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 Fractions
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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))).
Answers to Questions
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] $
Source code
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