Definition:Continued Fraction/Expansion of Real Number
Until this has been finished, please leave {{refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
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
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this point in more detail, feel free to use the talk page.
If you are able to explain it, then when you have done so you can remove this instance of {{Explain}} from the code.
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