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.

learn more… | top users | synonyms

2
votes
0answers
36 views

Continued fraction for $[1,2,3,4,5,6,\dots]$ [duplicate]

Any continued fraction that does not terminate or repeat can't be rational or a quadratic irrational. It is not hard to write something that does not fit these two categories. Can we still get a ...
2
votes
0answers
280 views

nth-root of continued fraction with Raney transducers

There are some algorithms for doing basic arithmetic by using regular continued fraction expansions. These algorithms are mainly due to Gosper (1972) and Raney (1973). These two approaches use (bi)...
1
vote
1answer
38 views

How to find periodic continued fraction expansion of $\frac{\sqrt{7}}3$

How to find periodic continued fraction expansion of $\frac{\sqrt{7}}3$ Using this formula here (it begins in the middle of the page), I obtained $\frac{\sqrt{7}}3=[0;1,\overline{7,2}]$ but how ...
2
votes
2answers
167 views

Fundamental unit in the ring of integers $\mathbb Z[\frac{1+\sqrt{141}}{2}]$

Find a fundamental unit in the ring of integers $\mathbb Z[\frac{1+\sqrt{141}}{2}]$ of $\mathbb Q(\sqrt{141})$ I have different corollaries for different numbers, the most appropriate for $141$ is ...
6
votes
2answers
253 views

What are the reduced elements of $\mathbb Q(\sqrt{30})$?

What are the reduced elements of $\mathbb Q(\sqrt{30})$ ? From the definition here(on page $32$); An element $\beta\in\mathbb Q(\sqrt{d})$ is said to be reduced, if $\beta>1$ and $-1<\beta'&...
2
votes
1answer
56 views

Continued fraction manipulation

I have the following continued fraction $$ \frac{1}{a_1x+}\;\;\frac{1}{b_1+}\;\;\frac{1}{a_2x+}\;\;\frac{1}{b_2} $$ The paper I am reading then converts this to the following continued z-fraction ...
1
vote
2answers
62 views

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 ...
1
vote
2answers
86 views

Continued Fraction for Root 5 [duplicate]

How can I find the continued fraction expansion for the square root of 5. Do this without the use of a calculator and show all the steps.
0
votes
1answer
103 views

continued fraction expansion for √7 [duplicate]

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}} $ ...
0
votes
1answer
35 views

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 $...
1
vote
2answers
64 views

Why is $\sqrt{3}=[1;1,2,1,2,\dots]$?

Why is $\sqrt{3}=[1;1,2,1,2,\dots]$ ? $\displaystyle[1;1,2,1,2,\dots]=1+\frac{1}{[1;2,1,2,\dots]}=1+\frac{1}{1+\frac{1}{2+\frac{1}{[1;2,1,2,\dots]}}}$ If I set $x=[1;2,1,2,\dots]$ then; $\frac{x+1}...
0
votes
3answers
123 views

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 ...
1
vote
1answer
63 views

continued fraction of $3 + 17\sqrt{3} $

On a computer, I tried to iterate the Euclidean algorithm on the number $3 + 17\sqrt{3}$ and here is what I got: \begin{array}{ccccrcrcrcr} 3 + 17\sqrt{3} &=& 32 &\cdot\;(& 1&+&...
1
vote
0answers
84 views

Integral of a Continued Fraction

How might one go about evaluating the following integral $\int_{-\infty}^{\infty}\mathrm{K}_{j=0}^{\infty}(F_{j}e^{-x^2})dx$? Where$\mathrm{K}$ denotes a continued fraction and $F_j$ is the jth ...
2
votes
1answer
157 views

Why are there no continued fraction representation for $\pi$ obeying mathematical rules?

There are several irrational numbers that can be represented with continued fraction such that a mathematical rule arises in this continued fraction. For example, the Euler number $e$ can be ...
1
vote
0answers
27 views

Continued Fraction summation representation

I have a rational fraction of the form: $$s=\frac{p_0+p_1x+p_2x^2+\cdots+p_Mx^M}{1+q_1x+q_2x^2+\cdots+q_Mx^M} $$ The paper I am reading converts this to the form: $$s = b_0+\sum^M_{m=1}\frac{b_m}{\...
7
votes
2answers
188 views

Simple Finite Continued Fraction

I am working on my senior thesis and have encountered, unexpectedly, a finite continued fraction that I would be interested in resolving. I already know the answer (by an informed guess based on where ...
5
votes
0answers
61 views

Is there any elegant formalization of fractional numbers?

