Enter a Number
[3; 7, 15, 1, 292, 1, 1, 1, 2, ...]
Convergents (Best Rational Approximations)
| n | aₙ | pₙ/qₙ | Error |
|---|
Stern-Brocot Tree Path
Approximation Quality
Continued Fractions
Every real number x has a unique expansion: x = a₀ + 1/(a₁ + 1/(a₂ + ...)) written [a₀; a₁, a₂, ...]
Rationals terminate. Quadratic irrationals (like φ, √2) are eventually periodic. Transcendentals like π, e have irregular patterns.
Golden ratio φ = [1;1,1,1,...] — all ones, hardest number to approximate rationally (slowest convergents).
π ≈ 355/113 — the convergent [3;7,15,1] is accurate to 7 decimal places. The next partial quotient 292 means this is an especially good approximation.
The Stern-Brocot tree contains every rational exactly once. The path to any rational is encoded in its continued fraction.