A real number is normal in base if every word of digits occurs in its base- expansion with limiting overlapping frequency , for every .
A number is a normal number in base exactly when is an equidistributed sequence. A word of length corresponds to a half-open interval of length ; these intervals form arbitrarily fine grids, which approximate any interval from inside and outside.
An absolutely normal number is a normal number in every integer base . The Birkhoff ergodic theorem and ergodicity of integer multiplication on the circle, followed by a countable intersection of full-measure sets, show that almost every real number is absolutely normal.
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A **normal number** is a real number whose individual digits, and in broader terms, digits of any base, are uniformly distributed. More formally, a number is said to be normal in base \( b \) if, in its expansion in that base, all digits from \( 0 \) to \( b-1 \) appear with equal frequency in the limit as you consider more and more digits.