Continued Fractions
Convert decimals to continued fractions and find best rational approximations. Useful for number theory puzzles.
Famous Continued Fractions
Notation
[a₀; a₁, a₂, a₃, ...] means: a₀ + 1/(a₁ + 1/(a₂ + 1/(a₃ + ...)))
Example: [3; 7, 15, 1] = 3 + 1/(7 + 1/(15 + 1/1)) = 355/113 ≈ π
What are Continued Fractions?
A continued fraction represents a number as a sequence of integers in a nested fraction form. Any real number can be written as a continued fraction, and rational numbers have finite continued fractions.
The Notation
A continued fraction [a₀; a₁, a₂, a₃, ...] represents:
a₀ + 1/(a₁ + 1/(a₂ + 1/(a₃ + ...)))
Convergents
Convergents are the best rational approximations you get by truncating the continued fraction. Each convergent is closer to the target number than any other fraction with a smaller denominator.
Famous Examples
- Golden ratio φ: [1; 1, 1, 1, ...] - all ones, the "most irrational"
- π: [3; 7, 15, 1, 292, ...] - gives 22/7 and 355/113
- √2: [1; 2, 2, 2, ...] - periodic after first term
- e: [2; 1, 2, 1, 1, 4, 1, 1, 6, ...] - interesting pattern
Continued Fractions in Puzzles
Continued fractions appear in puzzles involving:
- Number theory: Finding rational approximations
- Golden ratio: Fibonacci connections
- Periodic patterns: Square roots have repeating CF
- Historical math: Ancient approximations to π
Properties
- Rational numbers have finite continued fractions
- Quadratic irrationals (like √2) have periodic continued fractions
- The golden ratio has the simplest infinite continued fraction
- Convergents alternate between over and under estimates