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 
.
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. 
."
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. 
-Normality
in the Rational Fractions with Applications to Normal Numbers." Acta Arith. 22,
277-286, 1973. 
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.