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Borel's normal number theorem

Published online by Cambridge University Press:  14 February 2019

W. J. A. Colman
W. J. A. Colman*
Affiliation:
64 Shooters Hill Road, Blackheath, LondonSE3 7BG
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Extract

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.

Type
Articles
Information
The Mathematical Gazette , Volume 103 , Issue 556 , March 2019 , pp. 77 - 86
Copyright
Copyright © Mathematical Association 2019 

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References

1. Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers, Oxford University Press.10.1090/S0002-9904-1929-04793-1Google Scholar
2. Khoshnavisan, D., Normal numbers are normal, Clay Mathematics Institute (2006), available at www.claymath.org/library/annual_report/ar2006/06report_normalnumbers.pdfGoogle Scholar