Exercise 7.1.1 (Some finite continued fractions)

Question

Expand the rational fractions $17/3,\ 3/17$, and $8/1$ into finite simple continued fractions.

Richard Ganaye · Accepted Answer

\begin{proof} 
The Euclidean algorithm gives
\begin{align*}
17 &= 3\cdot 5 + 2,\
3 &= 2 \cdot 1 + 1,\
1 &= 1 \cdot 1 + 0.
\end{align*}
Therefore
\begin{align*}
\frac{17}{3} &= 5 +\frac{1}{\left(\frac{3}{2}ight)}\
&= 5 +\frac{1}{1 + \frac{1}{2}}\
&=\langle 5, 1,2 angle.
\end{align*}

With Sagemath:
\begin{verbatim}
sage: continued_fraction(17/3)
[5; 1, 2]
\end{verbatim}
Since
\begin{align*}
3 = 0\cdot 17 + 3,\
\end{align*}
we obtain
\begin{align*}
\frac{3}{17} &= 0 + \frac{1}{\left( \frac{17}{3}ight)}\
&= 0 + \frac{1}{5 +\frac{1}{1 + \frac{1}{2}}}\
&=\langle 0, 5, 1,2 angle.
\end{align*}
Finally
$$8 = 1\cdot 8 + 0,$$
so
$$\frac{8}{1} = \langle 8 angle.$$
\end{proof}

Exercise 7.1.1 (Some finite continued fractions)

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Exercise 7.1.1 (Some finite continued fractions)

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