Definition:Absolutely Normal Number

Definition

A real number $r$ is absolutely normal if it is normal with respect to every number base $b$.

That is, if and only if its basis expansion in every number base $b$ is such that:

no finite sequence of digits of $r$ of length $n$ occurs more frequently than any other such finite sequence of length $n$.

In particular, for every number base $b$, all digits of $r$ have the same natural density in the basis expansion of $r$.

Also see

• Results about absolutely normal numbers can be found here.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $0 \cdotp 12345 67891 01112 13141 51617 18192 02122 \ldots$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): absolutely normal number
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $0 \cdotp 12345 \, 67891 \, 01112 \, 13141 \, 51617 \, 18192 \, 02122 \ldots$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): absolutely normal number