Definition:Value of Continued Fraction
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Informally, the value of a finite continued fraction is the number which results from calculating out the fractions.
Note that formally the continued fraction and its value are considered to be distinct; a continued fraction is its sequence of partial quotients.
However, this is a nicety of interpretation and may usually be ignored — $x = \left[{a_1, a_2, a_3, \ldots, a_n}\right]$ is often used to mean that $x$ is the value of the given continued fraction.
Definition
Let $F$ be a field, such as the field of real numbers $\R$.
Finite Continued Fraction
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 $\dfrac{x_{11} }{x_{21} }$
Infinite Continued Fraction
Let $\struct {F, \norm {\,\cdot\,} }$ be a valued field.
Let $C = \sequence {a_n}_{n \mathop \ge 0}$ be a infinite continued fraction in $F$.
Then $C$ converges to its value $x \in F$ if and only if the following hold:
- For all natural numbers $n \in \N_{\ge 0}$, the $n$th denominator is nonzero
- The sequence of convergents $\sequence {C_n}_{n \mathop \ge 0}$ converges to $x$.
Also denoted as
The value of a continued fraction $C$ can also be denoted $\operatorname{Val}(C)$.