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Continued fractions: theory, convergents and best approximations

Continued fractions provide a powerful and classical way to represent real numbers. Unlike decimal expansions, continued fractions often reveal inherent Diophantine structure and produce the best rational approximations for their length. In this article we explore simple continued fractions, how they are computed, connections to the Euclidean algorithm, and why truncations (convergents) are often the optimal rational approximations to irrational numbers.

Simple continued fractions and convergents

A simple continued fraction has integer partial quotients and the nested form [a0; a1, a2, ...] = a0 + 1/(a1 + 1/(a2 + ...)). Convergents are rational approximations obtained by truncating the continued fraction after k terms. If p_k/q_k is the kth convergent, they satisfy recurrence relations p_k = a_k p_{k−1} + p_{k−2}, q_k = a_k q_{k−1} + q_{k−2} with initial values p_{−2}=0, p_{−1}=1, q_{−2}=1, q_{−1}=0.

Euclidean algorithm connection

For rational numbers p/q, performing the Euclidean algorithm (repeated integer division to find quotients and remainders) directly yields the partial quotients of the continued fraction. This close link explains why continued fractions terminate for rationals: the Euclidean algorithm terminates when the remainder becomes zero.

Why convergents are best approximations

One of the key theorems is that convergents give the best approximations in the sense that any rational r with denominator ≤ q_k cannot approximate the target number better than p_k/q_k. This property makes continued fractions fundamental in Diophantine approximation and helps explain the exceptional quality of approximations like 355/113 for π.

Periodic continued fractions and quadratic irrationals

Quadratic irrational numbers (roots of quadratic equations with integer coefficients) have eventually periodic continued fraction expansions. For example, √2 = [1; 2,2,2,...]. This periodicity ties directly into number theory and Pell equations.

Computation and rounding

Converting decimals to continued fractions requires deciding when to stop for irrational inputs; we allow a maximum number of terms to produce truncated expansions. Rational inputs are converted exactly by using fraction parsing and the Euclidean algorithm. Numeric stability is generally good because operations are integer-based for rationals; for decimal inputs use enough precision to avoid early truncation artifacts.

Applications

Continued fractions appear in approximation theory, cryptography (Lattice attacks use continued fraction approximations), computational number theory, and dynamical systems. They also serve as analytic tools: the continued fraction of e and π reveal interesting numerical structures and provide rapidly converging approximations.

Use this calculator to experiment with conversions, study Euclidean steps for rational inputs, and generate convergents for rational approximation problems.