Continued Fraction Calculator

A continued fraction is a way of representing a number as a sum of an integer part plus a fractional part whose denominator is itself another integer plus a fractional part, and so on. It is written as [a₀; a₁, a₂, …] where a₀ is the integer part and a₁, a₂, … are positive integers called partial quotients or coefficients.

The process uses repeated division similar to the Euclidean algorithm. At each step, you take the integer part of the current value as the next coefficient, then invert the fractional remainder and repeat. For example, 355/113 → [3; 7, 15, 1] because 355 ÷ 113 = 3 remainder 16, then 113 ÷ 16 = 7 remainder 1, and so on.

You work from the last coefficient backwards. Start with the last term aₙ and compute aₙ₋₁ + 1/aₙ, then take aₙ₋₂ + 1/(previous result), continuing until you reach a₀. The result is the exact rational number represented by those coefficients.

Convergents are the rational approximations obtained by truncating the continued fraction at each step. They are the best rational approximations to the original number given their denominator size, meaning no fraction with a smaller denominator is closer to the true value.

Yes. Every rational number (a ratio of two integers) has a finite continued fraction expansion that terminates after a finite number of steps. Irrational numbers like π, e, or √2 have infinite continued fraction expansions, though √2 = [1; 2, 2, 2, …] has a simple repeating pattern.

π ≈ [3; 7, 15, 1, 292, 1, 1, 1, 2, …] — the coefficients do not follow a known pattern. The number e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, …] does follow a beautiful repeating pattern. The golden ratio φ = [1; 1, 1, 1, …] has all coefficients equal to 1.

Continued fractions provide the best rational approximations to any real number, which is valuable in number theory, cryptography, gear ratio design, musical tuning theory, and computing rational approximations to irrational constants. For example, 355/113 is an excellent approximation to π derived from its continued fraction.

In the standard (canonical) form, all partial quotients after a₀ are positive integers, and a₀ may be any integer (including negative). Some generalized continued fractions allow negative coefficients, but the standard Euclidean algorithm produces non-negative terms except possibly the first.