Definition: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.
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 denominators, whose domain is the integer interval $\closedint 0 n$.
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 a sequence of partial denominators, whose domain is $\N_{\ge 0}$.
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 $n \ge 0$ be a natural number.
A simple finite continued fraction of length $n$ is a finite continued fraction in $\R$ of length $n$ whose partial denominators are integers that are strictly positive, except perhaps the first.
That is, it is a finite sequence $a: \closedint 0 n \to \Z$ 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 denominators are integers that are strictly positive, except perhaps the first.
That is, it is a sequence $a: \N_{\ge 0} \to \Z$ 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 $\paren {\floor {\alpha_n} }_{n \ge 0}$ where $\alpha_n$ is recursively defined as:
- $\alpha_n = \ds \begin{cases} x & : n = 0 \\
\dfrac 1 {\fractpart {\alpha_{n - 1} } } & : n \ge 1 \end{cases}$ where:
- $\floor {\, \cdot \,}$ is the floor function
- $\fractpart {\, \cdot \,}$ 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.
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Notation
A continued fraction can be denoted using ellipsis:
- $a_0 + \cfrac 1 {a_1 + \cfrac 1 {a_2 + \cfrac 1 {\ddots \cfrac {} {} } } }$
Another notation that can sometimes be seen is:
- $a_0 + \dfrac 1 {a_1 +} \dfrac 1 {a_2 +} \dfrac 1 {a_3 + \cdots}$
By definition, a continued fraction is its sequence of partial denominators and can thus be denoted:
- $\sequence {a_n}_{n \mathop \ge 0}$
- $\sqbrk {a_0; a_1, a_2, \ldots}$
- $\sqbrk {a_0, a_1, a_2, \ldots}$
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 denominators, 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
in order to distinguish such a continued fraction from a generalized continued fraction.
Also see
- Definition:Value of Continued Fraction
- Definition:Sequence of Partial Denominators
- Definition:Convergent of Continued Fraction
- Definition:Numerators and Denominators of Continued Fraction
- Results about continued fractions can be found here.
Other continued fraction expansions
- Definition:Continued Fraction Expansion of Real Number
- Definition:Continued Fraction Expansion of Laurent Series
- Definition:p-Adic Ruban Continued Fraction
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.
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): continued fraction