Talk:Continued fraction

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)

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 p² − nq² = ±1 if and only if ⁠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)

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)

Ordinary arithmetic operations on continued fraction representation

While the representation of numbers as continued fractions is very pretty, I wonder how we do basic arithmetic operations like addition, subtraction, multiplication, and division directly in this form, without having to resort to converting them to ordinary fractions. I feel that the usefulness of this article can be greatly enhanced by this inclusion. Manoguru (talk) 18:22, 17 December 2019 (UTC)

"Complete convergent"?

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

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

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