Continued Fraction Calculator

A continued fraction rewrites a ratio as a whole number plus one over another whole number plus one over another, nested as deep as the fraction requires, and this calculator finds that nesting the same way it has been found for centuries: the Euclidean algorithm. Each coefficient is the integer part of dividing the current numerator by the current denominator, after which the remainder becomes the new denominator and the process repeats — exactly the same steps used to find a greatest common divisor, just with every intermediate quotient recorded instead of thrown away. The default 355 over 113 is the classic approximation of pi, and it produces an unusually short continued fraction with a strikingly large third coefficient, which is precisely why 355/113 approximates pi to six decimal places despite having a denominator under 1,000 — a large coefficient partway through a continued fraction signals that the convergent just before it is an exceptionally good, disproportionately simple approximation.

Alongside the coefficient list, the calculator reconstructs the Final Convergent — the best rational approximation the expansion builds up to — by working the coefficients back into a fraction from the top down, and reports how far that convergent sits from the original ratio as Approximation Error. The expansion is capped at 20 coefficients regardless of how long the true continued fraction would run, so a numerator and denominator that share very few common factors partway through will be cut off before reaching an exact match.