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))).

Continued Fractions - dCode

Tag(s) : Series

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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.

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

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 $$ e = [2 ; 1, 2, 1, 1, 4, 1, 1, 6, \cdots] $$

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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Cite as source (bibliography):

*Continued Fractions* on dCode.fr [online website], retrieved on 2024-09-14,

continued,fraction,approximation,pade,reciprocal,sequence,convergent,representation

https://www.dcode.fr/continued-fractions

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