On normal numbers
DOI:
https://doi.org/10.4067/S0716-09172006000100002Keywords:
Normality, real numbers, normalidad, números reales.Abstract
A real number α is said to be normal to base 10 if, for every natural number L, each finite sequence of L digits appears in the decimals of α with frequency 1/10L. Even intuitive results concerning normal numbers presents complicated formalizations and to decide whether a given number is normal or not is sometimes almost impossible. In this paper we prove that if η = 0; a1a2a3a4... is a normal number, then
= 0, a1a1a2a1a2a3a1a2a3a4... is also normal. On the other hand, if η fails to be normal, there are some technical difficulties in order to decide whether is normal or not, and we also discuss the normality (or not) of when η fails to be normal.
References
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[7] M. W. Sierpnski, Démonstration élémentaire du théoréme de M.Borel sur les nombres absolument normeaux et détermination effective d’un tel nombre, Bull. Soc. Math. France 45, pp. 125-132, (1917).
[2] M. E. Borel, Les probabilités denomerables et leurs pplications arithmetiques, Rend. Circ. Mat. Palermo 27, (1909).
[3] D. G. Champernowne, The construction of decimals normal in the scale of ten, J. London Math. Soc., 8, pp. 254-260, (1933).
[4] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fourth Edition, Oxford University Press, London, (1975).
[5] I. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, John Wiley & Sons, (1974).
[6] A. Rényi, Foundations of Probability, Holden-Day Inc., (1970).
[7] M. W. Sierpnski, Démonstration élémentaire du théoréme de M.Borel sur les nombres absolument normeaux et détermination effective d’un tel nombre, Bull. Soc. Math. France 45, pp. 125-132, (1917).
Published
2017-05-08
How to Cite
[1]
D. M. Pellegrino, “On normal numbers”, Proyecciones (Antofagasta, On line), vol. 25, no. 1, pp. 19-30, May 2017.
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