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Moreover, it might be people someday will also realize that all the Means, Continued
Fractions, Daniel Bernoulli's method, Newton's method, Householder's method, and the whole Science of Quantity is ruled by an extremely simple arithmetical operation called: The Rational Mean. Domingo Gomez Morin
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A continued fraction algorithm is also nice for approximating real roots of polynomials of higher degree. The terms, however, are a chaotic succession of positive integers, and it is an open question (at least in 1973, probably still is) whether these are a bounded set. A few iterations throws all
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Fine, good article, but... I hope, someday, might be in a non distant future, people will finally realize that traditional continued fractions are just the same second-order Lineal
Homogeneous Recurrence Relations than those arising from the well-known Daniel Bernoulli's root-solving method, and
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Oh -- there was another error in the statement of the theorem. The trick with the discriminant only works when the equation has real coefficients. It's a bit more complicated when the coefficients are complex numbers, because the value of the discriminant doesn't have to be a real number in that
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We want to explain the simplest case of a periodic continued fraction first, so the article will be accessible to people with little mathematical sophistication. Any infinite periodic continued fraction can be transformed into a periodic cf of period 1, by a relatively straightforward algebraic
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case. For now the section has been adjusted to speak only of equations with real coefficients. The statement about complex coefficients will be added soon, along with a proof (in a separate article) of the theorem. Would you be kind enough to kick the tires on that stuff, too?
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Your general theorem cannot be true as stated; x - 2 has a perfectly good continued fraction for the positive root (and for that matter, the negative root) despite having
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formula for the solution to a quadratic equation. But on reflection, I do see what you mean. More verbiage has been added. Does it look better now?
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consequently such tradional concept constitutes just a particular case of the
Generalized Continued Fractions as can be seen at:
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substitution. But shouldn't something like that go in its own article, maybe "Periodic continued fraction"?
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the real conjugates of a real root into the open interval (-1,1); that greatly simplifies things.
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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The section about complex coefficients has been added. There's a red link to
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Well, I thought it was clear from the context that the theorem refers to the
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192:= 0; I think I see what is meant, but it's not what is said.
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Thanks for the help with this article, Septentrionalis!
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http://mipagina.cantv.net/arithmetic/gencontfrac.htm
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