Definition:Normal Number

Definition

A real number $r$ is normal with respect to a number base $b$ if and only if its basis expansion in 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 number base $b$, all digits of $r$ have the same natural density in the basis expansion of $r$.

Also known as

It is common to refer to a real number $r$ which is normal with respect to base $10$ merely as normal, without specifying the base.

Such usage can be confused with the concept of an absolutely normal number, so this usage is not to be used on $\mathsf{Pr} \infty \mathsf{fWiki}$

Also see

• Definition:Absolutely Normal Number

• Champernowne Constant is Normal
• Almost All Real Numbers are Normal

• Results about normal numbers can be found here.

Historical Note

In $1909$, Émile Borel demonstrated that almost all real numbers are normal.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Champernowne's number
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): normal number
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): normal number