1. Introduction
Continued fractions are a way of writing real numbers as a sequence of integers. They provide the best possible rational approximations of irrational numbers — a property that makes them fundamental in number theory, celestial mechanics and cryptography.
2. Definition
A continued fraction is an expression of the form:
Compact notation: x = [a₀; a₁, a₂, a₃, ...] where the aᵢ are positive integers.
Every real number has a continued fraction representation. Rational numbers have a finite representation. Irrational numbers have an infinite representation. Quadratic irrational numbers (like √2 and φ) have a periodic representation.
3. Convergents
The convergents of a continued fraction are the rational approximations obtained by truncating the continued fraction. The n-th convergent is p_n/q_n, where p_n and q_n satisfy the recurrences:
Hurwitz's theorem guarantees that convergents are the best rational approximations: no fraction p/q with q ≤ q_n approximates x better than p_n/q_n.
4. The golden ratio
The golden ratio φ = (1 + √5)/2 has the simplest possible continued fraction: φ = [1; 1, 1, 1, ...]. All its coefficients equal 1. This property makes φ the irrational number most difficult to approximate by rationals — it is the "most irrational" number.
The convergents of φ are exactly the ratios of consecutive Fibonacci terms: 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, ... This link between continued fractions and the Fibonacci sequence explains why φ appears in phyllotaxis.
5. Geometric representation
The convergence of continued fractions toward an irrational number can be represented geometrically. Successive convergents alternate above and below the target value, approaching in a spiral. This representation visually illustrates the optimal approximation property of convergents.
6. Conclusion
Continued fractions reveal the deep arithmetic structure of real numbers. They provide the best rational approximations, connect the Fibonacci sequence to the golden ratio, and have practical applications in cryptography and celestial mechanics. The spiral convergence of approximants illustrates how arithmetic and geometry meet in this domain.