---
tag: C98T2T5FJS
kind: concept
name: Normal number
msc: [11K16]
sources:
  - url: https://en.wikipedia.org/wiki/Normal_number
same_as:
  wikidata: Q902708
  wikipedia: Normal number
---

## Summary

A real number is normal in a base $b$ when every block of $k$ digits appears in its base-$b$ expansion with limiting frequency $b^{-k}$.
A number is absolutely normal when it is normal in every base $b \ge 2$.
Borel showed that almost every real number, in the sense of Lebesgue measure, is absolutely normal.
Yet for familiar constants such as $\sqrt{2}$, $\pi$ and $e$, normality is not known in any base.

## Explanations

### L1 human

Think of the digits of a number as the results of rolling a fair ten-sided die.
A normal number is one whose digits look random in this sense.
Each digit appears one tenth of the time, each pair of digits one hundredth of the time, and so on.
The number $0.123456789101112131415\ldots$, made by writing the positive integers one after the other, is normal in base $10$.
A [[rational number|0NCJ5WXN6Z]] is never normal, because its digits repeat with a fixed period.