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GeezerVison is showing mathworld.wolfram.com — Normal Number -- from Wolfram MathWorld

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Normal Number

A normal number to an integer base b>=2 is an irrational number for which every finite block of k digits occurs with limiting frequency b^(-k) in its base-b expansion, for every positive integer k. For example, for a normal decimal number, each digit 0-9 occurs with frequency 1/10, each pair of digits 00-99 with frequency 1/100, etc. A number that is normal in base b is often called b-normal.

A number is said to be simply normal to base b if its base-b expansion has each digit appearing with average frequency tending to b^(-1).

A number that is b-normal for every b=2, 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 alpha is b^k-normal, then it is also b^m-normal for positive integers k and m (Kuipers and Niederreiter 1974, p. 72; Bailey and Crandall 2001). Furthermore, if q and r are rational with q!=0 and alpha is b-normal, then so is qalpha+r, while if c=b^q is an integer with c>=2, then alpha is also c-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 ln2 (Bailey and Crandall 2002), Apéry's constant zeta(3) (Bailey and Crandall 2002), Pythagoras's constant sqrt(2) (Bailey and Crandall 2002), and e are normal, although the first 30 million digits of pi have approximately equal digit frequencies (Bailey 1988).

Tests of sqrt(n) for n=2 (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), but such tests do not prove normality. Isaac (2005) submitted a preprint claiming simple normality to base 2 for sqrt(s) when s 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 sqrt(2) 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

C_2=0.(1)(10)(11)(100)(101)(110)(111)..._2
(1)

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

Disjunctivity is weaker than normality: a base-b expansion is disjunctive if every finite base-b 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 pi, ln2, and zeta(3) would be 2-normal, where zeta(3) is Apéry's constant. Stoneham (1973) proved that the so-called Stoneham numbers

alpha_(b,c)=sum_(k=1)^infty1/(b^(c^k)c^k),
(2)

where b,c>1 are relatively prime positive integers, are b-normal whenever c is an odd prime and b is a primitive root of c^2. This result was extended by Bailey and Crandall (2002), who showed that alpha_(b,c) is b-normal for all positive integers b,c>1 provided only that b and c are relatively prime.

Korobov (1990) showed that the constants

beta_(b,c,d)=sum_(n=c,c^d,c^(d^2),c^(d^3),...)1/(nb^n)
(3)

are b-normal for positive integers b,c,d>1 and b and c 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 beta_(b,c,d).

Bailey and Crandall (2002) also established b-normality for constants of the form sum_(i)1/(b^(m_i)c^(n_i)) for certain sequences of integers (m_i) and (n_i).

See also

Absolutely Normal, b-Dense Number, Binary Champernowne Constant, Champernowne Constant, Copeland-Erdős Constant, e, Equidistributed Sequence, Pi, Reciprocal Fermat Constant, Stoneham Number, Strongly Normal Number

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References

Bailey, D. H. "The Computation of pi to 29360000 Decimal Digits Using Borweins' Quartically Convergent Algorithm." Math. Comput. 50, 283-296, 1988.Bailey, D. H. and Crandall, R. E. "On the Random Character of Fundamental Constant Expansions." Exper. Math. 10, 175-190, 2001. https://doi.org/10.1080/10586458.2001.10504441.Bailey, D. H. and Crandall, R. E. "Random Generators and Normal Numbers." Exper. Math. 11, 527-546, 2002. https://doi.org/10.1080/10586458.2002.10504704.Beyer, W. A.; Metropolis, N.; and Neergaard, J. R. "Square Roots of Integers 2 to 15 in Various Bases 2 to 10: 88062 Binary Digits or Equivalent." Math. Comput. 23, 679, 1969.Beyer, W. A.; Metropolis, N.; and Neergaard, J. R. "Statistical Study of Digits of Some Square Roots of Integers in Various Bases." Math. Comput. 24, 455-473, 1970a.Beyer, W. A.; Metropolis, N.; and Neergaard, J. R. "The Generalized Serial Test Applied to Expansions of Some Irrational Square Roots in Various Bases." Math. Comput. 24, 745-747, 1970b.Borel, É. "Les probabilités dénombrables et leurs applications arithmétiques." Rend. Circ. Mat. Palermo 27, 247-271, 1909.CaptainSude. "The Reciprocal Fermat Constant Is Nonnormal." Sep. 8, 2026. https://github.com/CaptainSude/reciprocal-Fermat-constant-Nonnormal/releases/tag/v1.0.0.Champernowne, D. G. "The Construction of Decimals Normal in the Scale of Ten." J. London Math. Soc. 8, 254-260, 1933.Copeland, A. H. and Erdős, P. "Note on Normal Numbers." Bull. Amer. Math. Soc. 52, 857-860, 1946.Gibbs, W. W. "A Digital Slice of Pi. The New Way to Do Pure Math: Experimentally." Sci. Amer. 288, 23-24, May 2003.Good, I. "Normal Recurring Decimals." J. London Math. Soc. 21, 167-169, 1946.Good, I. J. and Gover, T. N. "The Generalized Serial Test and the Binary Expansion of sqrt(2)." J. Roy. Statist. Soc. Ser. A 130, 102-107, 1967.Good, I. J. and Gover, T. N. "Corrigendum." J. Roy. Statist. Soc. Ser. A 131, 434, 1968.Isaac, R. "On the Simple Normality to Base 2 of sqrt(s), for s Not a Perfect Square." 16 Dec 2005. https://arxiv.org/abs/math/0512404.Kac, M. Statistical Independence in Probability, Analysis and Number Theory. Washington, DC: Math. Assoc. Amer., 1959.Korobov, N. "Continued Fractions of Certain Normal Numbers." Math. Zametki 47, 28-33, 1990. English translation in Math. Notes Acad. Sci. USSR 47, 128-132, 1990.Kuipers, L. and Niederreiter, H. Uniform Distribution of Sequences. New York: Wiley, 1974.Leśniak, K. "On Discrete Stochastic Processes with Disjunctive Outcomes." Bull. Austral. Math. Soc. 90, 149-159, 2014. https://doi.org/10.1017/S0004972714000124.Postnikov, A. G. "Ergodic Problems in the Theory of Congruences and of Diophantine Approximations." Proc. Steklov Inst. Math. 82, 1966.Sloane, N. J. A. Sequence A030190 in "The On-Line Encyclopedia of Integer Sequences."Stephan, R. "Isaac's Condition (TU) Is False." 15 Oct 2026. https://arxiv.org/abs/2610.21899.Stoneham, R. "A General Arithmetic Construction of Transcendental Non-Liouville Normal Numbers from Rational Functions." Acta Arith. 16, 239-254, 1970. https://doi.org/10.4064/aa-16-3-239-254.Stoneham, R. "On Absolute (j,epsilon)-Normality in the Rational Fractions with Applications to Normal Numbers." Acta Arith. 22, 277-286, 1973. https://doi.org/10.4064/aa-22-3-277-286.VibeMathed. "Reciprocal Fermat Constant Is Nonnormal." Sep. 8, 2026. https://vibemathed.com/problem/reciprocal-fermat-constant-is-nonnormal.Wagon, S. "Is pi Normal?" Math. Intel. 7, 65-67, 1985.Weisstein, E. W. "Bailey and Crandall Discover a New Class of Normal Numbers." MathWorld Headline News, Oct. 4, 2001. https://mathworld.wolfram.com/news/2001-10-04/normal/.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 26, 1986.

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Normal Number

Cite this as:

Weisstein, Eric W. "Normal Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NormalNumber.html

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