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.

Also see

• Continued Fraction Algorithm
• Definition:Simple Continued Fraction
• Correspondence between Rational Numbers and Simple Finite Continued Fractions
• Correspondence between Irrational Numbers and Simple Infinite Continued Fractions

Examples

• Continued Fraction Expansion/Example
• Continued Fraction Expansion of Irrational Square Root/Example

Generalizations

• Definition:Continued Fraction Expansion of Laurent Series
• Definition:p-Adic Ruban Continued Fraction Expansion