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.
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. Paris18, 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. Paris18, 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.
, 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.
into this equation gives 
instead of 0.
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.
