Definition:Numerators and Denominators of Continued Fraction

Definition
Let $k$ be a field.

Let $C = \left[{a_0, a_1, a_2, \ldots, a_n}\right]$ or $\left[{a_0, a_1, a_2, \ldots}\right]$ be a continued fraction in $k$, either finite or infinite.

Its numerators $p_0, p_1, p_2, p_3, \ldots$ and denominators $q_0, q_1, q_2, q_3, \ldots$ of $C$ are recursively defined as:

Also see

 * Value of Simple Finite Continued Fraction where it is shown that $C_k = \dfrac {p_k} {q_k}$, where $C_k$ is the $k$th convergent of the continued fraction.