Continued Fraction Calculator

Welcome to the Continued Fraction Calculator — a powerful tool that converts any decimal number, fraction, or square root into its continued fraction representation. See the famous notation [a₀; a₁, a₂, ...], explore rational approximations (convergents), and visualize the nested fraction structure interactively.

What is a Continued Fraction?

A continued fraction is a way of expressing a number as a nested sequence of integer parts and fractions:

Where a₀, a₁, a₂, ... are non-negative integers called partial quotients. The standard notation is [a₀; a₁, a₂, a₃, ...]. Some remarkable examples:

• π (pi) ≈ [3; 7, 15, 1, 292, 1, 1, ...] — the 292 means pi is extremely well approximated by 355/113
• φ (golden ratio) = [1; 1, 1, 1, ...] — the slowest-converging continued fraction
• √2 = [1; 2, 2, 2, ...] — periodic, as predicted by Lagrange's theorem
• e = [2; 1, 2, 1, 1, 4, 1, 1, 6, ...] — beautiful pattern

How the Algorithm Works

For Any Decimal x

1. Compute a₀ = ⌊x⌋ (floor of x)
2. Set x₁ = 1/(x − a₀), then compute a₁ = ⌊x₁⌋
3. Repeat: xₙ₊₁ = 1/(xₙ − aₙ), aₙ₊₁ = ⌊xₙ₊₁⌋
4. Stop when the fractional part is zero (rational) or you have enough terms

For a Fraction p/q (Euclidean Algorithm)

For a fraction, the algorithm is identical to the Euclidean algorithm for GCD:

Each division step of the Euclidean algorithm produces one partial quotient of the continued fraction.

Convergents: Best Rational Approximations

The convergents pₙ/qₙ are obtained by truncating the continued fraction at each step. They satisfy a remarkable property: pₙ/qₙ is the best rational approximation to x with denominator ≤ qₙ.

Periodic Continued Fractions

By Lagrange's theorem, a real number has a periodic continued fraction if and only if it is a quadratic irrational (solution to a quadratic equation with integer coefficients). This includes all square roots of non-perfect-square integers.

• √2 = [1; 2] — period of length 1
• √3 = [1; 1, 2] — period of length 2
• √7 = [2; 1, 1, 1, 4] — period of length 4
• √94 = [9; 1, 2, 3, 1, 1, 5, 1, 8, 1, 5, 1, 1, 3, 2, 1, 18] — period of length 16

How to Use This Calculator

1. Enter a value: decimal (e.g. 2.71828), fraction (e.g. 355/113), or square root (e.g. sqrt(7))
2. Set maximum terms: more terms give more partial quotients and convergents
3. Click Calculate: see CF notation, animated terms, nested visualization, convergents table, and Euclidean steps (for fractions)

Frequently Asked Questions

What is a continued fraction?

A continued fraction is an expression of the form a₀ + 1/(a₁ + 1/(a₂ + ...)) where a₀, a₁, a₂, ... are integers called partial quotients. Every real number has a continued fraction expansion. Rational numbers have finite expansions; irrational numbers have infinite ones. Quadratic irrationals (like square roots) have periodic expansions.

How do you convert a decimal to a continued fraction?

Take the floor (integer part) as the first term. Subtract it from the number, take the reciprocal, and repeat. For example, π ≈ 3.14159...: floor = 3, remainder = 0.14159..., reciprocal = 7.062..., floor = 7, remainder = 0.062..., reciprocal = 15.996..., floor = 15, giving [3; 7, 15, ...].

Why does sqrt(2) have a periodic continued fraction?

By Lagrange's theorem, a real number has a periodic continued fraction exactly when it is a quadratic irrational. √2 satisfies x² = 2, so it is quadratic irrational, giving [1; 2, 2, 2, ...]. The golden ratio φ = (1 + √5)/2 gives [1; 1, 1, 1, ...] — the simplest possible period.

What are convergents and why are they important?

Convergents are the fractions obtained by truncating the continued fraction. They are the best rational approximations — no fraction with a smaller denominator is closer to the target number. This is why 22/7 and 355/113 are famous approximations to π: they are convergents of π's continued fraction.

How is the continued fraction algorithm related to the Euclidean algorithm?

When the input is a fraction p/q, computing its continued fraction is identical to the Euclidean GCD algorithm. Each remainder-and-quotient step produces exactly one partial quotient. The continued fraction terminates exactly when the GCD is found.

Additional Resources

• Continued Fraction - Wikipedia
• Convergents - Wikipedia
• Euclidean Algorithm - Wikipedia