# Definition:Absolutely Normal Real Number

Jump to navigation
Jump to search

## 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.

## 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): Entry:**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): Entry:**absolutely normal number**