194 lines
4.6 KiB
TeX
194 lines
4.6 KiB
TeX
\section{}
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\definition{}
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A \textit{finite continued fraction} is an expression of the form
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\[
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a_0 + \cfrac{1}{a_1+\cfrac{1}{a_2 + \cfrac{1}{a_3 + ... + \cfrac{1}{a_{k-1} + \cfrac{1}{a_k}}}}}
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\]
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where $a_0, a_1, ..., a_k$ are all in $\mathbb{Z}^+_0$.
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We'll denote this as $[a_0, a_1, ..., a_k]$.
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\problem{}<num2cf>
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Write each of the following as a continued fraction. \par
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\hint{Solve for one $a_n$ at a time.}
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\begin{itemize}
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\item $5/12$
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\item $5/3$
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\item $33/23$
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\item $37/31$
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\end{itemize}
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\vfill
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\problem{}
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Write each of the following continued fractions as a regular fraction in lowest terms: \par
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\begin{itemize}
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\item $[2,3,2]$
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\item $[1,4,6,4]$
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\item $[2,3,2,3]$
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\item $[9,12,21,2]$
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\end{itemize}
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\vfill
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\pagebreak
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\problem{}<euclid>
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Let $\frac{p}{q}$ be a positive rational number in lowest terms.
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Perform the Euclidean algorithm to obtain the following sequence:
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\begin{align*}
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p \ &= \ q_0 q + r_1 \\
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q \ &= \ q_1 r_1 + r_2 \\
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r_1 \ &= \ q_2 r_2 + r_3 \\
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&\vdots \\
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r_{k-1} \ &= \ q_k r_k + 1 \\
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r_k \ &= \ q_{k+1}
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\end{align*}
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We know that we will eventually get $1$ as the remainder because $p$ and $q$ are relatively prime. \par
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Show that $p/q = [q_0, q_1, ..., q_{k+1}]$.
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\vfill
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\problem{}
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Repeat \ref{num2cf} using the method outlined in \ref{euclid}.
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\vfill
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\pagebreak
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\definition{}
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An \textit{infinite continued fraction} is an expression of the form
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\[
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a_0 + \cfrac{1}{a_1+\cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{a_4 + ...}}}}
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\]
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where $a_0, a_1, a_2, ...$ are in $\mathbb{Z}^+_0$.
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To prove that this expression actually makes sense and equals a finite number
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is beyond the scope of this worksheet, so we assume it for now.
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This is denoted $[a_0, a_1, a_2, ...]$.
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\problem{}<irrational>
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Using a calculator, compute the first five terms of the
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continued fraction expansion of the following numbers.
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Do you see any patterns?
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\begin{itemize}
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\item $\sqrt{2}$
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\item $\pi \approx 3.14159...$
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\item $\sqrt{5}$
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\item $e \approx 2.71828...$
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\end{itemize}
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\vfill
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\problem{}
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Show that an $\alpha \in \mathbb{R}^+$ can be written as a finite
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continued fraction if and only if $\alpha$ is rational. \par
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\hint{For one of the directions, use \ref{euclid}}
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\vfill
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\pagebreak
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\definition{}
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The continued fraction $[a_0, a_1, a_2, ...]$ is \textit{periodic} if it ends in a repeating sequence of digits. \par
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A few examples are below. We denote the repeating sequence with a line.
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\begin{itemize}
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\item $[1,2,2,2,...] = [1, \overline{2}]$ is periodic.
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\item $[1,2,3,4,5,...]$ is not periodic.
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\item $[1,3,7,6,4,3,4,3,4,3,...] = [1,3,7,6,\overline{4,3}]$ is periodic.
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\item $[1,2,4,8,16, ...]$ is not periodic.
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\end{itemize}
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\problem{}
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\begin{itemize}
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\item Show that $\sqrt{2} = [1, \overline{2}]$.
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\item Show that $\sqrt{5} = [1, \overline{4}]$.
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\end{itemize}
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\hint{use the same strategy as \ref{irrational} but without a calculator.}
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\vfill
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\problem{Challenge I}
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Express the following continued fractions in the form $\frac{a+\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers: \par
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\begin{itemize}
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\item $[~\overline{1}~]$
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\item $[~\overline{2,5}~]$
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\item $[~1, 3, \overline{2,3}~]$
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\end{itemize}
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\vfill
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\problem{Challenge II}
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Let $\alpha = [~a_0,~ ...,~ a_r,~ \overline{a_{r+1},~ ...,~ a_{r+p}}~]$ be any periodic continued fraction. \par
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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.
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\vfill
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\problem{Challenge III}
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Prove that any number of the form $\frac{a+\sqrt{b}}{c}$ where $a,b,c$ are integers
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and $b$ is not a perfect square can be written as a periodic continued fraction.
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%\begin{rmk}
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%Numbers of the form $\frac{a+\sqrt{b}}{c}$ where $a,b,c$ are integers and $b$ is not a perfect square are the ``simplest'' irrational numbers in the following sense. A number is rational if and only if it is the solution to a degree $1$ polynomial equation, $ax+b = 0$. Similarly, a number is of the form $\frac{a+\sqrt{b}}{c}$ if it is the solution to a degree $2$ polynomial equation, $ax^2 + bx + c = 0$ (Bonus exercise: prove this). Such numbers are called \textit{quadratic} irrational numbers or \textit{degree 2} irrational numbers.
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%\end{rmk}
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%\begin{rmk}
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%Notice that the results of this worksheet provide a very clean characterization of continued fraction expansions:
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%\begin{itemize}
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%\item $\alpha$ is a rational number if and only if it has a finite continued fraction expansion.
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%\item $\alpha$ is a degree $2$ irrational number if and only if it has an infinite periodic continued fraction expansion.
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%\end{itemize}
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%\end{rmk}
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\vfill
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\pagebreak |