Continued Fraction Calculator
Expand a decimal into its continued fraction and read off the best rational approximations (convergents) to the value.
How it works
x = a₀ + 1/(a₁ + 1/(a₂ + …)) where aₙ = ⌊xₙ⌋ and xₙ₊₁ = 1/(xₙ − aₙ) · convergents hₙ/kₙ via hₙ = aₙhₙ₋₁ + hₙ₋₂
A continued fraction expands a number as an integer plus one over another continued fraction, peeling off the whole part and inverting the remainder at each step. Truncating the expansion gives the convergents — the best possible rational approximations for their size, which is how 22/7 and 355/113 arise for π. The list of coefficients [a₀; a₁, a₂, …] is the fingerprint of the number.
Worked example
For 3.14159265, the coefficients start [3; 7, 15, 1, …]. Truncating after 3 and 7 gives 22/7 ≈ 3.142857; carrying further to [3; 7, 15, 1] yields 355/113 ≈ 3.1415929, accurate to six decimals.
Sources: Continued fraction — Wikipedia
Frequently asked questions
What is a convergent?
A convergent is the fraction you get by cutting the continued fraction off after a few terms. Each convergent is the closest rational approximation to the value for any denominator that size or smaller.
Why is the result only approximate?
A finite decimal is expanded to a limited number of terms, so the last convergent is a very close rational — but for irrational numbers like π no finite fraction is exact. The reported error shows how close it lands.
How do I read the bracket notation?
In [a₀; a₁, a₂, …], a₀ is the whole-number part and each following term is a denominator in the nested fraction. [3; 7] means 3 + 1/7 = 22/7.
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This calculator is for educational purposes. Double-check important results before acting on them.