In mathematics, a normal number is a real number whose infinite sequence of digits in every base b is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b, also all possible b2 pairs of digits are equally likely with density b−2, all b3 triplets of digits equally likely with density b−3, etc.
In lay terms, this means that no digit, or combination of digits, occurs more frequently than any other, and this is true whether the number is written in base10, binary, or any other base. A normal number can be thought of as an infinite sequence of coin flips (binary), dice rolls (base 6), or the combination of two dice as in backgammon — even though there will be sequences of 14, or 165,892, or 1234567890 tails/5s/double-6s, there will also be 'equally many' sequences of heads/2s/4&5s. No digit or sequence is 'favored'.
While a general proof can be given that almost all real numbers are normal (in the sense that the set of exceptions has Lebesgue measure zero), this proof is not constructive and only very few specific numbers have been shown to be normal. For example, it is widely believed that the numbers √2, π, and e are normal, but a proof remains elusive.
Read more about Normal Number: Definitions, Properties and Examples, Connection To Finite-state Machines, Connection To Equidistributed Sequences
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