Example 1 Find the finite simple continued fraction of $13 / 5$.

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We solve according to formula (12).
$$\begin{aligned}
13 / 5 & =\langle 2+3 / 5\rangle=\langle 2,5 / 3\rangle=\langle 2,\langle 1+2 / 3\rangle\rangle \\
& =\langle 2,\langle 1,3 / 2\rangle\rangle=\langle 2,1,1+1 / 2\rangle \\
& =\langle 2,1,1,2\rangle .
\end{aligned}$$

The characteristic of this finite simple continued fraction is: the numbers read from left to right and from right to left are the same (see Exercise 2, Question 5). $13 / 5$ can also be expressed as
$$13 / 5=\langle 2,1,1,1,1\rangle .$$