On normal numbers

Authors

  • Daniel M. Pellegrino Universidade Federal de Campina Grande.

DOI:

https://doi.org/10.4067/S0716-09172006000100002

Keywords:

Normality, real numbers, normalidad, números reales.

Abstract

A real number ? 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 ? 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 ? = 0,a1a2a3a4... is a normal number, then ?? = 0, a1a1a2a1a2a3a1a2a3a4... is also normal. On the other hand, if ? 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 ? fails to be normal.

Author Biography

Daniel M. Pellegrino, Universidade Federal de Campina Grande.

Depto de Matemática e Estatística.

References

[1] V. Becher and S. Figueira, An example of a computable absolutely normal number, Theoretical Computer Science 270 (2002), pp. 947-958, (2002).

[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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Section

Artículos