Continued Fraction Expansions, Pt. III

A fascinating generalization linking sequences, continued fractions, and polynomials.

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[JohnAbreu]

Thank you for your effort and your videos. Unfortunately, in this video (P. III), the final result in the case b=2k+1 leaves the integer part as a fraction: sqrt(13)= [ 3/2 ; 3, 3, 3, 3, .... ], which is conceptually wrong. I wish you had the time to edit/correct this mistake so students can watch/learn properly. Thanks.

[ptyamin]

I would like to add, the reason there are 2 continued fractions for a given rational number is because you can choose to reach the root 1, either from the left or the right. Thus only the beginning of the Calkin-Wilf path changes and only the end of the CFE changes.

[ffggddss]

Velly interessink!
as Arte Johnson used to say on Laugh-In. [I'm dating myself, aren't I?]

Just finished watching all 3 of your parts on CF's here, and there were numerous times I wanted to jump up in class and add some additional remarks, pointing out other connections; but alas, you wouldn't have called on me ;-)

I was rather hoping you were going to lead us into how to solve Pell's Equation (PE) by a method that's related to CF's; and you got awfully close to it.
You were also hovering around the solution of the general 2nd-order linear recursion (2o-1d-R), which leads to a quadratic whose solutions go into the closed formula for the n'th term. (The general solution of 2o-1d-R, also gives the formula for doing mortgage calculations, BTW.)

In an earlier part you remarked about algebraic numbers having infinitely-repeating CF's; actually, it's only algebraic numbers of degree 2 that do that; it isn't hard to show that x = any such repeating CF can be reduced to a quadratic equation in x.

But your mathematical trick for getting fractions that approximate square roots of non-square integers, n > 1, basically works because the num. & denom. — (p, q) — are PE solutions for n:
p² = nq² ± 1 . . . ["+" makes it PE; "-" makes it what might be called, the Associated Pell's Equation, or "APE"]
so that p/q is a "best" approximation of √n.

I guess I should slap together a video of my own to cover some of these things; I find them quite fascinating, even if no one else will.

BTW, I also have toyed with CF's on a some-time basis for many years, and I've homed in on an alternative form of them, that allows negative terms as well as the usual positive ones; but that restricts their magnitudes to 2 or more: |a| ≥ 2, with the additional requirement that any term of ±2, is followed by a term of like sign. This shortens the resulting CF's somewhat, and skips over some of the not-so-great approximations in the partial fractions (PF's). Doing this with π and e, gives:
π = [3; 7, 16, -294, 3, -4, 5, -15, -3, ...]
e = [3; -4, 2, 5, -2, -7, 2, 9, -2, -11, 2, 13, ...] = [1; 1, -2, -3, 2, 5, -2, -7, 2, 9, -2, -11, 2, 13, ...]
where liberty has been taken at the start, with e, to bring out the general pattern more fully; note that the pattern length here is reduced from 3 to 2.
π still has no discernible pattern, of course.

Interestingly, φ becomes
[2; -3, 3, -3, 3, -3, 3, ...]
The PF's of which, produce all the Fibonacci numbers again (after absolute-valuing the terms), but now with no repeats:
2/1, 5/3, 13/8, 34/21, ...

Anyway, thanks for doing these — as a math nerd myself, I had fun watching these. I take it the next segment you refer to, is in the works?