Continued Fraction Expansion
The simple continued fraction expansion of Champernowne's constant has been studied as well. Kurt Mahler showed that the constant is transcendental; therefore its continued fraction does not terminate (because it is not rational) and is aperiodic (because it is not an irreducible quadratic).
The terms in the continued fraction expansion exhibit very erratic behaviour, with huge terms appearing between many small ones. For example, in base 10,
- C10 = [0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15,
-
- 4 57540 11139 10310 76483 64662 82429 56118 59960 39397 10457 55500 06620 04393 09026 26592 56314 93795 32077 47128 65631 38641 20937 55035 52094 60718 30899 84575 80146 98631 48833 59214 17830 10987,
- 6, 1, 1, 21, 1, 9, 1, 1, 2, 3, 1, 7, 2, 1, 83, 1, 156, 4, 58, 8, 54, ...].
-
The large number at position 19 has 166 digits, and the next term of the continued fraction has 2504 digits. The fact that there are such large numbers as terms of the continued fraction expansion is equivalent to saying that the convergents obtained by stopping before these large numbers provide an exceptionally good approximation of the Champernowne constant. For example, truncating before the 4th partial quotient, we obtain the partial sum, which approximates Champernowne's constant with an error of about 1 × 10−9, while truncating just before the 18th partial quotient, we get
which approximates Champernowne's constant with error approximately 9 × 10−190.
Read more about this topic: Champernowne Constant
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