Best Rational Approximation Calculator
Find the best fraction approximation for any decimal number using continued fractions.
Shows the top 5 rational approximations within a denominator limit.
Rational Approximation
Every irrational number (like π or √2) can be approximated by a fraction p/q. The challenge is finding the best fraction — one that gives the smallest error for the smallest denominator.
Continued Fractions
Any real number can be written as a continued fraction: x = a₀ + 1/(a₁ + 1/(a₂ + 1/(a₃ + …)))
Written as [a₀; a₁, a₂, a₃, …]
The convergents (truncations of this expansion) give the best rational approximations — better than any other fraction with a smaller denominator.
Famous Examples
| Number | Decimal | Best Fraction | Error |
|---|---|---|---|
| π | 3.14159… | 355/113 | 0.000003% |
| π | 3.14159… | 22/7 | 0.04% |
| e | 2.71828… | 87/32 | 0.001% |
| √2 | 1.41421… | 99/70 | 0.0001% |
| φ (golden ratio) | 1.61803… | 89/55 | 0.0002% |
Why 355/113 is remarkable
355/113 = 3.1415929… — it matches π to 6 decimal places. Yet the next best fraction below denominator 113 is only 22/7 (accurate to 2 decimals). The Chinese mathematician Zu Chongzhi discovered 355/113 around 480 AD.
The Stern-Brocot Tree
Every positive fraction appears exactly once in this infinite binary tree. Traversing it efficiently finds the best approximation to any number.