Turing’s Normal Numbers: Towards Randomness

  • Verónica Becher
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7318)


In a manuscript entitled “A note on normal numbers” and written presumably in 1938 Alan Turing gave an algorithm that produces real numbers normal to every integer base. This proves, for the first time, the existence of computable normal numbers and it is the best solution to date to Borel’s problem on giving examples of normal numbers. Furthermore, Turing’s work is pioneering in the theory of randomness that emerged 30 years after. These achievements of Turing are largely unknown because his manuscript remained unpublished until its inclusion in his Collected Works in 1992. The present note highlights Turing’s ideas for the construction of normal numbers. Turing’s theorems are included with a reconstruction of the original proofs.


Initial Segment Unit Interval Block Length Computable Function Output Sequence 
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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Verónica Becher
    • 1
  1. 1.Departamento de Computación, Facultad de Ciencias Exactas y NaturalesUniversidad de Buenos AiresBuenos AiresArgentina

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