# Definition:Continued Fraction

## Contents

## 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 $\paren {\floor {\alpha_n} }_{n \ge 0}$ where $\alpha_n$ is recursively defined as:

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

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.

## 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 quotients 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 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

- Definition:Value of Continued Fraction
- Definition:Sequence of Partial Quotients
- Definition:Sequence of Complete Quotients
- 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): Entry:**continued fraction**