The question is just what is on the title, but I'll describe the context for completion: Natural numbers can be encoded quite elegantly on the Lambda Calculus as church numbers, that is, a function ...
2
votes
0answers
20 views

On the asymptotics of a continued fraction

I know the coefficients of the continued fraction representation of a function that diverges like $O(\sqrt{x})$, where the $a_k$ depend on $x$, $ f(x)=1+a_0/\left(1+\underset{k=1}{\overset{\infty }{K}...
4
votes
1answer
76 views

Continued Fraction, Closest Neighbours

For setting the proper divisor/multiplier registers in a phase locked loop, I use a continued fraction expansion, which I stop if either the numerator or denominator of the fraction becomes larger ...
1
vote
1answer
124 views

Good and best rational approximations

Lately I was reading a bit about continued fractions and came up with a question that I couldn't find an answer for. Here are the definitions I will use in the question: Let $x \in \mathbb{R}$. A ...
1
vote
1answer
71 views

Closest rational approximation of $\sqrt x$ with denominator having prime powers $\lt n$

I am representing denominators in rational numbers with powers of their prime factors for easy multiplication and division in lowest terms (by adding and subtracting the prime powers). I would like ...
2
votes
1answer
47 views

Good rational approximations using continued fractions?

My textbook claims that the best rational approximations (relative the size of the numerator and denominator) of an irrational number by using continued fraction are those whose expansions are ...
1
vote
0answers
44 views

Coefficients of the polynomials generated by $f_0=x,\ f_{i+1}=f_i\pm\dfrac1{f_i}$.

Let $f_0=x,\ f_{i+1}=f_i\pm\dfrac1{f_i}$ for $i\geq0$, i.e., $f_i=\dfrac{\sqrt{f_{i+1}^2\mp4}+f_{i+1}}2$ I have observed that $f_1=\dfrac{x^2\pm1}x$ $f_2=\dfrac{x^4\pm3x^2+1}{x(x^2\pm1)}$ $f_3=\...
0
votes
0answers
45 views

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 ...
0
votes
0answers
29 views

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 ...
7
votes
1answer
184 views

Continued fraction of $e^{-2\pi n}$

I found this problem on a well-known problem solving website. It is apparently from Ramanujan. With $$\LARGE{a = \frac{1}{1 + \frac{e^{-2\pi}}{1 + \frac{e^{-4\pi}}{1 + \ddots}}}},$$ what is $a$...
59
votes
3answers
696 views

Closed form solution for $\sum_{n=1}^\infty\frac{1}{1+\frac{n^2}{1+\frac{1}{\stackrel{\ddots}{1+\frac{1}{1+n^2}}}}}$.

Let $$ \text{S}_k = \sum_{n=1}^\infty\cfrac{1}{1+\cfrac{n^2}{1+\cfrac{1}{\ddots1+\cfrac{1}{1+n^2}}}},\quad\text{$k$ rows in the continued fraction} $$ So for example, the terms of the sum $\text{S}_6$ ...
0
votes
1answer
38 views

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 ...
6
votes
2answers
279 views

“Bizarre” continued fraction of Ramanujan! But where's the proof?

$$\frac{e^\pi-1}{e^\pi+1}=\cfrac\pi{2+\cfrac{\pi^2}{6+\cfrac{\pi^2}{10+\cfrac{\pi^2}{14+...}}}}$$ "Bizarre" continued fraction of Ramanujan! But where's the proof? i have no training in continued ...
1
vote
1answer
58 views

If $x=[a_0,a_1,\dots]$ show that $\mu$-almost every $x \in (0,1/N]$ is infinitely recurrent

Let $G$ be the Gauss map, $$G(x)= \begin{cases} 0 & \text{if} \ x=0 \\ \{\frac{1}{x} \}=\frac{1}{x} \ \mathrm{mod} \ 1 & \text{if $0<x\leq 1$}\end{cases}$$ and $\mu$ be the Gauss ...
6
votes
1answer
129 views

What is the value of this continued fraction?

