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 $n\geq0$ be a natural number.

Let $(a_k)_{0 \leq k \leq n}$ be a finite continued fraction in $F$.

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


Definition 1

The value $[a_0, a_1, \ldots, a_n] \in F \cup \{\infty\}$ is the right iteration of the binary operation:

$[\cdot,\cdot] : F \times \overline F \to \overline F$:
$[a, b] = a + \dfrac 1 b$.

That is, it is recursively defined as:

$[a_0, \ldots, a_n] = \begin{cases} a_0 & : n = 0 \\ a_0 + \dfrac 1 {\left[{a_1, \ldots, a_n}\right]} & : n > 0 \\ \end{cases}$

or as:

$[a_0, \ldots, a_n] = \begin{cases} a_0 & : n = 0 \\ \left[a_0, \ldots, a_{n-2}, a_{n-1} + \dfrac 1 {a_n}\right] & : n > 0 \\ \end{cases}$


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 $C = (a_n)_{n\geq 0}$ be a infinite continued fraction in $F$.


Then $C$ converges to its value $x \in F$ if and only if the following hold:

  1. For all natural numbers $n \in \N_{\geq 0}$, the $n$th denominator is nonzero
  2. The sequence of convergents $\sequence{C_n}_{n\geq 0}$ converges to $x$.


Also denoted as

The value of a continued fraction $C$ can also be denoted $\operatorname{Val}(C)$.


Also see