The continued fraction of a real number $x\in\mathbb R$ can be computed by the following algorithm.1

Algorithm: Continued Fraction (Python)

import math

def contFrac(x, k): cf = [] q = math.floor(x) cf.append(q) x = x - q i = 0 while x != 0 and i < k: q = math.floor(1 / x) cf.append(q) x = 1 / x - q i = i + 1 return cf



  1. will output
  2. will output

Proofs: 1

Definitions: 1

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  1. Hermann, D.: "Algorithmen Arbeitsbuch", Addison-Wesley Publishing Company, 1992
  2. Schnorr, C.P.: "Lecture Notes Diskrete Mathematik", Goethe University Frankfurt, 2001


  1. Because floating point arithmetic IEEE-754 “double precision”, python doubles contain 53 bits of precision. Therefore, the algorithm not always computes the write values of the continued fraction. The algorithm also limits the computation to 20 values of the continued fraction, since some continued fractions are not finite.