Number
| n | aₙ | pₙ | qₙ | error |
|---|
Continued fraction: α = a₀ + 1/(a₁ + 1/(a₂ + ···))
Convergents: p_{n}/q_{n} satisfies |α − pₙ/qₙ| < 1/qₙqₙ₊₁. They are the best rational approximations: no fraction with denominator ≤ qₙ is closer to α.
φ: All partial quotients = 1 → slowest converging CF (most irrational). Error ≈ 1/φⁿ.
π: Large partial quotient a₂=292 means 355/113 is exceptionally good (Milü).
Top chart: log₁₀|α − pₙ/qₙ| vs n — roughly linear (exponential convergence).
Bottom chart: convergents on number line, alternating above/below α.
Convergents: p_{n}/q_{n} satisfies |α − pₙ/qₙ| < 1/qₙqₙ₊₁. They are the best rational approximations: no fraction with denominator ≤ qₙ is closer to α.
φ: All partial quotients = 1 → slowest converging CF (most irrational). Error ≈ 1/φⁿ.
π: Large partial quotient a₂=292 means 355/113 is exceptionally good (Milü).
Top chart: log₁₀|α − pₙ/qₙ| vs n — roughly linear (exponential convergence).
Bottom chart: convergents on number line, alternating above/below α.