Champernowne Constant is Normal

Theorem

The Champernowne constant:

$0 \cdotp 12345 \, 67891 \, 01112 \, 13141 \, 51617 \, 18192 \, 02122 \ldots$

is normal with respect to base $10$.

Proof

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Historical Note

The Champernowne constant was constructed by David Gawen Champernowne as an example of a real number which was demonstrably normal.

He did this in the paper of 1933: The Construction of Decimals Normal in the Scale of Ten (J. London Math. Soc. Vol. 8: pp. 254 – 260), while still an undergraduate.

There is confusion in the literature as to whether he proved that the Champernowne constant is in fact absolutely normal.

It would be apparent from the title of the above paper that he did in fact prove normal only with respect to base $10$.

Research is ongoing.

Sources

• 1933: D.G. Champernowne: The Construction of Decimals Normal in the Scale of Ten (J. London Math. Soc. Vol. 8: pp. 254 – 260)
• 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,1234567891011 \ldots$
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $0 \cdotp 12345 67891 01112 13141 51617 18192 02122 \ldots$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $0 \cdotp 12345 \, 67891 \, 01112 \, 13141 \, 51617 \, 18192 \, 02122 \ldots$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Champernowne's number
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Champernowne's number