Exercise 7.1.3 (Convert $\langle 2,1,4 \rangle, \ldots$)

Question

Convert into rational numbers: $\langle 2,1,4 \rangle;\ \langle -3,2,12 \rangle; \ \langle 0,1,1,100\rangle$.

Richard Ganaye · Accepted Answer

\begin{proof} 
Since $\langle 1,4 \rangle = 1 + \frac{1}{4} = \frac{5}{4}$, 
$$\langle 2,1,4 \rangle = 2 + \frac{1}{\langle 1,4 \rangle} = 2 + \frac{4}{5} = \frac{14}{5}.$$
With Sagemath:
\begin{verbatim}
sage: cf = continued_fraction([2,1,4])
sage: cf.value()
14/5
\end{verbatim}
Similarly
$$\langle -3,2,12 \rangle = -3 + \frac{1}{\langle 2,12 \rangle} = -3 + \frac{12}{25} = -\frac{63}{25}.$$
Finally
$$\langle 1,1,100\rangle = 1 + \frac{1}{\langle 1,100 \rangle} = 1 + \frac{100}{101} = \frac{201}{101},$$
so
$$\langle 0, 1,1,100\rangle = \frac{101}{201}.$$
\end{proof}

Exercise 7.1.3 (Convert $\langle 2,1,4 \rangle, \ldots$)

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Exercise 7.1.3 (Convert $\langle 2,1,4 \rangle, \ldots$)

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