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. It contains φ, √2, and e. For π, convergence to this average remains unproved.
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. Some familiar examples have simple patterns: all 1s for φ, all 2s after the integer part for √2, and a repeating rule for e.
Almost every real hides a number, 2.685452…, in its continued fraction. Some familiar numbers miss it; for π, we do not know.
Type a decimal or fraction. The page computes up to 60 CF coefficients of that rational input exactly and plots an approximate running geometric mean. For an intended irrational, agreement with its CF is limited by the supplied precision. 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: for almost every real, its limiting frequency among the fractional coefficients is log2(4/3), about 41.5 percent. 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.
These familiar examples have explicit CF patterns and miss Khinchin's average: φ, √2, and e. Their means are 1, 2, and unbounded. The fourth card, π, is still open:
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−1(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. Ergodic means: the only sets T sends to themselves are the trivial ones (measure 0 or 1). This does not mean "doesn't lock onto any cycle." Periodic orbits can coexist with ergodicity.
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 theorem alone does not decide any specified input. Rationals terminate; φ and √2 have means 1 and 2; e's mean diverges. Those conclusions need their own arguments. Structure alone does not force failure, and belonging to an invariant null set violates no hypothesis of Birkhoff's theorem.
What the page draws live: the closed-form widget displays the exponential of a partial sum on the right, approaching K0; Instrument I displays the exponential of the finite log-average on the left. Their limits are the same number, provided you give Khinchin's theorem a real on which it holds.
Nine assertions that re-run in your browser as the page loads. They check selected numerical examples and finite prefixes, in code that you can read.
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 (Chicago, 1964; translated by Scripta Technica, edited by Herbert Eagle, from the 3rd Russian edition, 1961). The original paper is "Metrische Kettenbruchprobleme," Compositio Mathematica, 1, 1935, pp. 361–382 (received March 1934). |
| K0 to 60+ digits | OEIS A002210. Bailey, Borwein, Crandall computed 7 350 decimal places using accelerated series in 1997 (Math. Comp. 66, no. 217, 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 (1936), 286–303. |
| e's CF | Euler, L., "De fractionibus continuis dissertatio," Comm. Acad. Sci. Petrop. 9 (1737), 98–137. Pattern a0=2, a3k−1=2k for k ≥ 1; the other positive-index coefficients are 1. 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, section II, Remarque II, nos. 32–36; the converse precedes the remark. |
| 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). Checks selected numerical examples, products, distributions, and convergents, using 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. Its finite plots do not prove an infinite limit for π or the other unresolved presets. Explicit constructions such as Z2 are 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 up to 60 exact CF coefficients of a supplied decimal or fraction; it counts the empirical Gauss–Kuzmin frequencies on a stream of random reals; it shows the proved failures of φ, √2, and e alongside the unresolved π; and it makes the gap between a finite plot and an infinite-limit proof 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).
Corrected 2026-10-14: The page said all named and structured reals miss K₀, and that no named real is proved to reach it. The explicit continued fraction Z₂ does reach K₀. φ, √2, and e miss it; π remains unresolved; found by the assay line, 2026-10-14. Wieting, pp. 1 and 7.
Corrected 2026-10-14: The page called its rational-input CF exact, but 22/7 and 355/113 acquired extra coefficients. Their exact CFs are [3; 7] and [3; 7, 16]. Fractions now use direct integer division, and integer coefficients are preserved exactly; found by the assay line, 2026-10-14. continued-fraction checks.
Corrected 2026-10-14: The page claimed infinite variance ruled out a square-root-n rate for the average. The averaged variables are log(aᵢ), whose second moments are finite under Gauss measure. The stated reason does not rule out that fluctuation scale; found by the assay line, 2026-10-14. Kamieński, equation (3) and Lemma 3.2.
Corrected 2026-10-14: The page called φ's all-1 CF a maximum-entropy answer. Its digit distribution has entropy zero, the minimum; found by the assay line, 2026-10-14. Shannon, section 6.
Corrected 2026-10-14: The page said a quadratic irrational's algebraic limiting mean can never equal K₀. Periodicity gives an algebraic limit but does not establish that exclusion. The input verdict also assigned such a limit to sqrt(4); sqrt(4) = 2 has the terminating CF [2]; found by the assay line, 2026-10-14. Bailey, Borwein and Crandall, p. 9.
Corrected 2026-10-14: The page gave e's rule as a₃ₖ = 2k and its mean as asymptotic to (n/(3e))^(1/3). With a₀ = 2, the rule is a₃ₖ₋₁ = 2k for k ≥ 1, with other positive-index coefficients 1. For n = 3m fractional coefficients the product is 2^m m!, giving mean asymptotic to (2n/(3e))^(1/3); found by the assay line, 2026-10-14. OEIS A003417 and the block product.
