Continued Fraction Expansion Calculator

About this calculator

A continued fraction represents a real number as a nested sequence a₀ + 1/(a₁ + 1/(a₂ + 1/(a₃ + ...))), written compactly as [a₀; a₁, a₂, a₃, ...]. It's computed by repeatedly taking the integer part of the number, then continuing the expansion on the reciprocal of the remaining fractional part — a process closely related to the Euclidean algorithm for finding a greatest common divisor.

Continued fractions are prized in number theory, cryptography, and numerical analysis because truncating the expansion early produces the best possible rational approximation of a number for a given denominator size — this is how ancient and modern mathematicians derived accurate fractions for π (like 355/113) and how continued fractions play a role in Diophantine equation solving and certain integer-factorization and lattice-based cryptographic attacks.

This calculator expands a real number you enter into its continued fraction terms up to a chosen count, and also shows the resulting rational convergent — useful for math students, number theory enthusiasts, and programmers implementing rational approximation algorithms.