Continued fractions as matrix products

Let p_n / q_n be the nth convergent of a continued fraction:

$\frac{p_n}{q_n} = a_0 + \cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+ \ddots \cfrac{1}{a_n}}}}$

Then

$\begin{pmatrix} p_n & p_{n-1} \\ q_n & q_{n-1} \end{pmatrix} = \begin{pmatrix} a_0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} a_1 & 1 \\ 1 & 0 \end{pmatrix} \cdots \begin{pmatrix} a_n & 1 \\ 1 & 0 \end{pmatrix}$

Source: Julian Havil. The Irrationals. p. 212.

Calendars and continued fractions
Continued fractions of square roots
Normal hazard continued fraction

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