Editing Normal number (section)

{{Short description|Number with all digits equally frequent}}
{{for|the floating-point meaning in computing|normal number (computing)}}

In [[mathematics]], a [[real number]] is said to be '''simply normal''' in an [[integer]] [[radix|base]] <var>b</var><ref>The only bases considered here are [[natural number]]s greater than 1</ref> if its infinite sequence of [[positional notation|digits]] is distributed uniformly in the sense that each of the <var>b</var> digit values has the same [[natural density]]&nbsp;1/<var>b</var>. A number is said to be '''normal in base <var>b</var>''' if, for every positive integer <var>n</var>, all possible strings <var>n</var> digits long have density&nbsp;<var>b</var><sup>−''n''</sup>.

Intuitively, a number being simply normal means that no digit occurs more frequently than any other. If a number is normal, no finite combination of digits of a given length occurs more frequently than any other combination of the same length. A normal number can be thought of as an infinite sequence of coin flips ([[Binary number|binary]]) or rolls of a die ([[senary|base 6]]). Even though there ''will'' be sequences such as 10, 100, or more consecutive tails (binary) or fives (base 6) or even 10, 100, or more repetitions of a sequence such as tail-head (two consecutive coin flips) or 6-1 (two consecutive rolls of a die), there will also be equally many of any other sequence of equal length. No digit or sequence is "favored". 

A number is said to be '''normal''' (sometimes called '''absolutely normal''') if it is normal in all integer bases greater than or equal to 2.

While a general proof can be given that [[almost all]] real numbers are normal (meaning that the [[Set (mathematics)|set]] of non-normal numbers has [[Lebesgue measure]] zero),{{sfn|Beck|2009}} this proof is not [[Constructive proof|constructive]], and only a few specific numbers have been shown to be normal. For example, any [[Chaitin's constant]] is normal (and [[Uncomputable number|uncomputable]]). It is widely believed that the (computable) numbers [[square root of 2|{{sqrt|2}}]], [[pi|{{pi}}]], and ''[[e (mathematical constant)|e]]'' are normal, but a proof remains elusive.{{sfn|Bailey|Crandall|2002}}