikipedia users) to full technical coverage would be a big undertaking. Another obsticle is that I am not familiar with how to "typset" within the software of Wikipedia. Indeed, it took me the better part of 15 minutes to figure out how to put in my first comments. Maybe I need to look at the tutorials again, its been a few months. Anyway, I am certainly willing to help on this entry and would like to see it improved. Continued fractions are a great entry point into mathematical discovery and research, they are at once elementary enough that they can be understood by many people, while at the same time there are still a ton of unsolved problems about them vexing professional mathematicians today. Also, they have applications to a great many applied subjects.

Anyway, I would be happy to devote a few hours to this entry. I'm just not sure how to do that....

q-analogue

Jan 5, 2005

wormarmalade@gmail.com

Hi q-analogue.

I'll pitch in here. I wrote most of generalized continued fraction. I'm sure that anything you write will be of value. Just write what you want to, and if it's not brilliant prose (or probably even if it is!), wikipedians such as myself and linas will rehash it and revise it. Don't worry about formatting. Noone will mind if things aren't quite right.

Don't worry about any of your writing having bits missing, and don't worry about not writing a perfectly formed mathematical treatise off the bat. Just write a little bit and see what happens. Be bold! The beauty of wiki is that someone will fill in the bits. Just write what you want, when you get a minute.

(you can sign of with four tildes together to datestamp your comments).

best

Robinh 13:36, 5 Jan 2005 (UTC)

Hi, that sounds good to me.

I'll start with the usual definitions. I am used to useing LaTeX, so my mathematics will be in that form. Hope someone can read it. I will put side comments, not part of the article, in double parentheses which will contain further information that needs to be linked in, etc, or stuff I don't know how to do with the wiki format or html, or just comments on what I write.

A continued fraction is an expression of the form

(*) b_0 + \frac{a_1}{b_1 + \frac{a_2}{b_2+\frac{a_3}{b_3+\dots}}},

((Notes: The expression requires an equation number in place of the (*) I have used. Also, in the articles so far you have a_i=1 and b_i = a_i in my expression. Some books do this, but the case b_i=1 occurs just as often in continued fractions, and having the a's and b's inverted looks strange to the eye. The way I have written it is predominant.))

where the sequences \{a_i\} and \{b_i\} are usually taken to be elements of a field ((link to wikipedia article defining fields)), or are taken to be inderminates. Usually, the field is assumed to be complete ((wiki link needed)) and equipped with a norm ((wiki link needed)). In most cases, the sequences are complex numbers. ((wiki link needed)). The sequences \{a_i\} and \{b_i\} are refered to as the elements of the continued fraction (*).

Regular or simple continued fractions are the case in which a_i=1 and the b_i are natural numbers, with the exception of b_0, which is allowed to be 0.

((Note, the article on cfs so far in wiki is really about regular continued fractions.))

There exists a considerable body of research on how to give meaning to (*). The standard approach is to define the sequence of approximants \{f_n\} to (*) to be the rational functions ((wiki link needed))

(**) f_n= b_0 + \frac{a_1}{b_1 + \frac{a_2}{b_2+\frac{a_3}{b_3+\dots \frac{a_n}{b_n}}}}.

The continued fraction (*) is then said to converge if \lim_{n\to\infinity} f_n exists. When this limit exists, it is defined to be the value of the continued fraction (*). A more encompassing defintion of convergence in the case of complex elements was given by Lisa Lorentzen (ne Jacobsen) in 1986 who created the idea of general convergence of continued fractions. ((Her seminal paper "General convergence for continued fractions" was published in Transactions of the American Mathematical Society in Volume 294, no 2, pages 477-485. Her name on the paper is Lisa Jacobsen.))

An expression such as the right hand side of (**) is usually refered to as a finite continued fraction.

((Well, that covers the basic defintions with the exception of convergents. I'll try to add more in a bit.))

q-analogue Q-analogue 21:42, 5 Jan 2005 (UTC)

Couple of quick notes.

Diagonal dots should be used in place of \dots. Also, for regular continued fractions, b_0 is allowed to be negative when one is considering the regular continued fractions of negative real numbers.

q-analogue Q-analogue 21:52, 5 Jan 2005 (UTC)

A note on a topic of brackets for regular continued fractions. Square brackets are indeed more usual in publications today. Just look at a few issues of the Journal of Number Theory or Acta Arithmeticae. Angle brackets are used, but they are in the minority.