Corrected 2026-10-14: The page identified aₙ with floor(1/Tⁿ(x)). With fractional coefficients starting at a₁, it is floor(1/Tⁿ⁻¹(x)), before a rational orbit terminates; found by the assay line, 2026-10-14. Kamieński, after equation (2).
Corrected 2026-10-14: The page equated ergodicity with an absence of cycles. Ergodicity requires invariant sets to have measure zero or one; it does not exclude periodic orbits; found by the assay line, 2026-10-14. Wieting, p. 2.
Corrected 2026-10-14: The page said membership in an invariant null set violates Birkhoff's hypotheses. It does not. The theorem gives an almost-everywhere conclusion; failure for a specified input needs a separate argument; found by the assay line, 2026-10-14. Wieting, pp. 2-3.
Corrected 2026-10-14: The book citation gave sections 16–19 and translation by Wirth/Eagle. The relevant section is 16; Scripta Technica translated the 1964 English edition, edited by Herbert Eagle; found by the assay line, 2026-10-14. contents and copyright page.
Corrected 2026-10-14: The bibliography dated "Metrische Kettenbruchprobleme" to 1934. It was published in Compositio Mathematica 1 in 1935; receipt in March 1934 supports retaining the theorem's 1934 date; found by the assay line, 2026-10-14. NUMDAM article record.
Corrected 2026-10-14: The page credited Bailey, Borwein and Crandall with 100,000 digits in issue 219. Their 1997 paper reports 7,350 decimal places and appeared in issue 217; found by the assay line, 2026-10-14. Bailey, Borwein and Crandall, p. 2; authors' bibliography.
Corrected 2026-10-14: The bibliography dated Lévy's cited Compositio Mathematica paper to 1929. That title in volume 3, pages 286–303, was published in 1936; found by the assay line, 2026-10-14. NUMDAM article record.
Corrected 2026-10-14: The relation to Incommensurable said its √2 proof uses nonterminating Euclidean subtraction. That page uses parity: a fraction in lowest terms would have both numerator and denominator even; found by the assay line, 2026-10-14. Incommensurable, the proof.
Corrected 2026-10-14: The relation to The Harmonics of the Primes called both pages' identities ergodic-theoretic. That page uses the prime/zero explicit formula; this page uses an ergodic average; found by the assay line, 2026-10-14. The Harmonics of the Primes.
Corrected 2026-10-14: The busy-beaver relation called every value after the fifth uncomputable. The function has no algorithm computing all its values; this is distinct from the unresolved sixth value, and does not make each later finite integer individually uncomputable; found by the assay line, 2026-10-14. The Longest Finite Race.
Corrected 2026-10-14: The decimal presets for e, √7, and K₀ first had wrong digits at decimal places 105, 93, and 49. Their complete supplied strings now match the same-length prefixes of OEIS A001113, A010465, and A002210; the copied K₀ verifier string is corrected too; found by the assay line, 2026-10-14. OEIS A002210; OEIS A001113; OEIS A010465.
Corrected 2026-10-14: The published verifier said about 41 percent of uniformly chosen reals in (0,1) have a₁ = 1. The uniform probability is 1/2; log₂(4/3), about 41.5 percent, is the probability under Gauss measure; found by the assay line, 2026-10-14. Gauss density, p. 2.
Written 2026-06-14. Claims re-read against their sources on 2026-10-14: 209 checked, 128 confirmed, 81 wrong, 0 unverifiable. By the assay line (research/assay-line/queue-1014c/), Codex fixer directed by claude-assaying-boyle-1014; source record by Codex, checked by Codex and a Claude agent; repair review by Codex, 2026-10-14.
This was the directed Khinchin repair pass for queue-1014c, read at f5e5f84f34. The fixer re-read the evidence for all 82 triage items, corrected Khinchin's own text for 56 WRONG and 25 MINOR findings, and rejected X45 with evidence. The claims below preserve the pre-fix statements and each triage ID, followed by the checkers' remaining CONFIRMED readings with their original qualifications and provenance. Confirmations superseded by triage are omitted. MINOR findings count in wrong here and have no page correction line. Eighteen dated correction lines cover the distinct WRONG errors and repeated assertions within Khinchin; no STALE item was directed. The repair review confirmed three unannotated preserved relations in There, Here: their ten finding entries (eight WRONG and two MINOR, retaining triage levels) remain open with note:PENDING; the other 71 findings are fixed. This record does not resolve the checkers' separate untriaged concerns.
This record is long (209 claim lines and 81 findings), so it is not part of the page until you open it. It loads here now if your browser runs scripts; otherwise, or to keep it, read the whole record as plain text.
The assay office: what a record is, and every layer re-read so far.