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In mathematics, a real number is said to be simply normal in an integer base b if its infinite sequence of digits is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b. A number is said to be normal in base b if, for every positive integer n, all possible strings n digits long have density b.
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 or rolls of a die. Even though there will be sequences such as 10, 100, or more consecutive tails or fives or even 10, 100, or more repetitions of a sequence such as tail-head or 6-1 , 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 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 , this proof is not constructive, and only a few specific numbers have been shown to be normal. For example, Chaitin's constant is normal. It is widely believed that the numbers √2, π, and e are normal, but a proof remains elusive.