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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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 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 simple fraction, use the irreducible fraction tool.

Calculate an approximate value of the root (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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Source : https://www.dcode.fr/continued-fractions

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