Definition:Normal Real Number

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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 usual to assume that the number being described as normal is real, so to refer merely to a normal number.


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

Such usage can be confused with the concept of an absolutely normal number, so this practice is discouraged.


Also defined as

Some sources do not distinguish between a normal number and an absolutely normal number.

Such sources refer to an absolutely normal number merely as a normal number.


Also see