# 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:

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

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.

## Continued fraction expansion of a Laurent series

Let $k$ be a field.

Let $k((t^{-1}))$ be the field of formal Laurent series in the variable $t^{-1}$.

### Irrational Laurent series

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

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

- $\alpha_n = \displaystyle \begin{cases} f & : n = 0 \\ \dfrac 1 {f - \lfloor f \rfloor} & : n \geq 1 \end{cases}$

where $\lfloor \cdot \rfloor$ denotes the polynomial part.

### Rational Laurent series

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:

- $\left[{a_0, a_1, a_2, \ldots, a_n}\right]$

for the finite case, and:

- $\left[{a_0, a_1, a_2, \ldots}\right]$

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:

- $\left[{1, 2, 3}\right] = 1 + \cfrac 1 {2 + \cfrac 1 3}$