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.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
SuperGlobalCalculator is independently built and maintained. See how we build and verify our calculators.