# Definition:Normal Real 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 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.