Definition:Continued Fraction/Expansion of Real Number
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Contents
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.
how
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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
