Continued Fraction Calculator

A simple continued fraction expresses a number x as x = a0 + 1 / (a1 + 1 / (a2 + 1 / (a3 + ...))), where a0 is any integer and a1, a2, a3, ... are positive integers called the partial quotients. The standard shorthand is [a0; a1, a2, a3, ...], with a semicolon after the integer part and commas separating the rest. For example, 22 / 7 = 3 + 1 / 7 is written [3; 7], and the golden ratio is [1; 1, 1, 1, 1, ...] because phi = 1 + 1 / (1 + 1 / (1 + 1 / (1 + ...))). Every rational number has a finite continued fraction (the algorithm terminates exactly when the remainder hits zero), and every irrational number has a unique infinite continued fraction. Continued fractions reveal arithmetic structure that decimal expansions hide: a periodic continued fraction means the number is a quadratic irrational (a root of an integer quadratic), and the partial quotients tell you how easily the number can be approximated by simple fractions.

A convergent is formed by truncating the continued fraction. Convergents are best approximations in precise denominator-based senses, but an intermediate or semiconvergent fraction can be closest under an arbitrary denominator cap. For pi, 22 / 7 and 355 / 113 are famous convergents. The large partial quotient 292 after 355 / 113 explains its unusual accuracy.

The fraction-to-continued-fraction algorithm and the Euclidean algorithm for gcd are literally the same computation; you just keep different bookkeeping. Take 22 / 7. Divide: 22 = 3 * 7 + 1, quotient 3 (this is a0), remainder 1. Now invert and repeat with 7 / 1: 7 = 7 * 1 + 0, quotient 7 (this is a1), remainder 0. Stop. The terms are [3; 7], and the last non-zero remainder is gcd(22, 7) = 1. For 1071 / 462: quotients are 2, 3, 7, so the continued fraction is [2; 3, 7] and gcd(1071, 462) = 21 (the last non-zero remainder before the algorithm terminates). The reason this works is structural: each step computes a_i = floor(p / q), then replaces (p, q) with (q, p mod q). The quotients are the partial quotients of the continued fraction; the remainders are the intermediate gcd values. This calculator's fraction-to-cf mode is exactly the Euclidean algorithm, with the quotients reported instead of just discarded.

22 / 7 is the second convergent of pi's continued fraction [3; 7, 15, 1, 292, ...], after 3 / 1. It differs from pi by about 0.00126 and is easy to remember. Later convergents are more accurate: 333 / 106 differs by about 0.000083 and 355 / 113 by about 0.000000267.

A noble number has a continued fraction that eventually consists entirely of 1s. The golden ratio is [1; 1, 1, 1, ...]. Calling it the most irrational refers to a specific Diophantine-approximation sense: its convergents improve relatively slowly compared with numbers that have larger partial quotients. It is not a general ranking of every property of irrational numbers.