Continued Fraction Calculator

A continued fraction is a way to represent any real number as a nested sequence of integer additions and reciprocals. Every real number has a unique continued fraction expansion, and rational numbers always produce a finite continued fraction while irrational numbers produce an infinite one.

The standard form is written [a₀; a₁, a₂, a₃, …], which means:

a₀ + 1/(a₁ + 1/(a₂ + 1/(a₃ + …)))

All partial quotients aₙ (except possibly a₀) are positive integers. The integer part a₀ may be zero or negative, but every subsequent term must be a positive integer.

The continued fraction expansion of p/q is computed using the same idea as the greatest common divisor (GCD) algorithm:

1. Set a₀ = ⌊p/q⌋ (floor division). The remainder is r = p − a₀·q.
2. If r = 0, stop. The fraction is exact and the CF terminates.
3. Otherwise set p = q, q = r and repeat to find a₁, a₂, …

Example — 355/113:

• 355 = 3·113 + 16 → a₀ = 3
• 113 = 7·16 + 1 → a₁ = 7
• 16 = 16·1 + 0 → a₂ = 16, done

Result: 355/113 = [3; 7, 16]

For decimals, the calculator first converts the decimal to an exact rational number using the decimal-places method (multiply by 10^d, GCD-reduce), then applies the same algorithm.

The n-th convergent pₙ/qₙ is the best rational approximation to the original number using a denominator no larger than qₙ. Convergents are computed with the recurrence:

Each convergent alternately overshoots and undershoots the true value. The error column shows |pₙ/qₙ − x|, illustrating exponential convergence — each step typically gains multiple decimal digits of precision.

The continued fraction of π = [3; 7, 15, 1, 292, 1, 1, …] tells us something remarkable. After truncating at a₁ = 7 we get the convergent 22/7, which approximates π to 2.5 decimal places. Truncating at a₂ = 15 gives 333/106 (4 decimal places). Truncating at a₃ = 1 gives 355/113 — accurate to 6 decimal places despite a denominator of only 113.

The next coefficient a₄ = 292 is extraordinarily large. A large coefficient means the next convergent is a much better approximation — which is why 355/113 is so accurate and why there is no "nice" fraction that beats it until denominators reach the thousands.

This insight — that large partial quotients signal an especially good approximation just before them — is one of the key applications of continued fraction theory.