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The Constant of Champernowne

Pomstra, Gertjan (2018) The Constant of Champernowne. Bachelor's Thesis, Mathematics.

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Abstract

The constant of Champernowne first saw light when David G. Champernowne introduced it in 1933. He proved that the constant given by 0.123456789101112... is normal in base ten. Four years later, Kurt Mahler proved that this number also is transcendental by showing the transcendence of constants with a similar structure to that of the constant of Champernowne. In addition to these two properties, the constant also has a peculiar continued fraction expansion. It namely contains exceptionally large terms throughout the expansion. The contribution of this thesis will be to give a better understanding of the constant of Champernowne, its properties, the proofs and the techniques used for these proofs. The notion of normality will be explained and proven in the way Champernowne did and with the use of the notion of (ϵ,k)-normality. Thereafter we will introduce the notion of transcendence and this will be proven for the constant following an argument of Kurt Mahler. Lastly, we will consider the procedure of continued fraction expansions and apply it to the constant. Some general notions for continued fraction expansions will be shown. With these notions, we show the connection between the fractions used in the argument of Kurt Mahler and the large terms of the continued fraction in the case of the constant of Champernowne.

Item Type: Thesis (Bachelor's Thesis)
Supervisor:
Supervisor nameSupervisor E mail
Top, J.J.Top@rug.nl
Sterk, A.E.A.E.Sterk@rug.nl
Degree programme: Mathematics
Thesis type: Bachelor's Thesis
Language: English
Date Deposited: 17 Jul 2018
Last Modified: 17 Jul 2018 14:28
URI: http://fse.studenttheses.ub.rug.nl/id/eprint/17907

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