Continued Fraction Calculator

How to Use This Calculator

1. Enter a numerator and denominator for p/q → CF conversion.
2. Use Decimal → CF tab to convert a decimal like 3.14159 to its CF.
3. Use CF → Rational to reconstruct a fraction from CF terms.
4. Use Professional to explore convergents and famous approximations.

Formula

Example

Frequently Asked Questions

• What is a continued fraction? ▼
  A continued fraction writes a number as a₀ + 1/(a₁ + 1/(a₂ + …)) where a₀ is an integer and all subsequent aᵢ are positive integers. Every rational number has a finite CF; irrationals have infinite CFs.
• What are convergents? ▼
  Convergents are the fractions obtained by truncating the CF. They are the best rational approximations — no fraction with a smaller denominator is closer to the original value. Example: 22/7 and 355/113 are convergents of π.
• Why is 355/113 such a good approximation of π? ▼
  355/113 ≈ 3.1415929, accurate to 6 decimal places. It is a convergent of π's CF [3;7,15,1,292,…]. The next term 292 is large, meaning the approximation is especially good.
• Which numbers have periodic continued fractions? ▼
  By Lagrange's theorem, a number has a periodic CF if and only if it is a quadratic irrational: √2=[1;2,2,2,…], golden ratio φ=[1;1,1,1,…], √3=[1;1,2,1,2,…].
• What are real-world applications of CFs? ▼
  Best gear ratios in mechanical engineering, musical scale tuning (5/4≈major third), calendar systems (leap years balance 365.2422…), and Diophantine approximation in number theory.

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