# Definition:Continued Fraction/Expansion of Real Number

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## 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/Examples
- Continued Fraction Expansion of Irrational Square Root/Examples