Continued Fraction Converter - Best Rational Approximations

How to use this tool

1. Enter a number. Type a decimal (e.g. 3.14159) or a fraction (e.g. 355/113) into the input box. Decimal digits are treated as an exact base-10 rational.
2. Click Convert. Press Convert to compute the continued fraction expansion.
3. Read the CF notation. The result is shown as [a₀; a₁, a₂, …]. Each coefficient aᵢ is a positive integer (a₀ may be any integer).
4. Explore convergents. The convergents table shows the best rational approximations from coarse (small denominator) to fine (large denominator), with their decimal error.

What is a continued fraction?

A continued fraction is an expression of the form a₀ + 1/(a₁ + 1/(a₂ + 1/(a₃ + …))), written compactly as [a₀; a₁, a₂, …]. Every real number has a continued fraction expansion, and every rational number has a finite one.

When you enter a decimal, this tool reads its written digits as an exact base-10 rational rather than converting it to a floating-point value first. For example, 0.0000001 is normalized exactly to 1/10000000. The displayed decimal columns are approximations for readability; the normalized fraction and continued-fraction arithmetic use BigInt integers. Inputs are capped at 1,000 digits and expansions at 20 terms to keep the calculation practical in the browser.

Famous continued fractions

These patterns describe the exact mathematical constants. The π, √2, φ, and e buttons enter 50-digit finite decimal approximations, which are rational numbers; this tool displays at most 20 terms. Their prefixes match the familiar expansions at that precision, but a truncated decimal is not the symbolic constant itself.

• π = [3; 7, 15, 1, 292, 1, 1, 1, 2, …]
• e = [2; 1, 2, 1, 1, 4, 1, 1, 6, …] (beautiful pattern!)
• √2 = [1; 2, 2, 2, 2, …] (periodic)
• φ (golden ratio) = [1; 1, 1, 1, …] (all ones: "worst" approximable number)
• 355/113 = [3; 7, 16]: an unusually precise approximation of π

A typed decimal is also interpreted as an exact base-10 rational. If its expansion does not finish within 20 terms, the result says it was truncated at that limit; it does not mean the entered decimal is irrational.

Why are convergents the best approximations?

A convergent pₙ/qₙ is closer to the target than any other fraction with denominator ≤ qₙ. This makes continued fractions the theoretically optimal tool for rational approximation: it's important in gear design, music theory (equal temperament), and calendar design (when is a year divisible by a simple fraction?).

The significance of large coefficients

A large coefficient aₖ (like 292 in π's expansion) means the previous convergent is an exceptionally good approximation. The fraction 355/113 is so accurate because the next coefficient is 292, meaning the next improvement requires a denominator roughly 292 times larger.

Connection to the Euclidean algorithm

The continued fraction expansion of a rational number a/b (where a > b) produces exactly the same sequence of quotients as running the Euclidean GCD algorithm on a and b. For example, to find gcd(34, 21):

34 = 1×21 + 13  -> quotient 1
21 = 1×13 +  8  -> quotient 1
13 = 1× 8 +  5  -> quotient 1
 8 = 1× 5 +  3  -> quotient 1
 5 = 1× 3 +  2  -> quotient 1
 3 = 1× 2 +  1  -> quotient 1
 2 = 2× 1 +  0  -> quotient 2

So 34/21 = [1; 1, 1, 1, 1, 1, 2] in continued fraction notation - the Fibonacci quotients, which is why consecutive Fibonacci numbers are the "worst case" for the Euclidean algorithm (requiring the most steps for their size). This deep connection between continued fractions and the GCD algorithm was known to Euler.