A real number that is -normal for every base 2, 3,
4, ... is said to be absolutely normal. As proved by Borel (1922, p. 198), almost
all real numbers in
are absolutely normal (Niven 1956, p. 103; Stoneham 1970; Kuipers and Niederreiter
1974, p. 71; Bailey and Crandall 2002).
The first specific construction of an absolutely normal number was by Sierpiński (1917), with another method presented by Schmidt (1962). These results were both obtained by complex constructive devices (Stoneham 1970), and are by no means easy to construct (Stoneham 1970, Sierpiński and Schinzel 1988).
Bailey, D. H. and Crandall, R. E. "Random Generators and Normal Numbers." Exper. Math.11, 527-546, 2002.Borel,
E. "Les probabilités dénombrables et leurs applications arithmétiques."
Rend. Circ. Mat. Palermo27, 247-271, 1909.Borel, E. Leçons
sur la théorie de fonctions. Paris, pp. 197-198, 1922.Borwein,
J. and Bailey, D. Mathematics
by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A
K Peters, p. 143, 2003.Kuipers, L. and Niederreiter, H. Uniform
Distribution of Sequences. New York: Wiley, 1974.Niven, I. M.
Irrational
Numbers. New York: Wiley, 1956.Schmidt, W. "Über die
Normalität von Zahlen zu verschiedenen Basen." Acta Arith.7,
299-309, 1962.Sierpiński, W. "Démonstration élémentaire
d'un théorème de M. Borel sue les nombres absolutment normaux
et détermination effective d'un tel nombre." Bull. Soc. Math. France45,
125-144, 1917.Sierpiński, W. and Schinzel, A. Elementary
Theory of Numbers, 2nd Eng. ed. Amsterdam, Netherlands: North-Holland, 1988.Stoneham,
R. "A General Arithmetic Construction of Transcendental Non-Liouville Normal
Numbers from Rational Functions." Acta Arith.16, 239-253, 1970.
-normal for every base 2, 3,
4, ... is said to be absolutely normal. As proved by Borel (1922, p. 198), almost
all real numbers in 
are absolutely normal (Niven 1956, p. 103; Stoneham 1970; Kuipers and Niederreiter
1974, p. 71; Bailey and Crandall 2002).
