# Definition:Numerators and Denominators of Continued Fraction

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*Not to be confused with Definition:Partial Numerator or Definition:Partial Denominator.*

## Contents

## Definition

Let $F$ be a field.

Let $n\in \N\cup \{\infty\}$ be an extended natural number.

Let $C = \left[{a_0, a_1, a_2, \ldots}\right]$ be a continued fraction in $F$ of length $n$.

### Definition 1: recursive definition

The **sequence of numerators** of $C$ is the sequence $(p_k)_{0 \leq k \leq n}$ that is recursively defined by:

- $p_k = \begin{cases} a_0 & : k = 0 \\ a_0 a_1 + 1 & : k = 1 \\ a_k p_{k - 1} + p_{k - 2} & : k \geq 0 \end{cases}$

The **sequence of denominators** of $C$ is the sequence $(q_k)_{0 \leq k \leq n}$ that is recursively defined by:

- $q_k = \begin{cases} 1 & : k = 0 \\ a_1 & : k = 1 \\ a_k q_{k - 1} + q_{k - 2} & : k \geq 0 \end{cases}$

### Definition 2: using matrix products

Let $k \geq 0$, and let the indexed matrix product:

- $\displaystyle \prod_{i = 0}^k\begin{pmatrix} a_i & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} x_{11}^{(k)} & x_{12}^{(k)} \\ x_{21}^{(k)} & x_{22}^{(k)} \end{pmatrix}$

The $k$th **numerator** is $x_{11}^{(k)}$ and the $k$th **denominator** is $x_{21}^{(k)}$.