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Borel's normal number theorem

  • W. J. A. Colman (a1)


We shall present some new proofs concerning the distribution of the digits in decimals. We finish with a proof of Borel's normal number theorem. The proof given here is more direct and straightforward than the one given in [1]. This theorem is concerned with the distribution of the digits within infinite decimals. The actual statement of the theorem will be given later after some intermediate results have been established. The theorem is not only unexpected but seemingly counter-intuitive. In order to follow the arguments given here it is useful to have a numerical example as we will count the decimals and then their digits. We shall deduce the results for a general base b.



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1. Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers, Oxford University Press.10.1090/S0002-9904-1929-04793-1
2. Khoshnavisan, D., Normal numbers are normal, Clay Mathematics Institute (2006), available at

Borel's normal number theorem

  • W. J. A. Colman (a1)


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