A normal number to an integer base is an irrational number for which every finite block of digits occurs with limiting frequency in its base- expansion, for every positive integer . For example, for a normal decimal number, each digit 0-9 occurs with frequency , each pair of digits 00-99 with frequency , etc. A number that is normal in base is often called -normal.

A number is said to be simply normal to base if its base- expansion has each digit appearing with average frequency tending to .

A number that is -normal for every , 3, ... is said to be absolutely normal (Bailey and Crandall 2002).

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 positive integers and (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 with , 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 2002), the natural logarithm of 2 (Bailey and Crandall 2002), Apéry's constant (Bailey and Crandall 2002), Pythagoras's constant (Bailey and Crandall 2002), and e are normal, although the first 30 million digits of have approximately equal digit frequencies (Bailey 1988).

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. 1970a,b), but such tests do not prove normality. Isaac (2005) submitted a preprint claiming simple normality to base 2 for when is not a square number. The author acknowledged an incorrect lemma in the original argument and supplied an alternative argument in a September 2006 revision.

Stephan (2026) proved that Condition (TU), the tail-invariance assertion used in Isaac's proposed proof, is false. The counterexample concerns the supremum limit of the relative frequency of binary digits equal to 1 in square roots of numbers whose binary expansions differ in only finitely many places. It invalidates that proposed route to simple normality, not the normality conjecture itself. Simple normality of to base 2 remains open. Stephan (2026) supplies a Lean formalization of the elementary proof using only standard logical axioms.

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 2002), 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

(1)

(OEIS A030190) is 2-normal (Bailey and Crandall 2001).

Disjunctivity is weaker than normality: a base- expansion is disjunctive if every finite base- digit word occurs at least once (Leśniak 2014). The reciprocal Fermat constant is the subject of an AI-generated result claiming that it is disjunctive but not normal in base 2 (CaptainSude 2026, VibeMathed 2026).

Bailey and Crandall (2001) showed that, subject to an unproven but reasonable hypothesis related to generators of pseudorandom numbers, the constants , , and would be 2-normal, where is Apéry's constant. Stoneham (1973) proved that the so-called Stoneham numbers

(2)

where 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 (2002), who showed that is normal for all positive integers provided only that and are relatively prime.

Korobov (1990) showed that the constants

(3)

are -normal for positive integers and and relatively prime, a result reproved using completely different techniques by Bailey and Crandall (2002). Amazingly, Korobov (1990) also gave an explicit algorithm for computing terms in the continued fraction of .

Bailey and Crandall (2002) also established -normality for constants of the form for certain sequences of integers and .