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Continued Fractions

Tool to compute continued fractions. A continued fraction is the representation of a number N in a form of a series of integers (a0, a1, ..., an) such as N = (a0+1/(a1+1/(a2+1/(...1/(an))).

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Continued Fractions -

Tag(s) : Series

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# Continued Fractions

## Continued Fraction to Number Converter

 Display value as Fraction or exact number Numeric Value (approximation)

### How to calculate a continued fraction?

Continued fraction expansion is close to algorithm of euclidean division, as for PGCD.

Example: If the fraction approximating pi is $355/113 = 3.14159292035...$

$$355 = 3 \times 113 + 16 \\ 113 = 7 \times 16 + 1 \\ 16 = 16 \times 1 + 0$$

The continued fraction is [3,7,16]

Some developments of continuous fractions are infinite

To find the corresponding fraction, use the irreducible fraction tool.

### How to calculate the continued fraction of a root?

Calculate an approximate value of the root (approximation as accurate as possible) and dCode will provide the corresponding continuous fraction.

### How to write a continued fraction in LaTex?

The easiest way is to use cfrac: $$e=2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{ 1+\cfrac{1}{1+\cfrac{1}{4+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{6+\cdots}}}}}}}}$$

But the shortest way is to write $$e = [2 ; 1, 2, 1, 1, 4, 1, 1, 6, \cdots]$$

### Which are the most remarquable continued fractions?

Most known continued fractions are:

Square Root of 2: $\sqrt{2} = [1;2,2,2,2,2,\cdots]$

— Golden Ratio: $\Phi = [1;1,1,1,1,1,\cdots]$

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Continued Fractions on dCode.fr [online website], retrieved on 2023-12-05, https://www.dcode.fr/continued-fractions

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