Definition:Continued Fraction/Expansion of Real Number

Definition

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.