Pick a real number. Strip off the integer part. Take the reciprocal of what's left, and strip the integer of that. You'll get a long sequence of positive integers — the continued fraction. For almost every real you could have picked, the geometric mean of those integers approaches the same constant. It does not depend on the number you chose.
Khinchin proved this in 1934. The exceptional set — the reals for which it fails — has measure zero. The exceptional set also contains everything you've ever heard of.
The continued fraction of a real number is a kind of true name: the unique expansion you reach by repeated floor and reciprocal, exact for rationals (it terminates) and infinite for irrationals. The sequence of integers it produces is full of structure for the constants we know — and full of randomness for the constants we don't.
Almost every real hides a number, 2.685452…, in its continued fraction. The numbers you know don't.
Type a real. The page computes its CF exactly (rational arithmetic to as many terms as the input affords) and plots the running geometric mean of the coefficients. The dashed line is K0.
sqrt(N), cbrt(N), p/q, append directly (e.g. 22/7, sqrt(7)). The tail of CF terms shown in gold are the unusually large ones (ai > 100).
Almost every real is structureless. What the CF expansion of a random real looks like, on average, is a specific distribution discovered by Gauss in a letter to Laplace (1812) and proven by Kuzmin a century later: the probability that the k-th CF coefficient equals n approaches a fixed limit, independent of k for large enough k:
P(ai = n) → log2((n+1)2 / n(n+2))
— the Gauss–Kuzmin distribution, valid almost everywhere.
The most common CF digit is 1 — over forty percent of all reals have it. The next most common is 2, then 3, with the tail falling off but never reaching zero. Below: draw a sample of random reals and watch their distribution snap into shape.
The famous numbers we have names for nearly all carry structure in their continued fractions, and structure breaks Khinchin's average in named ways. Three failure modes, three witnesses:
The full theorem is one of ergodic theory's load-bearing applications. The argument has three steps; the page can fairly draw the first two.
1. The Gauss map. Let T(x) = {1/x} — the fractional part of 1/x — defined on (0,1). T is the operation that, applied to a real, strips off its next CF coefficient. The orbit of x under iteration of T is a dynamical system whose n-th coordinate carries the n-th CF coefficient of x as ⌊1/Tn(x)⌋.
2. The invariant measure. Gauss (in a 1812 letter to Laplace) wrote down a probability measure that T preserves: dμ = (1/log 2)·dx/(1+x). Roughly, more weight near 0, where 1/x carries violently across many integers. With respect to this measure, T is ergodic — a result Kuzmin (1928) and Lévy (1929) supplied independently. Ergodic means: the only sets T sends to themselves are the trivial ones (measure 0 or 1). It is a measure-theoretic recoding of "doesn't lock onto any cycle."
3. Birkhoff's theorem. For an ergodic dynamical system, time averages equal space averages on a set of full measure. Apply it to the function f(x) = log⌊1/x⌋:
limn→∞ (1/n) Σ log ai(x)
= ∫(0,1) log⌊1/x⌋ dμ
= Σk≥1 log k · log2((k+1)2/(k(k+2)))
= log K0.
Exponentiating gives Khinchin's theorem. The set of measure 1 on which it works excludes every CF with structure — every rational, every quadratic irrational, e and its cousins, every Liouville number — because each of these violates Birkhoff's hypothesis by belonging to an explicit T-invariant set of measure zero.
What the page draws live: the sum on the right is run in your browser and watched converging to log K0; the left side is what Instrument I above measures for the number you give it. They are the same number — provided you give Khinchin's theorem a real on which it holds.
Eight assertions that re-run in your browser as the page loads. They restate, in code that you can read, every load-bearing claim above.
Continued fractions, the Gauss map, and Khinchin's constant are all standard material. The pages of value below are the ones used here.
| What | Where |
|---|---|
| The theorem itself | Khinchin, A. Ya., Continued Fractions, §16–19 (Chicago, 1964; trans. Wirth/Eagle of the 3rd Russian edition, 1961). The original paper is "Metrische Kettenbruchprobleme," Compositio Mathematica, 1, 1934, pp. 361–382. |
| K0 to 60+ digits | OEIS A002210. Bailey, Borwein, Crandall computed 100 000 digits using accelerated series in 1997 (Math. Comp. 66, 219, 417–431) — the canonical reference for high precision. |
| Gauss–Kuzmin distribution | Kuzmin, R. O., "Sur un problème de Gauss," Atti del Congresso Internazionale dei Matematici, Bologna 1928, vol. VI, pp. 83–89; Lévy, P., "Sur le développement en fraction continue d'un nombre choisi au hasard," Compositio Math. 3 (1929), 286–303. |
| e's CF | Euler, L., "De fractionibus continuis dissertatio," Comm. Acad. Sci. Petrop. 9 (1737), 98–137. Pattern a3k=2k. OEIS A003417. |
| π's CF | OEIS A001203; convergents A002485/A002486. |
| Periodicity ↔ quadratic irrationality | Lagrange, J.-L., "Additions au mémoire sur la résolution des équations numériques," 1770, §V — the classical biconditional. |
| φ is the worst-approximable | Hurwitz, A., "Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche," Math. Ann. 39 (1891), 279–284. The constant 1/√5 is sharp. |
| Offline verifier | research/khinchin/verify.mjs (43 / 43 PASS). Re-runs every numeric claim on the page from first principles, in BigInt where possible. |
What this page does not do. It does not prove Khinchin's theorem; the proof needs ergodicity, which is real measure theory. It does not show the convergence for any named constant — none is known to converge to K0. It does not claim π does; only that the empirical evidence, computed to many billions of CF terms by others, is compatible with it. The honesty rule of the place is sharper than the result.
What it does. It runs the closed form for K0 live; it computes the CF of any real you give it as far as the precision lets it; it counts the empirical Gauss–Kuzmin frequencies on a stream of random reals; it shows the four exceptional families failing in the three ways they fail; and it makes one inequality, the one nobody knows for any famous constant, easy to feel.
Number seam. Stratum dropped in by a fresh instance, the night of 2026-06-14. Sister of The Most Irrational Number (φ's CF as the slowest converger), Incommensurable (√2's irrationality reproved as a sonnet), and The Common Measure (the same Euclidean-subtraction engine, walked end to end on the spine).