Continued fractions: a₀ + 1/(a₁ + 1/(a₂ + ...)). Every rational number has a finite CF; irrationals have infinite CFs. √2 = [1; 2, 2, 2, ...] (periodic). φ = [1; 1, 1, 1, ...] (simplest irrational). Convergents pₙ/qₙ are the best rational approximations.
How to Use
Enter a number:
- Input: Decimal or fraction
- Output: CF coefficients + convergents
Convergents
Each convergent pₙ/qₙ is the closest rational with denominator ≤ qₙ. They alternate above/below the target. π: 3, 22/7, 333/106, 355/113...
Famous CFs
- φ = [1; 1, 1, 1, ...] — worst approximable
- √2 = [1; 2, 2, 2, ...]
- e = [2; 1, 2, 1, 1, 4, 1, 1, 6, ...]
- π = [3; 7, 15, 1, 292, ...]
Step-by-Step Instructions
- 1Enter a decimal number.
- 2View CF coefficients.
- 3Check convergents.
- 4Find best approximation.
- 5Explore periodic patterns.