Definition:Continued Fraction

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Not to be confused with Definition:Generalized Continued Fraction.

Definition

Let $F$ be a field, such as the field of real numbers $\R$.


Finite Continued Fraction

Let $n \geq 0$ be a natural number.


Informally, a finite continued fraction of length $n$ in $F$ is an expression of the form:

$a_0 + \cfrac 1 {a_1 + \cfrac 1 {a_2 + \cfrac 1 {\ddots \cfrac {} {a_{n-1} + \cfrac 1 {a_n}} }}}$

where $a_0, a_1, a_2, \ldots, a_n \in F$.


Formally, a finite continued fraction of length $n$ in $F$ is a finite sequence, called sequence of partial quotients, whose domain is the integer interval $\left[0 \,.\,.\, n\right]$.


A finite continued fraction should not be confused with its value, when it exists.


Infinite Continued Fraction

Informally, an infinite continued fraction in $F$ is an expression of the form:

$a_0 + \cfrac 1 {a_1 + \cfrac 1 {a_2 + \cfrac 1 {\ddots \cfrac {} {a_{n-1} + \cfrac 1 {a_n + \cfrac 1 {\ddots}}} }}}$

where $a_0, a_1, a_2, \ldots, a_n, \ldots \in F$.


Formally, an infinite continued fraction in $F$ is a sequence, called sequence of partial quotients, whose domain is $\N_{\geq 0}$.


An infinite continued fraction should not be confused with its value, when it exists.


Simple Continued Fraction

Let $\R$ be the field of real numbers.


Simple Finite Continued Fraction

Let $n\geq 0$ be a natural number.


A simple finite continued fraction of length $n$ is a finite continued fraction in $\R$ of length $n$ whose partial quotients are integers that are strictly positive, except perhaps the first.

That is, it is a finite sequence $a : \left[0 \,.\,.\, n\right] \to \Z$ with $a_n > 0$ for $n >0$.


Simple Infinite Continued Fraction

A simple infinite continued fraction is a infinite continued fraction in $\R$ whose partial quotients are integers that are strictly positive, except perhaps the first.

That is, it is a sequence $a : \N_{\geq 0} \to \Z$ with $a_n > 0$ for $n >0$.


Continued fraction expansion of a real number

Irrational Number

Let $x$ be an irrational number.


The continued fraction expansion of $x$ is the simple continued fraction $(\lfloor \alpha_n \rfloor)_{n\geq0}$ where $\alpha_n$ is recursively defined as:

$\alpha_n = \displaystyle \begin{cases} x & : n = 0 \\ \dfrac 1 {\{\alpha_{n-1}\}} & : n \geq 1 \end{cases}$

where:

$\lfloor \cdot \rfloor$ is the floor function
$\{ \cdot \}$ is the fractional part function.


Rational Number

Let $x$ be a rational number.


The continued fraction expansion of $x$ is found using the Euclidean Algorithm.


Notation

A continued fraction can be denoted using ellipsis:

$a_0 + \cfrac 1 {a_1 + \cfrac 1 {a_2 + \cfrac 1 {\ddots \cfrac {} {}}}}$

which suggests the definition of its value, but it should be noted that this is only a notation.

By definition, a continued fraction is its sequence of partial quotients and can thus be denoted:

$(a_n)_{n\geq 0}$
$[a_0; a_1, a_2, \ldots ]$
$[a_0, a_1, a_2, \ldots ]$

where the last two notations are usually reserved for its value.


Also defined as

When one is primarily concerned with simple continued fractions of real numbers, it is common to require a continued fraction to have strictly positive partial quotients, except perhaps the first.


Also known as

A continued fraction is also known as a:

continued fraction in canonical form
regular continued fraction
simple continued fraction

as opposed to a generalized continued fraction.


Also see


Other continued fraction expansions


Generalizations

Historical Note

The concept of the continued fraction has been around a long time, since Euclid at least, and a great deal of research has been done and terminology developed.

Much of its modern theory was established by Leonhard Paul Euler.