Continued Fraction Calculator — Expansion + Convergents

Real-world use cases

• Find the simplest fraction that approximates a decimal

  You measured a ratio as 1.61803 and want a clean fraction for a CAD drawing or a gear train. Expand it and the convergents hand you 13/8, 21/13, 34/21 in rising accuracy. Pick the one whose denominator your shop can cut, and you have the tightest possible fraction for that tooth count, not a denominator you picked at random.

• Reproduce the classic pi approximations for a lesson

  Teaching where 22/7 and 355/113 come from? Type 3.14159265, watch the expansion start [3;7,15,1], and the convergent table prints 22/7, 333/106 and 355/113 with the error against pi shrinking from 1e-3 to 1e-7 row by row. The signed error column makes the jump from 333/106 to 355/113 land as a concrete number, not hand-waving.

• Build gear ratios and clock trains

  Horologists and machinists need two gears whose tooth ratio matches a target like the sidereal day factor 1.0027379. The convergents give ratios of small integers you can actually manufacture — 366/365 is right there — so you trade a tiny accuracy loss for a buildable pair instead of chasing a 7-digit ratio no gear can hold.

• Sanity-check a continued fraction by hand

  You ran the Euclidean algorithm on paper and got [4;2,6,7] for 415/93. Paste 415/93 here, confirm the coefficients match, and read the convergents 4/1, 9/2, 58/13, 415/93 to verify the last one folds back exactly to your starting fraction — the built-in proof that your hand expansion was right.

Common pitfalls

• Writing the bracket with a comma after the integer part. The convention puts a semicolon there, giving [3;7,15] rather than [3,7,15]. The semicolon flags that 3 is the whole-number part and 7, 15 are the reciprocal layers. Tools and textbooks that use a comma everywhere are still correct in value, but the semicolon form is what you will see in number-theory references.

• Expecting a finite expansion from a decimal that only approximates an irrational. Feeding 3.14159265 gives a finite continued fraction because you typed a rational truncation of pi, not pi itself. The deep coefficients reflect your digits, not pi, so do not read the tail of a truncated-decimal expansion as the true continued fraction of the irrational.

• Confusing the convergent with the next coefficient. A coefficient like 15 is one term of the expansion; the convergent 333/106 is the fraction you get by truncating there. Reading the coefficient list as if it were the approximations gives nonsense — the approximations live in the convergent column, built by the recurrence, not in the bracket.

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