Definition:Value of Continued Fraction/Finite

Definition
Let $F$ be a field, such as the field of real numbers $\R$. Let $n \ge 0$ be a natural number.

Let $\sequence {a_k}_{0 \mathop \le k \mathop \le n}$ be a finite continued fraction in $F$.

Let $\overline F = F \cup \set \infty$ be extended by infinity.

Definition 1
The value $\sqbrk {a_0, a_1, \ldots, a_n} \in F \cup \set \infty$ is the right iteration of the binary operation:
 * $\sqbrk {\cdot, \cdot}: F \times \overline F \to \overline F$:
 * $\sqbrk {a, b} = a + \dfrac 1 b$.

That is, it is recursively defined as:
 * $\sqbrk {a_0, \ldots, a_n} = \begin{cases}

a_0 & : n = 0 \\ a_0 + \dfrac 1 {\sqbrk {a_1, \ldots, a_n} } & : n > 0 \\ \end{cases}$ or as:
 * $\sqbrk {a_0, \ldots, a_n} = \begin{cases}

a_0 & : n = 0 \\ \sqbrk {a_0, \ldots, a_{n - 2}, a_{n - 1} + \dfrac 1 {a_n} } & : n > 0 \\ \end{cases}$

Definition 2
Let the matrix product:
 * $\begin{pmatrix} a_0 & 1 \\ 1 & 0 \end{pmatrix} \cdots \begin{pmatrix} a_n & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{pmatrix}$

The value of the finite continued fraction is $\dfrac{x_{11} }{x_{21} }$

Also see

 * Value of Finite Continued Fraction equals Numerator Divided by Denominator
 * Properties of Value of Finite Continued Fraction