# Definition:Continued Fraction Expansion

## Definition

## 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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## Continued fraction expansion of a Laurent series

Let $k$ be a field.

Let $\map k {\paren {t^{-1} } }$ be the field of formal Laurent series in the variable $t^{-1}$.

### Irrational Laurent series

Let $f \in \map k {\paren {t^{-1} } }$ be an irrational formal Laurent series.

The **continued fraction expansion** of $f$ is the infinite continued fraction $\sequence {\floor {\alpha_n} }_{n \mathop \ge 0}$ where $\alpha_n$ is recursively defined as:

- $\alpha_n = \ds \begin {cases} f & : n = 0 \\ \dfrac 1 {f - \floor f} & : n \ge 1 \end {cases}$

where $\floor \cdot$ denotes the polynomial part.

### Rational Laurent series

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By definition, a continued fraction equals its sequence of partial quotients.

Therefore, to reduce the cumbersome nature of its representation, the continued fraction in the definitions are usually written as:

- $\sqbrk {a_0, a_1, a_2, \ldots, a_n}$

for the finite case, and:

- $\sqbrk {a_0, a_1, a_2, \ldots}$

for the infinite case.

Such an expression is known as the **continued fraction expansion** of the continued fraction, especially in the case of the infinite version.

For example:

- $\sqbrk {1, 2, 3} = 1 + \cfrac 1 {2 + \cfrac 1 3}$