Champernowne Constant is Transcendental

Theorem

The Champernowne constant:

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

is transcendental.

Proof

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Historical Note

The transcendental nature of the Champernowne constant was demonstrated by Kurt Mahler in 1937: Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen (Proc. Konin. Neder. Akad. Wet. Ser. A Vol. 40: pp. 421 – 428).

Sources

• 1937: Kurt Mahler: Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen (Proc. Konin. Neder. Akad. Wet. Ser. A Vol. 40: pp. 421 – 428)
• 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): $0 \cdotp 12345 \, 67891 \, 01112 \, 13141 \, 51617 \, 18192 \, 02122 \ldots$