I am curious about the value of the continued fraction $$1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{5+\cfrac{1}{6+\dots}}}}}.$$ Can we evaluate it ? Is it a nice value ? Clearly it should ...
1
vote
2answers
79 views

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-\...
10
votes
1answer
84 views

Multiply all terms in continued fraction by a constant

I noticed that continued the fraction for $\sqrt{12}$ is $3;2,6,2,6,2,\ldots$ and the continued fraction for $\sqrt{7\times12}$ is $9;6,18,6,18,6,\ldots$ all the terms in the continued fraction are ...
3
votes
2answers
204 views

Continued fraction to irrational number

Let $[1;\overline{2,1}$] be a continued fraction. I want to find the corresponding number. I know how to transform fractions of the form $[a;\overline{b}]$ but I am having a hard time here. Thank you.
4
votes
0answers
115 views

Different types of transcendental numbers based on continued-fraction representation

I've been reading Wikipedia's article on continued fractions. A few examples are given for the continued-fraction representation of irrational numbers: $\sqrt{19}=[4;2,1,3,1,2,8,2,1,3,1,2,8,\dots]$ ...
4
votes
2answers
118 views

How find this Continued fraction $[1,3,5,7,9,11,\cdots]$ value.

show this: $$\alpha=[1,3,5,7,9,11,\cdots]=1+\dfrac{1}{3+\dfrac{1}{5+\dfrac{1}{7+\dfrac{1}{\cdots}}}}=\dfrac{e^2+1}{e^2-1}$$ I found wiki Continued fraction also not have this problem,maybe this ...
15
votes
1answer
271 views

Closed-form of infinite continued fraction involving factorials

Is there a closed form of this: $$ 1!+\dfrac{1}{2!+\dfrac{1}{3!+\dfrac{1}{4!+\ldots}}} $$
1
vote
1answer
58 views

What is a counterexample for this one?

Let $x$ be an irrational number. Let $\{a_0\}$ be the sequence of positive integers except for $a_0$ such that $x=a_0+K(1/a_n)$. Let $a,b$ be integers such that $b>0$ and $gcd(a,b)=1$ and $|x-a/b|...
2
votes
0answers
46 views

Continued fraction approximation to a function and its derivative

I am recently working on fitting a model with incomplete beta function. In order to put it into my optimization algorithm, I must find out the derivatives of the incomplete beta function $B_p(x,y)$ ...
2
votes
1answer
61 views

Convergents of continued fraction proof

Let $\frac{P_n} {Q_n} and \frac{P_{n+1}} {Q_{n+1}}$ be two consecutive continued fraction convergents for $b$. Then prove that: $$\left|{\frac{P_n} {Q_n}-b}\right|< \frac{1}{2Q_n^2}$$ or $$\...
11
votes
0answers
179 views

Divergent continued fractions?

The solutions to $$ x^2-6x+10=0 \tag 1 $$ are $$ 3\pm i\tag2. $$ Rearranging $(1)$ just a bit, we get $$ x = 6 -\frac{10}x \tag3 $$ and then substituting the right side of $(3)$ for $x$ within the ...
1
vote
1answer
77 views

Approximate irrational numbers with the same denominator

Let $\alpha$ be a irrational number, then using the continued fraction expansion we can find two sequences $\{p_n\}$ and $\{q_n\}$ with $q_n\rightarrow\infty$ as $n\rightarrow\infty$ such that $$|\...
0
votes
1answer
54 views

Continued Fraction Form of sqrt(6) [duplicate]

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!
1
vote
1answer
51 views

Is a finite continued fraction a closed-form expression?

We had a discussion regarding this answer on Electrical Engineering. The answer in question discussed a finite continued fraction. We're wondering whether it's a closed-form expression or not. ...
0
votes
1answer
259 views

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 ...
0
votes
0answers
159 views

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 ...
1
vote
1answer
38 views

Comparation between continued fractions

I'm trying to solve the following problem but I'm having some difficulties.. Let $a_0,a_1,\dots,a_n$ and $b_0,b_1,\dots,b_n,b_{n+1}$ be positive integers. Give conditions that make the following ...
0
votes
1answer
101 views

Properties of continuity

Let $f,g :[a,b]\to\mathbb{R}$ be continuous functions such that $$\int\limits_c^df(x)\leq \int\limits_c^dg(x)dx$$ whenever a$\leq$c$<$d$\leq$b. I need to show that $f(x)\leq g(x)$. I have the ...
1
vote
1answer
119 views

Solution of Pell equation over field of p-adic numbers

Right now I am studying Pell equation. Using continued fractions, we can find the solution of Pell equation. Now my question, is it possible for me to find a solution in the field of p-adic numbers ...