Continued fraction

The calculator below represents a given rational number as a finite continued fraction. It also shows the continued fraction coefficients (the first coefficient is the integer part). Read more on continued fractions just below the calculator.

Rational number to continued fraction

Fraction

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Continued fraction

 

Coefficients

 

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The calculator below converts the continued fraction coefficients back to the rational number.

Continued fraction to rational number

Continued fraction coefficients

Calculation precision

Exact

Rounded

Digits after the decimal point: 5

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Rational number

 

Continued fraction

 

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Continued (recurring) fraction

Continued or recurring fraction is a number representation kind as a sum of the number integer part and the fractional part. The fractional part numerator is always one, the denominator is the sum of the integer part and the fractional part. The fractional part denominator may again contain the sum of integer and fractional part and so on.

a₀,a₁,a₂...a_n is the continued fraction coefficients.
We use the following algorithm to calculate continued fraction coefficients:

// n - the fraction numerator
// d - the fraction denominator

loop while d ≠ 0
        r ⟵  n mod d;
        output ⟵ (n-r)/d;
        n ⟵ d;
        d ⟵ r;
 end loop  

The reverse transformation algorithm:

// f[] - the continued fraction coefficient array with indexes 0...k-1
// k - number of the coefficients
n ⟵ f[k-1];
d ⟵ 1;
loop while k greater than 1 
        r ⟵  d;
        d ⟵ n;
        k ⟵ k-1;
        n ⟵ f[k-1]*n+r;
end loop  
output ⟵ n/d;