Q-analogue 09:17, 6 Jan 2005 (UTC)

Gandalf claims that 3.245 is [ 3 ; 4 , 12 , 4]

While I claimed that 3.245 is [ 3 ; 4 , 12 , 4, 1]

Can someone please resolved this dispute.

Please ignore this. I did the calculations myself and Gandalf was right and I was wrong. [ 3 ; 4 , 12 , 4] = 649/200 = 3.245 exact [ 3 ; 4 , 12 , 4, 1] = 808/249 = 3.24497992

Ohanian 23:49, 2005 Apr 5 (UTC)

Is there an algorithm for converting continued fractions into fractions?

For example: [ 3 ; 4 , 12 , 4 ] -> 649/200

The reason I asked is that I wanted to do addition, subtraction , multiplication and division with continued fractions.
Ohanian 05:06, 2005 Apr 6 (UTC)

There is such an algorithm, but you can do arithmetic on continued fractions without first turning them into fractions. I have been meaning to write a Wikipedia article about this for some time. Until then, you could take a look at [Arithmetic With Continued Fractions], which also has references to more detailed explanations. Dominus 16:02, 6 Apr 2005 (UTC)

	$X_{old} = \frac{N_{old}}{D_{old}}$	(1)	Let $X o l d$ be a rational number. ie $X_{old}=\frac{649}{200}$
	$X_{new} = \frac{N_{new}}{D_{new}}$	(2)	Let $X n e w$ be a different rational number.
	$X_{old} = P + \frac{1}{X_{new}}$	(3)	Let $X o l d$ be related to $X n e w$ by an integer value $P\,\!$
(3)+(2)	$X_{old} = P + 1 \div \frac{N_{new}}{D_{new}}$
	$X_{old} = P + \frac {D_{new}}{N_{new}}$
	$X_{old} = \frac { N_{new} \cdot P + D_{new} }{N_{new}}$	(4)
	$P = N_{old} \,\, \mathbf{intdiv} \,\, D_{old}$	(5)	Let $P\,\!$ be integer value of $X o l d$ ie. $P = 3$ if $X_{old}=\frac{649}{200}$
(1) + (4)	$N_{old}= N_{new} \cdot P + D_{new}$	(6)
(1) + (4)	$D o l d = N n e w$
	$N n e w = D o l d$	(7)
(6) + (7)	$N_{old}= D_{old} \cdot P + D_{new}$
	$N_{old} - D_{old} \cdot P = D_{new}$
	$D_{new} = N_{old} - D_{old} \cdot P$	(8)
(2) + (7) + (8)	$X_{new} = \frac{D_{old}}{ N_{old} - D_{old} \cdot P }$	(9)

This then leads to a recursive algorithm of turning a rational number (649/200) into continued fraction [3;4,12,4] by recursively calculating P using (5) and the new value of X using (9).

Ohanian 08:57, 2005 Apr 14 (UTC)

More fun with continued fractions of $π$

The continued fractions of $π$ is

[3; 7, 15, 1, 292, 1, 1, 1, 2, 1, ... ]

I notice that the good rational representation of the continued fractions of $π$ are those just before a term of a large number. For example just before 292.

[3; 7, 15, 1 ] = 355/113

So I wanted to find where (which terms) the large numbers are in the continued fraction terms of $π$

So far, my python program returns....

$ python pi.py
=======================================
Calculating the terms of pi using 4/pi = [(1,1),(3,4),(5,9),(7,16), ... ]
New record  3  at  0  term
New record  7  at  1  term
New record  15  at  2  term
New record  292  at  4  term
New record  436  at  307  term
New record  20776  at  431  term
New record  78629  at  28421  term
New record  179136  at  156381  term
New record  528210  at  267313 term
New record  12996958  at  453293 term

Ohanian 03:58, 2005 Apr 17 (UTC)

I believe it is still an open question whether the terms in the regular continued fraction for pi are bounded. It looks from your computation that they are unbounded. A close compadre of pi is e, which has a very nice regular continued fraction, whose terms are clearly unbounded. I wonder whether Euler's constant is like pi. Scott Tillinghast, Houston TX 21:08, 6 April 2007 (UTC)

[edit] Simple and not simple

Please expand on the simple continued fraction (unity numerators) to include non unity numerators.

