Normal Number
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
 |
(1)
|
(OEIS A030190) is 2-normal (Bailey and Crandall 2001).
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
 |
(2)
|
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
 |
(3)
|
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 
.
Bailey, D. H. "The Computation of 
to 
Decimal
Digit using Borwein's' 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. http://www.nersc.gov/~dhbailey/dhbpapers/baicran.pdf.
Bailey, D. H. and Crandall, R. E. "Random Generators and Normal Numbers." Exper. Math. 11, 527-546, 2002.
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.
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 
." 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 
, for 
Not a Perfect Square." 16 Dec 2005. http://arxiv.org/abs/math.NT/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.
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."
Stoneham, R. "A General Arithmetic Construction of Transcendental Non-Liouville Normal Numbers from Rational Functions." Acta Arith. 16, 239-253, 1970. http://matwbn.icm.edu.pl/ksiazki/aa/aa16/aa1631.pdf.
Stoneham, R. "On Absolute 
-Normality
in the Rational Fractions with Applications to Normal Numbers." Acta Arith. 22,
277-286, 1973. http://matwbn.icm.edu.pl/ksiazki/aa/aa16/aa1632.pdf.
Wagon, S. "Is 
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. http://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.
Weisstein, Eric W. "Normal Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NormalNumber.html
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