A remarkable theorem of Khinchin states that for almost all real numbers
, the simple continued fraction of
, has the property that the geometric mean
approaches a universal constant, the Khinchin constant ≈ 2.69, as
tends to infinity. This constant is shown as a horizontal line in the graph. Interestingly, not a single naturally occuring number
is known for which one can verify this property. (Rational numbers, quadratic surds, and powers of ⅇ
are known exceptions.) This Demonstration explores the convergence behavior of the first 1000 continued fraction terms for simple multiples of powers of various transcendental numbers.