Continued Fraction Converter
Turn a decimal or fraction into a continued fraction and read off convergents — the best short rational approximations.
Continued Fraction Converter
Convert between decimal numbers, fractions, and continued fraction representations. Explore the beautiful mathematical structure of continued fractions.
Quick Examples
Advanced Settings
What you get
A continued fraction writes a number as nested integer quotients: a₀ + 1/(a₁ + 1/(a₂ + …)). Truncating that list gives convergents — fractions that are unusually accurate for their denominator size.
- π ≈ [3; 7, 15, 1, 292, …] — early convergents include 22/7 and 355/113
- √2 = [1; 2, 2, 2, …] — periodic because √2 is a quadratic irrational
- φ (golden ratio) = [1; 1, 1, 1, …] — all 1s
Engineers use the same idea for gear ratios and other "simple fraction close to this real number" problems.
How to use the converter
- Enter a decimal, a fraction, or pick a sample constant
- Read the partial quotients and the convergent table
- Stop when the error is small enough for your use case
Frequently Asked Questions
Why not just use decimals?
Decimals are fine for display. Continued fractions spotlight best rational approximations and often show clean patterns that long decimals hide.
How many terms do I need?
A few convergents are often enough for everyday accuracy. Add more only if you need tighter error.
Why do some expansions repeat?
Periodic continued fractions are exactly the quadratic irrationals (numbers like (a + √b)/c). That's why √2 is all 2s after the first term.
Any real-world uses?
Gear teeth, musical temperament ratios, calendar approximations, and solving some Diophantine equations (including Pell-type problems).
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