Check out Generalized continued fraction.--Niels Ø 19:47, May 23, 2005 (UTC)

[edit] Arithmetic sequences

Are there any formulas for the continued fractions when the numbers are in a special sequence such as a Arithmetic progression or Geometric progression (Examples: [0;1,2,3,...],[1;2,4,8,...])?--SurrealWarrior 03:16, 14 Jun 2005 (UTC)

According to Richard Schroeppel in HAKMEM, when the terms of the continued fraction are an arithmetic sequence, say [a+d, a+2d, a+3d, ...], the value of the fraction is

$I_{a/d}(2/d) \over I_{1+a/d}(2/d)$ ,

where the I(x) are modified Bessel functions. Hope this helps! -- Dominus 13:29, 14 Jun 2005 (UTC)

Thanks--SurrealWarrior 19:34, 14 Jun 2005 (UTC)

[edit] Mistakes in article

When I was to translate it to chinese, I spotted some strange lines.

Under theorem 3 reads

Corollary 1: Each convergent is in its lowest terms (for if $h n$ and $k n$ had a common divisor it would divide $h n q n - 1 - q n h n - 1$ , which is impossible).

What is the q_n? Is it a typo?

Theorem 4 Each convergent is nearer to the nth convergent than any of the preceding convergents. In symbols, if the rth convergent is considered to $[a_0;a_1,a_2,\ldots a_n]=x$ , then

$\left|[a_0; a_1, a_2, \ldots a_r]-x\right|> \left|[a_0; a_1, a_2, \ldots a_s]-x\right|$

for all $r < s .$

What does "considered to" means exactly? And should there be a condition that r,s<n?

I suspect there are more mistakes in the article, but I've forgotten all about the continued fractions. So please can anyone proofread it? --Small potato 30 June 2005 13:47 (UTC)

[edit] General comments

"Longer expressions are defined analogously." Second sentence. It looks to me that the scheme already suggests an infinitely long expression.

I think that the article is becoming inconsistent. For example, Theorem 1... x represents any real, but in the lead paragraph the cascade of denominators (of which x is one) is to consist of positive integers. And I think that "x" to this point has had a special meaning, namely, it's the number which is generating the fraction. A symbol other than x (like "k") would perhaps be better, and I believe that having "it" be a positive integer suffices both for consistency and the generality of the result. For consistency, the first comma should be a semicolon ... [a_0; a_1, ... ]

Theorem 4, I think, needs some work. Again, "x" seems to represent a fixed quantity generating the continued fraction in most of the article. To me, this meaning for "x" is necessarily what the Corollaries to Theorem 4 are referring to... if you think about an infinite continued fraction with the convergents see-sawing above and below x (this being the limit and number that has generated the fraction) and gradually converging to x (and only x), then this interpretation of x is the only one which makes the corollaries true as written. So, the choice of the letter x to represent something else in this theorem and the corollaries is perhaps unfortunate on the one hand, and leads to falsehood on the other (unless the fraction is a finite one, terminating with the quotient a_n).

(Thinking out loud for a minute... I'm not sure yet that Theorem 4 is true, for either interpretation of "x". Supposing the "x" means the limiting value (= the number generating the infinite continuing fraction = the continuing fraction)... one may approach a value in a see-saw fashion without having the "too small and getting bigger" things related much to the "too big and getting smaller" things in terms of closeness to the limiting value... though some of the inequalities may have implications I don't yet see. That is... in principle at least... even if the convergents are getting closer to one another, they may be approaching the limit in a rather lopsided fashion... with "see"-ing being far more pronounced than "saw"-ing, to continue the visual metaphor. For the other meaning of "x", I'm not sure either. One meaning may be true, one false. I'll have to really think about this, of course. As for this thinking... it seems to me that Theorem 5 has considerable bearing, and in fact Theorem 5 might be better placed before Theorem 4.)

I've added the word "nontrivial" to the parenthesized proof of Corollary 1 to Theorem 3... I don't think this is controversial.

I'm not following the text for Semiconvergents at all (but the concept is interesting). What exactly is to be between what? I would guess that ... the semiconvergent fraction, the one involving the a's, is to be between the successive convergents, the ones explicitly mentioned. What exactly is being claimed ... what exactly is the theorem?

Pell's equation... no, p=1 and q=0 is a solution. I'll need to think before editing this, for now I'll just edit to indicate both p and q are positive.

I've noted that there are actually four, rather than three, convergents displayed in the Infinite Continued Fractions section, and that these may be considered numbers 0, 1, 2, and 3.

