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Questions tagged [continued-fractions]

A is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number.

In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on.

In a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers ai are called the coefficients or terms of the continued fraction.

Links:

Continued Fraction at Wolfram MathWorld

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Do you know of any instance when a continued fraction is really of use?

Are continued fractions just an abstract ruse, or a more natural way to describe a ratio? Is there any paractical use? when sucha fraction is really necessary?
user177880
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Prove $q_n \alpha - p_n = \frac{(-1)^n}{\alpha_{n+1}q_n +q_{n-1}}$

I would like to prove that given any $\alpha = [a_0; a_1 \ldots] \in \mathbb{R}$, with: \begin{equation} \alpha_n = [a_n; a_{n-1}, \ldots, a_0]\\ \frac{p_n}{q_n} = [a_0; a_1, \ldots, a_n] \end{equation} Then we have the…
sc636
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Characterisation of continued fractions for algebraic numbers

Rational numbers have finite continued fractions, and quadratic algebraic numbers over $\mathbb Q$ have eventually periodic continued fraction representations. Is there a way to recognise a different kind of algebraic number (e.g. third-degree…
tomsmeding
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How to find other Ramanujan-type continued fractions

In a letter to G.H Hardy on January $13^{\text{th}}$, Ramanujan posted this interesting continued fraction:$$\cfrac {1}{1+\cfrac {e^{-2\pi}}{1+\cfrac {e^{-4\pi}}{1+\cfrac {e^{-6\pi}}{1+\cfrac {e^{-8\pi}}{1+\ddots}}}}}=\left(\sqrt{\dfrac…
Frank
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Why either of odd or even convergents lie on a line?

On the geometrical interpretation of continued fractions we can see the following facts ($\alpha$ is an irrational number) : 1- The line $y=\alpha x$ never passes through a lattice point $\alpha$ is an irrational number. 2- Considering the…
user200918
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Write the following as a fraction

$1+\dfrac{1}{6x}+\dfrac{1}{x^2+3x}$ I keep getting answer like $\dfrac{6x^2 + 19x + 9}{6x(x+3)}$ but I think it's wrong because I get different answers. Please help me Greetings from Russia!
Vladimir
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Estimation of numerators and denominators of convergents of continued fractions

I was going through C.Odd's textbook on continued fractions and in the introductory chapter it introduced the formula for the numerator and denominator of the $\ k$ convergent in terms of the numerator and denominator of the $\ k-1$ convergent .…
Mikhail Tal
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Transformation of this log-type continued fraction

I've learned that…
tyobrien
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Only valid for Pythagoraean triples $\sqrt2+\frac{b}{\sqrt2+\frac{b}{\sqrt2+\frac{b}{\sqrt2\cdots}}}=\sqrt{c+a}$?

$$\sqrt2+\frac{b}{\sqrt2+\frac{b}{\sqrt2+\frac{b}{\sqrt2\cdots}}}=\sqrt{c+a}$$ Where (a,b,c) are the Pythagoraean Triples and are satisfy by the Pythagoras theorem $a^2+b^2=c^2$ An example of Pythagoraean triple (3,4,5) It is true that this…
user334593
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Recommended reading for continued fractions? And some results

First of all, I apologize for my amateurness and inexperience. Although I always enjoyed math, only two years ago I started experimenting with continued fractions and gained a deep reverence for them. I first "discovered" the trivial but beautiful…
Mohamed Sabba
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Continued Fraction Algorithm for 113/50

The numbers $a_k$ can be found for $\frac{113}{50}$ by using a continued fraction algorithm. Note that $\frac{113}{50}$ is rational, and as a result it will have to terminate. Can anyone help me determine the numbers $a_k$ for $\frac{113}{50}$? What…
Jessie
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continued fraction expansion for √7

Can someone help me find the continued fraction expansion for $\sqrt{7}$ just like I did for below. For $\sqrt{3}$ I did this: I was given that $x = \sqrt{3} -1 $ $x = \frac{1}{1+\frac{1}{2+x}} $ take the second $x = \frac{1}{1+\frac{1}{2+x}} …
Jessie
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From the continued fraction

What would be the irrational number $\dfrac{a+b\sqrt{c}}{d}$, where $a,b,c,d$ are integers given by this expression: $$ \left( \begin{array}{@{}c@{}}2207-\cfrac{1}{2207-\cfrac{1}{2207-\cfrac{1}{2207-\dotsb}}}\end{array} \right)^{1/8} $$
Afonso Henriques
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Continued fraction : How to find the first 3 terms

I can't calculate the exact first tree terms $F_0$, $F_1$ and $F_2$ of this continued fraction : $$F_n=\cfrac{1}{-\text{i$\omega $}\,+A\,\cfrac{(n+1)^2}{{4 (n+1)^2-1}}F_{n+1}}$$ $A$ and $\omega$ are reals. Please, how to obtain these terms ?
Betatron
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Is there an integer-arithmetic algorithm to compute the continued fraction expansion of the square root of a rational number?

There is a well known algorithm (Bhaskara-Brouncker) to compute the continued fraction expansion of the square root of an integer, using integer arithmetic only, no floating point. Is such an algorithm known also for square roots of fractions of…
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