# Definition:Absolutely Normal Real 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 known as

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

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.