(I've returned to this to edit, expand, and tidy up some grammar... Mathguy2, later in the day, Nov 5 2005)

Mathguy2 20:08, 5 November 2005 (UTC)

[edit] History of continued fractions

Can someone supply a brief section on the history of continued fractions?--Niels Ø 16:24, 19 February 2006 (UTC)

[edit] Examples of periodic cont'd fractions

I think an example like sqrt(13)=[3;1,1,1,1,6,1,1,1,1,6,...] or sqrt(14)=[3;1,2,1,6,1,2,1,6,....] would better illustrate what "periodic" means.

[edit] Simple continued fraction

Shouldn't the name of this article be Simple continued fraction? Since this type of continued fraction is only one of many forms of continued fraction. Please refer to http://mathworld.wolfram.com/ContinuedFraction.html.

[edit] Overly restrictive definition

I've read through the discussion of this page, and I notice that "q-analogue" has already pointed this out. The definition of a continued fraction in this article is ridiculously restrictive. I imagine that whoever set this up in the first place wanted to avoid dealing with the very messy problem of convergence – that's a lot tougher than dealing with the convergence of series, especially when the cf is taken over the complex numbers.

Anyway, I'm intending to fix this definition fairly soon, and also in the associated article generalized continued fraction. I want to add more content about continued fractions, and I really think the definition in this article has to be fixed, or future authors won't be able to build on it. DavidCBryant 12:52, 30 November 2006 (UTC)

AFAIK, the definition given in this article (in which the co-efficients are restricted to positive integers) is the standard definition in number theory for a simple continued fraction. Although the concept of a continued fraction can clearly be extended in various ways, we do need to be careful that when we use terms in Wikipedai we use them with their usual meanings. If you intend to extend or change the definition given in this article, please cite sources for your new definition, so we can be sure it is not original research. Gandalf61 15:47, 30 November 2006 (UTC)

Citing a few references is not enough to justify changing the definition. I think the article should have a sort of disambiguation near the top, pointing out that "continued fraction" has several more-or-less restrictive/inclusive meanings in different mathematical fields, and pointing the reader to the article covering the desired meaning. Then, this article should go on to describe the most common meaning.--Niels Ø 18:13, 30 November 2006 (UTC)

Well, I also kicked this around a bit over on the Wikiproject:Mathematics page, and everybody seems to feel the same way. So all I've done to this definition is insert the word "regular" at the beginning. I also cleaned up the TeX code so the picture that's displayed looks more like the way most American authors write about continued fractions. And I talked to RobinH, who apparently originated the article on generalized continued fractions. I have put a slightly more general definition over there that applies to continued fractions which can be evaluated using arithmetic in any number field.

"... so we can be sure it is not original research" -- Now that's funny! Continued fractions (in the form described in this basic article) were apparently known to the Egyptians. At least, they had discovered the infinite sequence of convergents to √2 -- 3/2, 7/5, 17/12, 41/29, ... something like 4,000 years ago. Euler (d. 1783) produced a lot of analytical results using continued fractions, which he apparently regarded as nothing more than another way to express an infinite series. Gauss (d. 1855) used continued fractions in many sorts of calculations, and developed formulas (with complex z sprinkled liberally throughout them) using cf's to facilitate the calculation of tables of logarithms, tables of trigonometric functions, etc.

When Weierstrass and his crew started insisting on rigorous proofs in calculus, the analytical forms of the continued fraction fell into disrepute, except among the people who had to make real calculations ... in general, well-formed continued fractions for functions like ln and sin and arctan converge a whole lot faster than the best of their series counterparts. But they're pretty hard to work with from the pure math viewpoint. Proving convergence is a whole lot tougher with continued fractions than it is with infinite series. Weierstrass never got far with them, and therefore a whole generation of math students never learned much about them.

It remained for the 20th century to put the theory of continued fractions on a rigorous basis. While it's true that some convergence problems are still too tough, it's now pretty easy to show that many of Gauss's formulas, dating back to 1825, at least, are rigorously true.

Most people aren't aware of this, but when you ask your computer to calculate a number like the natural logarithm of 3, or the arctangent of 175.687, it doesn't use a Taylor series. It uses a continued fraction, often one that's been optimized to converge very quickly within a limited range. A whole lot of the pure math work that supports modern computers got coded into the library subroutines for FORTRAN, and we've been carrying that stuff forward ever since. Why reinvent the wheel?

