Continued Fractions Calculator

Definition and Notation

A continued fraction represents a real number x as: x = a_0 + \cfrac1a_1 + \cfrac1a_2 + \cfrac1a_3 + ·s where a_0 is an integer (the integer part) and a_1, a_2, ... are positive integers. Compact notation: x = [a_0; a_1, a_2, a_3, ...]. For finite continued fractions (rational numbers), x = [a_0; a_1, ..., a_n]. For infinite continued fractions (irrational numbers), the sequence continues forever. The algorithm to compute continued fraction representation is the Euclidean algorithm: take integer part, invert remainder, repeat. Every rational number has a finite continued fraction; every irrational has an infinite one. Periodic continued fractions (eventually repeating pattern) represent quadratic irrationals like √(2) = [1; 2, 2, 2, ...].

x = a_0 + \cfrac1a_1 + \cfrac1a_2 + \cfrac1a_3 + ·s

Computing Continued Fractions

The algorithm for finding the continued fraction of x is straightforward. Step 1: a_0 = \lfloor x \rfloor (integer part). Step 2: If x = a_0 (integer), stop. Otherwise, compute r_0 = x - a_0 (fractional part). Step 3: Set x_1 = 1/r_0 and a_1 = \lfloor x_1 \rfloor. Step 4: Repeat with x_2 = 1/(x_1 - a_1), and so on. For example, π ≈ 3.14159: a_0 = 3, x_1 = 1/0.14159 = 7.063, a_1 = 7, x_2 = 1/0.063 = 15.996, a_2 = 15, giving π ≈ [3; 7, 15, 1, ...]. This algorithm terminates for rational numbers and continues indefinitely for irrationals. The connection to the Euclidean algorithm is profound: finding \gcd(a,b) uses the same sequence of divisions.

a_n = \lfloor x_n \rfloor, \quad x_n+1 = (1)/(x_n - a_n)

Convergents and Best Approximations

The n-th convergent is the rational number obtained by truncating the continued fraction after n terms: C_n = [a_0; a_1, ..., a_n] = (p_n)/(q_n) Convergents can be computed recursively: p_-1=1, p_0=a_0, and p_n = a_n p_n-1 + p_n-2. Similarly for denominators: q_-1=0, q_0=1, and q_n = a_n q_n-1 + q_n-2. Remarkably, convergents are the best rational approximations in a precise sense: if |x - p/q| < |x - p_n/q_n| for some p/q, then q > q_n. In other words, convergents give the most accurate approximation for a given denominator size. This makes them optimal for applications like fraction approximations, musical intervals, gear ratios, and calendar systems (e.g., 365/366 day year fractions).

p_n = a_n p_n-1 + p_n-2, \quad q_n = a_n q_n-1 + q_n-2

Periodic Continued Fractions

A continued fraction is eventually periodic if its sequence of partial quotients repeats from some point onward: x = [a_0; a_1, ..., a_k, \overlineb_1, ..., b_m] where the overline indicates repetition. A fundamental theorem states that x has an eventually periodic continued fraction if and only if x is a quadratic irrational (solution to ax^2 + bx + c = 0 with integer coefficients and x irrational). For example, √(2) = [1; \overline2] (purely periodic), √(7) = [2; \overline1,1,1,4], and the golden ratio \phi = [1; \overline1] (the simplest infinite continued fraction). This characterization connects continued fractions deeply to algebraic number theory and Pell's equation.

√(2) = [1; \overline2], \quad \phi = [1; \overline1]

Properties and Inequalities

Continued fractions satisfy beautiful inequalities. Convergents alternate around the true value: C_0 < C_2 < C_4 < ·s < x < ·s < C_5 < C_3 < C_1 (for x > 0). The error in approximation satisfies: (1)/(q_n(q_n+1) + q_n) < |x - (p_n)/(q_n)| < (1)/(q_n q_n+1) This bounds how good the approximation is. Consecutive convergents satisfy: p_n q_n-1 - p_n-1q_n = (-1)^n-1, implying \gcd(p_n, q_n) = 1 (convergents are always in lowest terms). These properties make continued fractions powerful for proving irrationality, finding Diophantine approximations, and analyzing dynamical systems.

(1)/(q_n(q_n+1) + q_n) < |x - (p_n)/(q_n)| < (1)/(q_n q_n+1)

Golden Ratio and Fibonacci Connection

The golden ratio \phi = (1+√(5))/2 has the simplest infinite continued fraction: \phi = [1; 1, 1, 1, 1, ...] = 1 + \cfrac11 + \cfrac11 + \cfrac11 + ·s Its convergents C_n = F_n+1/F_n are ratios of consecutive Fibonacci numbers: 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, ..., which approach \phi as n \to ∞. This connection explains why Fibonacci ratios appear in nature (phyllotaxis, spiral patterns)—they provide optimal packing. The slow convergence of \phi (all partial quotients are 1, the smallest possible) makes it the "most irrational" number, hardest to approximate by rationals. This has applications in dynamical systems: perturbations with frequencies in golden ratio avoid resonances, explaining its appearance in planetary orbits and atomic spectra.

\phi = [1; \overline1], \quad C_n = \fracF_n+1F_n \to \phi

Applications and Computational Uses

Continued fractions have numerous applications. In computer arithmetic, they provide exact fraction representation (unlike floating point). Calendar design uses continued fractions: the year length 365.2422 days = [365; 4, 7, 1, 3, 4, ...] gives the Gregorian calendar (convergent 365 + 97/400 = 365.2425 days, accurate to 1 day/3300 years). Gear ratio design uses convergents for fractional speed ratios with small gears. Numerical analysis uses continued fractions to evaluate functions and solve equations (Gauss continued fractions for hypergeometric functions). In quantum chaos, continued fraction expansion of rotation numbers characterizes quasiperiodic motion. The Lenstra elliptic curve factorization algorithm uses continued fractions for integer factorization. They appear in moment problem solutions, Padé approximants, and orthogonal polynomials.

\textApplications: fraction arithmetic, calendars, gears, numerical analysis, chaos theory

Worked Example

Inputs

• numberInput: 3.14159
• maxTerms: 5

Result: [3; 7, 15, 1, 292] ≈ 355/113

Explanation

π ≈ 3.14159 has continued fraction [3;7,15,1,292,...]; convergent 355/113 is accurate to 6 decimal places

Golden Ratio

Inputs

• numberInput: 1.618034
• maxTerms: 8

Result: [1; 1, 1, 1, 1, 1, 1, 1]

Explanation

φ = (1+√5)/2 has all partial quotients equal to 1, the simplest infinite continued fraction