FoldUnfold Table of Contents Iff Criterion For An Irrational Number to be Purely Periodic Iff Criterion For An Irrational Number to be Purely Periodic Recall from the Periodic and Purely Periodic Continued Fractions that an irrational number $\theta = \langle a_0; a_1, a_2, ... \rangle$ is said to be purely periodic if there exists an $n \in \mathbb{N}$ such that $a_{n+r} … [Read more...]

## Quadratic Irrationals Have Periodic Continued Fractions

FoldUnfold Table of Contents Quadratic Irrationals Have Periodic Continued Fractions Quadratic Irrationals Have Periodic Continued Fractions On the Periodic Continued Fractions are Quadratic Irrationals page we proved that if $\theta$ is an irrational and $\theta = \langle a_0; a_1, a_2, ... \rangle$ is a periodic continued fraction then $\theta$ is a quadratic … [Read more...]

## Periodic Continued Fractions are Quadratic Irrationals

FoldUnfold Table of Contents Periodic Continued Fractions are Quadratic Irrationals Periodic Continued Fractions are Quadratic Irrationals Theorem 1: Let $\theta \in \mathbb{R} \setminus \mathbb{Q}$ with $\theta = \langle a_0; a_1, a_2, ... \rangle$. If $\theta$'s infinite continued fraction is periodic then $\theta$ is a quadratic irrational. Proof: Since $\theta$ … [Read more...]

## Iff Criterion For An Irrational Number to be Periodic

FoldUnfold Table of Contents Iff Criterion For An Irrational Number to be Periodic Iff Criterion For An Irrational Number to be Periodic Recall from the Periodic and Purely Periodic Continued Fractions page that an infinite continued fraction $\langle a_0; a_1, a_2, ... \rangle$ is said to be periodic if there exists an $n \in \mathbb{N}$ such that $a_{n+r} = a_{r}$ for … [Read more...]

## Periodic and Purely Periodic Continued Fractions

FoldUnfold Table of Contents Periodic and Purely Periodic Continued Fractions Periodic and Purely Periodic Continued Fractions Definition: An infinite continued fraction $\langle a_0; a_1, a_2, ... \rangle$ is said to be Periodic if there exists an $n \in \mathbb{N}$ such that $a_{n+r} = a_r$ for all $r$ sufficiently large and $n$ is called the Period of the infinite … [Read more...]

## Criterion for a/b to be a Convergent of an Infinite Continued Fraction

FoldUnfold Table of Contents Criterion for a/b to be a Convergent of an Infinite Continued Fraction Criterion for a/b to be a Convergent of an Infinite Continued Fraction Theorem 1: Let $\epsilon \in \mathbb{R} \setminus \mathbb{Q}$. If there exists $\frac{a}{b} \in \mathbb{Q}$ with $(a, b) = 1$ such that $\displaystyle{\left | \epsilon - \frac{a}{b} \right | then for … [Read more...]

## Irrational Number Approximation by Infinite Continued Fractions

FoldUnfold Table of Contents Irrational Number Approximation by Infinite Continued Fractions Irrational Number Approximation by Infinite Continued Fractions Consider the infinite continued fraction $\langle 1; 2, 2, 2, 2, ... \rangle$. We already know that this infinite continued fraction represents an irrational number $x$ and that: (1) \begin{align} \quad x = 1 + … [Read more...]

## Examples of Computing Infinite Continued Fractions

FoldUnfold Table of Contents Examples of Computing Infinite Continued Fractions Example 1 Example 2 Examples of Computing Infinite Continued Fractions We have seen that every infinite simple continued fraction $\langle a_0; a_1, a_2, ... \rangle$ with $a_0 \in \mathbb{Z}$ and $a_n \in \mathbb{N}$ for $n \geq 1$ converges to an irrational number. We will now look at some … [Read more...]

## Infinite Continued Fractions and Irrational Numbers

FoldUnfold Table of Contents Infinite Continued Fractions and Irrational Numbers Infinite Continued Fractions and Irrational Numbers Recall from the Convergence of Infinite Continued Fractions page that if $\langle a_0; a_1, a_2, ... \rangle$ is an infinite simple continued fraction with $a_0 \in \mathbb{Z}$ and $a_n \in \mathbb{N}$ for $n \geq 1$ then $\langle a_0; a_1, … [Read more...]

## The nth Convergent of an Infinite Continued Fraction

FoldUnfold Table of Contents The nth Convergent of an Infinite Continued Fraction The nth Convergent of an Infinite Continued Fraction Recall from The Sequences (hn) and (kn) page that if $\langle a_0; a_1, a_2, ... \rangle$ is a simple continued fraction (finite or infinite) and if $a_0 \in \mathbb{Z}$, $a_n \in \mathbb{N}$ for all $n \geq 1$ then the corresponding … [Read more...]