A number is said to be simply normal to base if its base- expansion has each digit appearing with average frequency tending to .
A normal number is an irrational number for which any finite pattern of numbers occurs with the expected limiting frequency in the expansion in a given base (or all bases). For example, for a normal decimal number, each digit 0-9 would be expected to occur 1/10 of the time, each pair of digits 00-99 would be expected to occur 1/100 of the time, etc. A number that is normal in base- is often called -normal.
A number that is -normal for every , 3, ... is said to be absolutely normal (Bailey and Crandall 2003).
As stated by Kac (1959), "As is often the case, it is much easier to prove that an overwhelming majority of objects possess a certain property than to exhibit even one such object....It is quite difficult to exhibit a 'normal' number!" (Stoneham 1970).
If a real number is -normal, then it is also -normal for and integers (Kuipers and Niederreiter 1974, p. 72; Bailey and Crandall 2001). Furthermore, if and are rational with and is -normal, then so is , while if is an integer, then is also -normal (Kuipers and Niederreiter 1974, p. 77; Bailey and Crandall 2001).
Determining if numbers are normal is an unresolved problem. It is not even known if fundamental mathematical constants such as pi (Wagon 1985, Bailey and Crandall 2003), the natural logarithm of 2 (Bailey and Crandall 2003), Apéry's constant (Bailey and Crandall 2003), Pythagoras's constant (Bailey and Crandall 2003), and e are normal, although the first 30 million digits of are very uniformly distributed (Bailey 1988).
While tests of for (Pythagoras's constant digits, 3 (Theodorus's constant digits, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15 indicate that these square roots may be normal (Beyer et al. 1970ab), normality of these numbers has (possibly until recently) also not been proven. Isaac (2005) recently published a preprint that purports to show that each number of the form for not a perfect square is simply normal to the base 2. Unfortunately, this work uses a nonstandard approach that appears rather cloudy to at least some experts who have looked at it.
While Borel (1909) proved the normality of almost all numbers with respect to Lebesgue measure, with the exception of a number of special classes of constants (e.g., Stoneham 1973, Korobov 1990, Bailey and Crandall 2003), the only numbers known to be normal (in certain bases) are artificially constructed ones such as the Champernowne constant and the Copeland-Erdős constant. In particular, the binary Champernowne constant
(OEIS A030190) is 2-normal (Bailey and Crandall2001).
Bailey and Crandall (2001) showed that, subject to an unproven but reasonable hypothesis related to pseudorandom number generators, the constants , , and would be 2-normal, where is Apéry's constant. Stoneham (1973) proved that the so-called Stoneham numbers
where and are relatively prime positive integers, are -normal whenever is an odd prime and is a primitive root of . This result was extended by Bailey and Crandall (2003), who showed that is normal for all positive integers provided only that and are relatively prime.
Korobov (1990) showed that the constants
are -normal for positive integers and and relatively prime, a result reproved using completely different techniques by Bailey and Crandall (2003). Amazingly, Korobov (1990) also gave an explicit algorithm for computing terms in the continued fraction of .
Bailey and Crandall (2003) also established -normality for constants of the form for certain sequences of integers and .