Liouville's Constant
Liouville's constant, sometimes also called Liouville's number, is the real number defined by
(OEIS A012245). Liouville's constant is a decimal fraction with a 1 in each decimal place corresponding to a factorial 
, and zeros everywhere
else. Liouville (1844) constructed an infinite class of transcendental
numbers using continued fractions, but
the above number was the first decimal constant to be proven transcendental
(Liouville 1850). However, Cantor subsequently proved that "almost all"
real numbers are in fact transcendental.
A recurrence plot of the binary digits is illustrated
above.
Liouville's constant nearly satisfies
which has solution 0.1100009999... (OEIS A093409), but plugging 
into this equation gives 
instead of 0.
Liouville's constant has continued fraction [0, 9, 11, 99, 1, 10, 9, 999999999999, 1, 8, 10, 1, 99, 11, 9, 999999999999999999999999999999999999999999999999999999999999999999999999,
...] (OEIS A058304; Stark 1994, pp. 172-177),
which shows sporadic large terms. The numbers of digits 
in the 
th term is plotted above on a semilog plot, which
shows a nested structure (E. Zeleny, pers. comm., Aug. 17, 2005). Interestingly,
the 
th incrementally largest term (considering
only those entirely of 9s in order to exclude the term 
) occurs precisely
at position 
, and this term consists of 
9s.
SEE ALSO: Exponential Factorial,
Liouville Number,
Transcendental
Number
REFERENCES:
Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag,
p. 147, 1997.
Conway, J. H. and Guy, R. K. "Liouville's Number." In The
Book of Numbers. New York: Springer-Verlag, pp. 239-241, 1996.
Courant, R. and Robbins, H. "Liouville's Theorem and the Construction of Transcendental Numbers." §2.6.2 in What
Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, pp. 104-107, 1996.
Liouville, J. "Mémoires et communications des Membres et des correspondants de l'Académie." C. R. Acad. Sci. Paris 18, 883-885,
1844.
Liouville, J. "Nouvelle démonstration d'un théor'eme sur les irrationalles algébriques, inséré dans le Compte rendu de la dernière
séance." C. R. Acad. Sci. Paris 18, 910-911, 1844.
Liouville, J. "Sur des classes très-étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à
des irrationelles algébriques." J. Math. pures appl. 16,
133-142, 1851.
Sloane, N. J. A. Sequences A012245, A058304, and A093409
in "The On-Line Encyclopedia of Integer Sequences."
Stark, H. M. An
Introduction to Number Theory. Cambridge, MA: MIT Press, 1994.
Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England:
Penguin Books, p. 26, 1986.
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Liouville's Constant
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