Continued Fraction Calculator
Convert decimals to continued fractions and find the best rational approximations (convergents).
Input
Continued Fraction
[3; 7, 15, 1, 288, 1, 2, 1, 3, 1]
This value is approximately π (pi)
Best Rational Approximation
1953857 / 621932
= 3.141592650000
Error: 3.2152e-13
Convergents
3/1error: 1.42e-1
22/7error: 1.26e-3
333/106error: 8.32e-5
355/113error: 2.70e-7
102573/32650error: 6.89e-10
102928/32763error: 2.46e-10
308429/98176error: 6.52e-11
411357/130939error: 1.26e-11
1542500/490993error: 2.95e-12
1953857/621932error: 3.22e-13
Terms
a0 = 3a1 = 7a2 = 15a3 = 1a4 = 288a5 = 1a6 = 2a7 = 1a8 = 3a9 = 1
Continued Fractions
Notation
x = a₀ + 1/(a₁ + 1/(a₂ + 1/(a₃ + ...)))
Written as [a₀; a₁, a₂, a₃, ...]
Properties
- Rational numbers have finite continued fractions
- Quadratic irrationals have periodic continued fractions
- Convergents are best rational approximations
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Editorial Note
MyCalcBuddy Editorial Team
This page is maintained as an educational calculator reference.
📚
Formula Source: Handbook of Mathematical Functions
by Abramowitz & Stegun
🔄Last reviewed: May 2026
✓Formula checks are based on standard references and internal QA review.