Every real number has best rational approximations: fractions p/q that are closer to x than any fraction with smaller denominator. The denominators q₁, q₂, q₃, … grow, but at what rate? Paul Lévy proved in 1935 that for almost every real number, qₙ^(1/n) converges to e^β ≈ 3.27582, where β = π²/(12 ln 2).
For almost all real numbers, ln(qₙ) grows linearly at slope β ≈ 1.1865. The denominators of π's convergents (1,7,106,113,33102…) grow faster on average due to the anomalous partial quotient 292.
The golden ratio φ = [1;1,1,1,…] has Fibonacci denominators 1, 1, 2, 3, 5, 8, 13, … growing at rate φ ≈ 1.618 per step. This is far slower than e^β ≈ 3.276, which is why φ is the "most irrational" number: its approximations improve the most slowly. Most numbers have denominators growing much faster, at rate e^β.
Comparison of denominator growth rates for golden ratio versus typical number
| φ = [1;1,1,1,…] | Typical number |
|---|---|
| qₙ grows as φⁿ ≈ 1.618ⁿ | qₙ grows as (e^β)ⁿ ≈ 3.276ⁿ |
| Slowest possible growth | Lévy's theorem |
The value β = π²/(12 ln 2) emerges from integrating the Gauss-Kuzmin distribution. The ln 2 comes from working in base 2 (binary), and π² arises from the same sources as ζ(2) = π²/6. Lévy's constant: 1.1865691104156254… e^β = 3.275822918721811159787681882…
The partial quotient 292 at step 5 makes π's denominators grow much faster than average. For a "typical" number the ratio ln(qₙ)/n → β ≈ 1.187.
| n | Partial quotient aₙ | Convergent pₙ/qₙ | Denominator qₙ | ln(qₙ)/n |
|---|---|---|---|---|
| 1 | 3 | 3/1 | 1 | 0.00 |
| 2 | 7 | 22/7 | 7 | 0.97 |
| 3 | 15 | 333/106 | 106 | 1.55 |
| 4 | 1 | 355/113 | 113 | 1.19 |
| 5 | 292 | 103993/33102 | 33102 | 2.52 |
| 6 | 1 | 104348/33215 | 33215 | 1.74 |
| 7 | 1 | 208341/66317 | 66317 | 1.54 |
Levy's constant beta = pi^2/(12 ln 2) ≈ 1.18657. For almost every real number, the nth convergent denominator qn satisfies qn^(1/n) to e^beta ≈ 3.27582. Proved by Paul Levy in 1935. The golden ratio, with Fibonacci denominators growing at rate phi ≈ 1.618, is far below average, confirming it as the hardest number to approximate. The formula combines pi and ln 2, connecting circle geometry with logarithms through the Gauss-Kuzmin distribution.