From 043736b26e7ab7850fe53571114bc8f828a18636 Mon Sep 17 00:00:00 2001 From: Mark Date: Mon, 13 Apr 2026 16:18:23 -0700 Subject: [PATCH] `Continued Fractions` edits --- .../Continued Fractions/parts/01 part A.tex | 8 ++++---- .../Continued Fractions/parts/02 part B.tex | 16 ++++++++-------- 2 files changed, 12 insertions(+), 12 deletions(-) diff --git a/src/Advanced/Continued Fractions/parts/01 part A.tex b/src/Advanced/Continued Fractions/parts/01 part A.tex index 9843452..ce1da27 100644 --- a/src/Advanced/Continued Fractions/parts/01 part A.tex +++ b/src/Advanced/Continued Fractions/parts/01 part A.tex @@ -78,7 +78,7 @@ An \textit{infinite continued fraction} is an expression of the form a_0 + \cfrac{1}{a_1+\cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{a_4 + ...}}}} \] where $a_0, a_1, a_2, ...$ are in $\mathbb{Z}^+_0$. -To prove that this expression actually makes sense and equals a finite number +Showing that this expression converges to a finite number is beyond the scope of this worksheet, so we assume it for now. This is denoted $[a_0, a_1, a_2, ...]$. @@ -133,7 +133,7 @@ A few examples are below. We denote the repeating sequence with a line. \problem{} \begin{itemize} \item Show that $\sqrt{2} = [1, \overline{2}]$. - \item Show that $\sqrt{5} = [1, \overline{4}]$. + \item Show that $\sqrt{5} = [2, \overline{4}]$. \end{itemize} \hint{use the same strategy as \ref{irrational} but without a calculator.} @@ -159,7 +159,7 @@ Express the following continued fractions in the form $\frac{a+\sqrt{b}}{c}$ whe \problem{Challenge II} Let $\alpha = [~a_0,~ ...,~ a_r,~ \overline{a_{r+1},~ ...,~ a_{r+p}}~]$ be any periodic continued fraction. \par -Prove that $\alpha$ is of the form $\frac{a+\sqrt{b}}{c}$ for some integers $a,b,c$ where $b$ is not a perfect square. +Show that $\alpha$ is of the form $\frac{a+\sqrt{b}}{c}$ for some integers $a,b,c$ where $b$ is not a perfect square. @@ -168,7 +168,7 @@ Prove that $\alpha$ is of the form $\frac{a+\sqrt{b}}{c}$ for some integers $a,b \problem{Challenge III} -Prove that any number of the form $\frac{a+\sqrt{b}}{c}$ where $a,b,c$ are integers +Show that any number of the form $\frac{a+\sqrt{b}}{c}$ where $a,b,c$ are integers and $b$ is not a perfect square can be written as a periodic continued fraction. diff --git a/src/Advanced/Continued Fractions/parts/02 part B.tex b/src/Advanced/Continued Fractions/parts/02 part B.tex index f492086..3c2e918 100644 --- a/src/Advanced/Continued Fractions/parts/02 part B.tex +++ b/src/Advanced/Continued Fractions/parts/02 part B.tex @@ -65,7 +65,7 @@ Verify the recursive formula for $1\leq j\leq 3$ for the convergents $C_j$ of: \ \problem{Challenge IV} -Prove that $p_n = a_np_{n-1} + p_{n-2}$ and $q_n = a_nq_{n-1} + q_{n-2}$ by induction. +Show that $p_n = a_np_{n-1} + p_{n-2}$ and $q_n = a_nq_{n-1} + q_{n-2}$ by induction. \begin{itemize} \item As the base case, verify the recursive formulas for $n=1$ and $n=2$. \item Assume the recursive formulas hold for $n\leq m$ and show the formulas hold for $m+1$. @@ -97,7 +97,7 @@ we will show that $p_n q_{n-1} - p_{n-1}q_n = (-1)^{n-1}$. -\problem{Challenge VI} +\problem{Challenge V} Similarly derive the formula $p_nq_{n-2}-p_{n-2}q_n = (-1)^{n-2}a_n$. @@ -141,7 +141,7 @@ We will show that $|\alpha-C_n|<\frac{1}{q_n^2}$. We are now ready to prove a fundamental result in the theory of rational approximation. \problem{Dirichlet's approximation theorem} Let $\alpha$ be any irrational number. -Prove that there are infinitely many rational numbers $\frac{p}{q}$ such that $|\alpha - \frac{p}{q}| < \frac{1}{q^2}$. +Show that there are infinitely many rational numbers $\frac{p}{q}$ such that $|\alpha - \frac{p}{q}| < \frac{1}{q^2}$. @@ -154,8 +154,8 @@ Prove that there are infinitely many rational numbers $\frac{p}{q}$ such that $| -\problem{Challenge VII} -Prove that if $\alpha$ is \emph{rational}, then there are only \emph{finitely} many rational numbers $\frac{p}{q}$ +\problem{Challenge VI} +Show that if $\alpha$ is \emph{rational}, then there are only \emph{finitely} many rational numbers $\frac{p}{q}$ satisfying $|\alpha - \frac{p}{q} | < \frac{1}{q^2}$. @@ -195,8 +195,8 @@ Let $\frac{a}{b}$ and $\frac{c}{d}$ be consecutive elements of the Farey sequenc -\problem{Challenge VIII} -Prove that $bc-ad=1$ for $\frac{a}{b}$ and $\frac{c}{d}$ consecutive rational numbers in Farey sequence of order $n$. +\problem{Challenge VII} +Show that $bc-ad=1$ for $\frac{a}{b}$ and $\frac{c}{d}$ consecutive rational numbers in Farey sequence of order $n$. \begin{itemize}[itemsep=2mm] \item In the plane, draw the triangle with vertices (0,0), $(b,a)$, $(d,c)$. @@ -237,7 +237,7 @@ $|\alpha - \frac ab| \geq |\alpha - \frac{p_n}{q_n}|$ -\problem{Challenge X} +\problem{Challenge VIII} Prove the following strengthening of Dirichlet's approximation theorem. If $\alpha$ is irrational, then there are infinitely many rational numbers $\frac{p}{q}$ satisfying $|\alpha - \frac pq| < \frac{1}{2q^2}$.