What is a continued fraction?

A continued fraction writes a number as a0 + 1/(a1 + 1/(a2 + ...)), where each ai (called a partial quotient) is a non-negative integer. Every real number has a unique continued fraction representation; rational numbers terminate, while irrationals go on forever.

What are convergents?

Convergents are the rational numbers p_n/q_n obtained by truncating the continued fraction after n terms. They are provably the best rational approximations to the original value for any denominator up to q_n.

Why does the continued fraction for a rational number terminate?

For a rational number p/q, the process is identical to the Euclidean algorithm on p and q, which terminates in finitely many steps.

How accurate are the convergents?

Each convergent p/q satisfies |x − p/q| < 1/q², which is better than any other fraction with a denominator ≤ q. Convergents improve rapidly — just a few terms typically give very tight approximations.

What is the precision limit?

JavaScript's 64-bit floating-point arithmetic limits meaningful terms to roughly 15–17 significant digits. After that, rounding errors in the remainder cause spurious partial quotients.

Continued Fraction Calculator

Enter a decimal number to see its continued fraction [a0; a1, a2, ...] and the best rational approximations at each convergent.

Input

Decimal number Max terms

Examples

Convergents (rational approximations)

n	a_n	Convergent p/q	Decimal value	Error

Enter a number above and click Compute.

Copied!

Summary

A continued fraction expresses a number as a0 + 1/(a1 + 1/(a2 + ...)), where each ai is a non-negative integer called a partial quotient. This calculator computes the partial quotients for any decimal input and lists the convergents — the sequence of rational numbers that get progressively closer to the original value. Convergents are the best rational approximations for a given denominator size, making them useful in gear ratio design, music theory, and approximating irrational numbers like π and φ.

How it works

Enter a decimal number (e.g. 3.14159 or 1.41421).
Choose how many terms (partial quotients) to compute, up to 20.
The calculator repeatedly takes the integer part as the next partial quotient, then inverts the remainder.
Each row in the convergents table shows the running rational approximation p/q.
Copy the notation or any convergent with the copy buttons.

Use cases

Approximate π, φ (golden ratio), √2, or any irrational constant as a simple fraction.
Find the closest gear ratio from a limited set of tooth counts.
Music theory: derive just-intonation intervals from frequency ratios.
Cryptography: continued fractions underlie the Wiener attack on RSA.
Calendar design: understand why a solar year is approximated as 365 + 1/4 − 1/100 + 1/400.
Classroom demonstrations of number theory and Euclidean algorithm connections.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu