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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Uniqueness of finite continued fractions expansions

Under which condition on $a_0, \dots , a_n$ the continued fraction expansion of a positive rational number $$r= a_0- \frac{1}{ \ddots -\frac{1}{a_n}}$$ is unique?
Antonio Alfieri
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Farey Sequence implemenatation

I'm trying to use the Farey sequence to get the next lowest reduced fraction in a list. For example, for $n = 8$, we have $\dots, \frac13, \frac38, \frac25, \frac37, \frac12, \dots$ So let's take $\frac38$ and $\frac25$ and attempt to get…
Ryan
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Continued Fraction Expansions Confusion

Let $\theta$ be an irrational number with continued fraction expansion $[a_0; a_1, a_2, \cdots]$. Suppose $P_n/Q_n = [a_0; a_1, \cdots , a_n]$ is the $n^{th}$ convergent. Then how do I show that $P_0=a_0$. I have that $P_0/Q_0= [a_0]$ but where do…
cf12418
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Narrowing a Stern-Brocot tree

Say I only wanted to enumerate the rational numbers between 0 and $a$. Is there a way to "narrow" a Stern-Brocot tree to provide this? I tried keeping my left bound at "$\frac{0}{1}$" and setting my right-bound to "$a$" (where $a$ obviously is a…
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modification of Dedekind cuts

Dedekind defining real numbers as equivalence classes of Cauchy sequences of rational numbers. $x=y$ means $x-y=0$ ie $x_n - y_n \to 0$. addition and multiplication are defined for each coordinate. motivated by continued fractions. what if we…
cactus314
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Continued Fraction

I am working on the following question "Use the continued fraction $[1;0,1,1,2,1,1,4,1,1,6,1,1]$ to get an estimate for $e$." But I got stuck when I tried to compute $q_i$, since $a_1=0$ , $q_1 =0$. Therefore, $S_1$ is undefined
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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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Geometric Proof for Slopes (Contined Fractions)

I just started learning about continued fractions, and my lecture had a theorem that estimated the slope $a$ of a given line $L$. This was done in terms of the slope of the point $P$ with coordinates $(q,p)$. The proof in the lecture went through…
Jessie
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continued fraction of the roots of $x^2 - \frac{53793390359}{1088391168}x + \frac{823543}{12230590464} = 0$

I would like to find the continued fraction expansion of the roots of: $$x^2 - \frac{53793390359}{1088391168}x + \frac{823543}{12230590464} = 0$$ Eq 1.6 from [1] What makes this problem so difficult is obtaining numerical accuracy. However, we…
cactus314
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Continued fraction approximation

Let $\theta\in\Bbb{R}_{\gt0}$. A) Prove that the convergents for the continued fraction expansion of $\theta$ give us better and better rational approximations to $\theta$. B) Suppose $\theta\notin \Bbb{Q}$. Prove that the convergents $a_n/b_n$…
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How is Lagrange's $2\sqrt{D}$ bound on partial denominators proven for periodic regular continued fractions of quadratic irrationals

For the quadratic surd: $$ \zeta = \dfrac{P + \sqrt D}Q $$ the wikipedia article on periodic continued fractions mentions that Lagrange proves that the largest partial denominator of a regular continued fraction for $\zeta$ is less than $2\sqrt D$. …
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Question Mark Function and continued fraction representations

One could defined Minkowki's question mark question by : $$?(x) = a_0 + 2 \sum_{n= 1}^\infty \dfrac{(-1)^{n+1}}{2^{a_0 +\dots +a_k}},$$ with $x = [a_0;a_1,a_2,\dots]$. Is Minkowski's question mark function (as defined above) independent of the…
Joelafrite
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Continued Fraction Form of sqrt(6)

I have to find the continued fraction form of sqrt(6). I have tried it, and have the answers but I can't get to the correct answer. If someone could help me that would be much appreciated. Thank you!
Patrick Feltes
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The radius of image of a circle under mobius transformation

A Mobius transformation of the plane takes $z \mapsto \frac{az+b}{cz+d}$. These are known to take circles to circles, but given an explicit circle, how do we compute the radius. Let's parameterize our circle by $z(t) = z_0 + r e^{2\pi i n t}$. …
cactus314
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How to find convergents/approximate ratios for 3 (or more) numbers - (3 number Euclidean algorithm?)

It is easy to find approximate ratios between 2 numbers by using the Euclidean algorithm to calculate continued fractions. However I can not find a method to do this for 3 numbers. I have tried a shared Euclidean algorithm (dividing by the lowest of…
Richard
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