# Tagged Questions

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.

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+50

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### Is the inverse of Minkowski's question mark function continuous on the dyadic fractions?

I'm looking for a continuous function from the dyadic fractions between 0 and 1 to the rational numbers between 0 and 1. The inverse of Minkowski's question mark (also known as Conway's box function) ...
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### Chinese estimate for $\pi$. Were they lucky?

The famous chinese estimate $\pi\approx\frac{355}{113}$ is good. I think that is too good. As a continued fraction: $$\pi=[3:7,15,1,292,\ldots]$$ That $292$ is a bit too big. Is there a reason for a ...
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### Using Maple for continued fraction expansions

I can find the continued fraction expansion of a value using Maple. Is there a simple way for finding the sequence of convergents (approximants) of the continued fraction expansion in Maple? Currently ...
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### Can every transcendental number be expressed as an infinite continued fraction?

Every infinite continued fraction is irrational. But can every number, in particular those that are not the root of a polynomial with rational coefficients, be expressed as a continued fraction?
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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 ...
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### Almost everywhere growth of continued fraction partial quotients

What is an upper bound for the growth of the largest partial quotient (i.e. the 'digit') among the first $n$ partial quotients in the continued fraction expansion of almost all real numbers as $n$ ...
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### Proving a particular infinite continued-fraction identity

By iterating the basic relation $$\forall z \in \mathbb{C} \setminus \{ -1 \}: \quad z = 1 + \frac{z^{2} - 1}{z + 1},$$ one obtains the following finite continued-fraction identities: \begin{alignat}...
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### Continued fraction $1 + \frac 2{3 + \frac 4 {5 + \cdots}} = \frac 1 {\sqrt{e} - 1}$?

I saw this link (written in Japanese) and found an interesting problem: Calculate $1 + \frac 2{3 + \frac 4 {5 + \cdots}}$. The link provides the answer ($\frac 1 {\sqrt e - 1}$) and a hint that one ...
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### Pell's Equation $x^2-11y^2=1$ through Continued Fractions

Use continued fractions to find the minimal solution to $x^2-11y^2=1$. I know that $\sqrt{11}=3+\frac{1}{3+\frac{1}{6+\frac{1}{3+...}}}$ I took $\sqrt{11}=3+\frac{1}{3+\frac{1}{6+\sqrt{11}}}$ and I ...
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### Curious about an empirically found continued fraction for tanh

First of all, and since this is my first question in this forum, I would like to specify that I am not a professional mathematician (but a philosophy teacher); I apologize by advance if something is ...
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### Two complementary continued fractions that are algebraic numbers

Define the two similar continued fractions, $$x=\cfrac{1}{km\color{blue}+\cfrac{(m-1)(m+1)} {3km\color{blue}+\cfrac{(2m-1)(2m+1)}{5km\color{blue}+\cfrac{(3m-1)(3m+1)}{7km\color{blue}+\ddots}}}}\tag1$$...
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### Use the simple continued fraction of $\sqrt{27323}$ to factor $27323$…

Use the simple continued fraction of $\sqrt{27323}$ to factor $27323$. So far I have: $\sqrt{27323} = 1 + (\sqrt{27323} - 1)$ which gives... $= 1 + \frac{1}{(\frac{1}{164.2967029})}$ I'm ...