# Category:Continued Fractions

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This category contains results about Continued Fractions.

Definitions specific to this category can be found in Definitions/Continued Fractions.

Let $n \geq 0$ be a natural number.

Informally, a **finite continued fraction of length $n$** in $F$ is an expression of the form:

- $a_0 + \cfrac 1 {a_1 + \cfrac 1 {a_2 + \cfrac 1 {\ddots \cfrac {} {a_{n-1} + \cfrac 1 {a_n}} }}}$

where $a_0, a_1, a_2, \ldots, a_n \in F$.

Formally, a **finite continued fraction of length $n$** in $F$ is a finite sequence, called **sequence of partial quotients**, whose domain is the integer interval $\left[0 \,.\,.\, n\right]$.

A finite continued fraction should not be confused with its **value**, when it exists.

## Subcategories

This category has the following 3 subcategories, out of 3 total.

### C

### E

### S

## Pages in category "Continued Fractions"

The following 54 pages are in this category, out of 54 total.

### A

### C

- Condition for Rational to be a Convergent
- Continued Fraction Algorithm
- Continued Fraction Algorithm/Proof 1
- Continued Fraction Expansion of Euler's Number
- Continued Fraction Expansion of Euler's Number/Convergents
- Continued Fraction Expansion of Euler's Number/Proof 1
- Continued Fraction Expansion of Euler's Number/Proof 1/Lemma
- Continued Fraction Expansion of Fourth Power of Pi
- Continued Fraction Expansion of Golden Mean
- Continued Fraction Expansion of Golden Mean/Rate of Convergence
- Continued Fraction Expansion of Golden Mean/Successive Convergents
- Continued Fraction Expansion of Irrational Number Converges to Number Itself
- Continued Fraction Expansion of Irrational Square Root
- Continued Fraction Expansion of Irrational Square Root/Examples
- Continued Fraction Expansion of Irrational Square Root/Examples/13/Convergents
- Continued Fraction Expansion of Irrational Square Root/Examples/2
- Continued Fraction Expansion of Irrational Square Root/Examples/29/Convergents
- Continued Fraction Expansion of Irrational Square Root/Examples/61
- Continued Fraction Expansion of Irrational Square Root/Examples/61/Convergents
- Continued Fraction Expansion of Limit of Simple Infinite Continued Fraction equals Expansion Itself
- Continued Fraction Expansion of Pi
- Continued Fraction Expansion of Pi/Convergents
- Continued Fraction Expansion of Root 2
- Continued Fraction Identities/First/Infinite
- Convergents are Best Approximations
- Convergents are Best Approximations/Corollary
- Convergents of Simple Continued Fraction are Rationals in Canonical Form
- Correspondence between Rational Numbers and Simple Finite Continued Fractions