Definition talk:Continued Fraction

Reorganizing

I was thinking, in order to make this page more organized, that we could place the sections Partial Quotient, Value, Convergent, Numerators and Denominators and (to be added) Terms, Complete Quotient in a separate section Terminology. I'm not sure though, because, after all, the entire page is terminology. It just doesn't look very nice now with so many short sections.

Another idea is to make a section Types of Continued Fractions where we could place Simple continued fraction, Canonical Form and General Form.

That should make it easier for visitors to find what they're looking for. What do you think; any other suggestions? --barto (talk) 08:13, 12 July 2017 (EDT)

Anything else added needs to go in its own page and included in the Continued Fractions category. --prime mover (talk) 08:36, 12 July 2017 (EDT)

Ok. So Definition:Continued Fraction/Expansion of Real Number gets its own page? I'm fine with that. (It's cleaner, but a bit more obscure because the definition of continued fractions we care most about is then hidden.) --barto (talk) 09:26, 12 July 2017 (EDT)

More to the point I'm not happy with Definition:Continued Fraction/Expansion of Real Number for 2 reasons.

1. It's placed wrong. If it has to be included as a sub-page of a definition, then it needs to be a sub-page of SCF as that's what it is. This page Definition:Continued Fraction gives a general definition of Continued Fractions (not the most general one, but pretty general) in which the coefficients are real. The definition you provide is specifically an implementation of the simple version, the one with integer coefficients.

2. I'm not sure it's a definition page in the first place. It's a consequence of the properties of a SCF: that an infinite one defines an irrational number and a finite one defines a rational one, as it proven. So given that, I'm not sure what it actually defines. --prime mover (talk) 10:23, 12 July 2017 (EDT)

You can place it where you think is most appropriate. What it defines is the simple continued fraction of a real number every theorem talks about; a particular choice of CF constructed in an algorithm/theorem. --barto (talk) 10:33, 12 July 2017 (EDT)

Much like Definition:Power Series defines what a power series is, and Definition:Taylor Series says which power series we we associate with a function. --barto (talk) 10:37, 12 July 2017 (EDT)

Okay, and the other thing to concern ourselves about is the terminology. As we have established, we prefer not to chop and change the names for things. As we've started using "simple", that's what we will stick with, and "regular" is relegated to an "also known as". --prime mover (talk) 10:57, 12 July 2017 (EDT)

I agree, that was also one of my points of concern. I guess some sources only consider continued fractions with integer coefficients, which is where the confusion starts.

The "regular" I added between parentheses at Definition:Continued Fraction/Expansion of Real Number was meant in the sense of "usual"; I might as well have written "usual" instead.

I thus don't claim "regular" is used as a synonym for "simple". FYI, Definition:Regular Continued Fraction appears to redirect to Definition:Continued Fraction/Canonical Form. --barto (talk) 11:27, 12 July 2017 (EDT)

Nope, I still can't see the point in Definition:Continued Fraction/Expansion of Real Number. If it's the continued fraction it's defining, then we have Definition:Simple Continued Fraction. If it's the value it's defining, then we have Definition:Value of Continued Fraction. The fact that each of one maps one-to-one with each of the other is proved in the results. I must have a brain like a brick becase I can't comprehend the subtlety of what you're taking about. --prime mover (talk) 11:05, 12 July 2017 (EDT)

Yes yes yes, they are in bijection. But suppose we delete Definition:Continued Fraction/Expansion of Real Number. Then what is the meaning of: "the continued fraction of $x$"?

I think the answer would be "It is the continued fraction given by Irrational Number is Limit of Unique Simple Infinite Continued Fraction."

That means, for example, in Condition for Rational to be Convergent,

Then $\dfrac a b$ is a convergent of $x$.

has to be replaced by

Then $\dfrac a b$ is a convergent of the continued fraction of $x$ given by Irrational Number is Limit of Unique Simple Infinite Continued Fraction.

Okay, one may say that's still better than creating a definition page. But consider that the phrase "the continued fraction of $x$" is actually a used and defined concept in mathematics, so why whould we refrain from defining it and hiding its definition in a theorem? --barto (talk) 11:27, 12 July 2017 (EDT)

Why add an extra page when we already have a perfectly good page defining it. Add an "also see" if you must, referencing the proof that demonstrates that one leads to the other, but adding a new definition to redefine an existing definition for a specific context does not work for me.

Unless you have a direction to go with all this, like: you're going to construct a new field of stuff and add a whole new section of work, I recommend you leave it as it is, because it has been more-or-less carefully crafted to be in this form. --prime mover (talk) 12:32, 12 July 2017 (EDT)

Which perfectly good page defines what is meant by "the continued fraction of $x$" or "a convergent of $x$"? --barto (talk) 12:43, 12 July 2017 (EDT)

Ahh, I think I see where you're going. It's another point of view. Perhaps you want to say that "a continued fraction of $x$" is one whose value is $x$, and then, by uniqueness, call it the continued fraction of $x$? Hmm.. Somewhat indirect, but acceptable. --barto (talk) 13:00, 12 July 2017 (EDT)

Either way, we need a definition for the continued fraction of a real number, simply because there are theorems about the continued fraction of a real number, and we don't want to cite the algorithm every time, but a proper definition instead. --barto (talk) 16:27, 17 July 2017 (EDT)

All we do is cite "the continued fraction of a real number" and because it has an "also see" on that page to a page that proves uniqueness and existence (if there isn't it ought to), that's all we need. --prime mover (talk) 17:17, 17 July 2017 (EDT)

Numbering

Note that the partial quotients traditionally start with $a_0$. We better change this before we encounter incompatibility problems. --barto (talk) (contribs) 15:34, 17 January 2018 (EST)

Yes it is a bit inconsistent. Suppose we ought to. There might be a fair number of knock-on effects of this. Pell's Equation in particular and all points off it will need to be reviewed. --prime mover (talk) 15:49, 17 January 2018 (EST)

I think it's worth the effort. --barto (talk) (contribs) 15:51, 17 January 2018 (EST)

Feel free. I'm busy on something else right now. --prime mover (talk) 18:12, 17 January 2018 (EST)

Formalization

I added a formal definition of cf's as the sequence of partial quotients, which is the only sensible way to formalize it. I haven't found any reference that does better than expression. --barto (talk) (contribs) 18:07, 17 January 2018 (EST)