Continued Fraction Calculator – Decimal to Fraction

A continued fraction peels off the integer part, inverts the remainder, and repeats. For 5/8 = 0.625: integer part 0, invert 8/5 = 1.6 → a₁ = 1, remainder 3/5 → invert 5/3 → a₂ = 1, remainder 2/3 → invert 3/2 → a₃ = 1, remainder 1/2 → a₄ = 2. So 5/8 = [0; 1, 1, 1, 2]. Truncating the expansion gives convergents — best-possible rational approximations at each size: [0;1] = 1, [0;1,1] = 1/2, [0;1,1,1] = 2/3, and the full value 5/8. Irrationals yield infinite patterns: √2 = [1; 2, 2, 2, …] with convergents 1, 3/2, 7/5, 17/12 (17/12 ≈ 1.41667), and the golden ratio is [1; 1, 1, …] forever — worst-case for rational approximation. The calculator computes coefficients from any decimal or fraction and tabulates convergents with their errors.

1. 1

   Enter a decimal or fraction

   e.g. 0.625 or 5/8

2. 2

   Peel and invert

   Integer parts a₀, a₁, … are extracted step by step

3. 3

   Build the convergents

   Each truncation [a₀;…aₖ] becomes a fraction

4. 4

   Watch errors shrink

   Convergent errors drop faster than any comparable denominator

• 1

  Convergents are the best rational approximations by denominator size

• 2

  Terminating decimals give terminating expansions; irrationals never do

• 3

  The golden ratio converges slowest of all — all coefficients are 1

• Forgetting the leading integer part a₀ in the bracket notation

• Stopping the peel before the remainder hits exactly zero for rationals

• Reading convergents as exact values; 2/3 ≈ 0.6667 ≠ 0.625

• Confusing simple with generalized (non-unit numerators) continued fractions