I don't want to take credit for KFG's discoveries -- I just want to be able to write about them without having to redefine a simple term like "continued fraction". DavidCBryant 21:13, 4 December 2006 (UTC)

More opinions. DavidCBryant 00:05, 6 December 2006 (UTC)

To DavidCBryant: You said "...well-formed continued fractions for functions like ln and sin and arctan converge a whole lot faster than the best of their series counterparts.". I would like to know more about this. I was working on the pages for Natural logarithms and Inverse trigonometric functions and I was disappointed at how slowly the series converge. JRSpriggs 07:33, 6 December 2006 (UTC)

[edit] Inaccuracy

The article states that, "Every finite continued fraction is rational, and every rational number can be represented in precisely two different ways as a finite continued fraction, which agree except at the very end." However, this is inaccurate: there are an infinite number of ways to represent any given rational number as a continued fraction:
[a0; a1, a2, a3, ..., an + 1]
[a0; a1, a2, a3, ..., an, 1]
[a0; a1, a2, a3, ..., an, 0, 1]
[a0; a1, a2, a3, ..., an, 0, 0, 1]
...
Granted, these are trivial, but saying that there are exactly two ways to represent a continued fraction is misleading. This is implied in other parts of the article as well. 66.32.24.171 19:20, 28 January 2007 (UTC)

Hi, 66.32.24.171! That's an astute observation, but you're overlooking something.

The definition of a "continued fraction" in this article (as it now stands) says that "...all the other numbers a_n are positive integers." Zero is not a positive integer. So the cases with the extra "0"s are ruled out by the definition.

Is it less misleading now? Or does this point require additional clarification? DavidCBryant 19:59, 28 January 2007 (UTC)

PS I got into an argument with some other editors a while back because I wanted to call the "continued fractions" in this article "regular continued fractions in canonical form", or "simple continued fractions in canonical form", mainly because the restriction of partial denominators to positive integers rules out a lot of interesting continued fractions from complex analysis. But I lost that argument, so for the time being the things I naturally think of as "continued fractions" are not defined as such in Wikipedia – they're "generalized continued fractions" instead. dcb

Yes, you're right. I'm sorry for overlooking that. (I am the same person, just on a school network instead of home.) I guess that closes the matter. 152.23.77.52 18:53, 29 January 2007 (UTC)

[edit] Assessment

I added an assessment of this article as "Start" class, importance "High". The reason I didn't rate it higher on the quality scale was that, although it seems to have an appropriate depth of content, the organization of that content is lacking. In addition I found the motivation section shallow; to my mind the application to diophantine approximation is what gives this subject its high importance, but one has to pore through the whole article to find that. —David Eppstein 04:44, 8 February 2007 (UTC)

[edit] Clarification

In "Best rational approximations":

Added subscripts to clarify which term was being used in the expressions. Otherwise it would be difficult to be sure which term of series is meant. JKSellers 18:07, 5 March 2007 (UTC)

Added k index values to table to be consistent with a _k used in a couple paragraphs down. JKSellers 18:23, 5 March 2007 (UTC)

Minor adjustment to make comments referring to the changes I made at 18:07, 5 March 2007 more consistent. Strictly speaking, the article is correct without the clarifications I added. However, the changes I made were in an area I had have dealt with a number of times and in a number of different applications with programmers new to the concepts, and the clarification I added will save effort to for most new readers wishing to understand this small part. JKSellers 18:52, 5 March 2007 (UTC)

[edit] Contradiction in article

In the bullet-pointed, desirable features of continued fraction notation in the Motivation section, it is stated that "The terms of a continued fraction will repeat if and only if it is the continued fraction representation of a quadratic irrational, that is, a real solution to a quadratic equation with integer coefficients". But in the Periodic continued fractions subsection of the Other continued fraction expansions section, it is stated that "The numbers with periodic continued fraction expansion are precisely the solutions of quadratic equations with rational coefficients.". Which is it? This needs fixing Djr36 21:46, 27 March 2007 (UTC)

There is no contradiction, because they are the same thing: a number is a solution of some quadratic equation with integer coefficients if and only if it is also the solution of some quadratic equation with rational coefficients.

To see this, suppose x is a root of px² + qx + r, where p, q, and r are rational. Let d be the product of the denominators of p, q, and r. Then dpx² + dqx + dr is a quadratic polynomial with integer coefficients, and x is also a root of this polynomnial.

For example, consider the equation (1/3)x² + (1/2)x + 1/6 = 0. The solutions of this equation are precisely the same as the solutions of 12x² + 18x + 6 = 0. (Here I multiplied everything by 36. Of course multiplying everything by 6 would have worked too.)

I believe that this should be clear to almost anyone with enough mathematical background to understand the rest of the article. -- Dominus 22:46, 27 March 2007 (UTC)

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