Difference between revisions of "Normal number"
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− | Almost-all numbers are normal with respect to every base | + | Almost-all numbers are normal with respect to every base $g$ (see e.g. Theorem 8.11 in {{Cite|N}}). It is not known whether familiar numbers like $\sqrt2,\,e,',\pi$ are normal or not. Normal numbers are potentially interesting in the context of random number generators. A normal number to a base $g$ is necessarily irrational. The weakly-normal number (to base $10$) $0\cdot12345678901234567890\ldots$ is of course rational. The number $x = 0\cdot1234567891011121314\ldots, obtained as $x = 0 \cdot \alpha_1 \alpha_2 \alpha_3 \ldots$ where $\alpha_i$ stands for the group of digits representing $i$ to base $10$, is normal to base $10$ {{Cite|C}}. The same recipe works to obtain normal numbers to any given base. |
====References==== | ====References==== |
Revision as of 09:05, 12 October 2014
2020 Mathematics Subject Classification: Primary: 11K16 [MSN][ZBL]
A real number ,
, having the following property: For every natural number
, any given
-tuple
consisting of the symbols
appears with asymptotic frequency
in the sequence
![]() | (1) |
obtained from the expansion of in an infinite fraction in base
,
![]() |
In more detail, let be a natural number and let
![]() | (2) |
be the infinite sequence of -tuples corresponding to (1). Let
denote the number of occurrences of the tuple
among the first
tuples of (2). The number
![]() |
is said to be normal if for any number and any given
-tuple
consisting of the symbols
,
![]() |
The concept of a normal number was introduced for by E. Borel (see [B], [B2], p. 197). He called a real number
weakly normal to the base
if
![]() |
where is the number of occurrences of
,
, among the first
terms of the sequences
and normal if
are weakly normal to the bases
. He also showed that for a normal number
![]() |
for any and any given
-tuple
. Later it was proved (see [Pi], [NZ], and also [Po]) that the last relation is equivalent to Borel's definition of a normal number.
A number is called absolutely normal if it is normal with respect to every base
. The existence of normal and absolutely-normal numbers was established by Borel on the basis of measure theory. The construction of normal numbers in an explicit form was first achieved in [C]. Earlier (see [S], [L]) an effective procedure for constructing normal numbers was indicated. For other methods for constructing normal numbers and for connections between the concepts of normality and randomness see [Po].
Uniform distribution of the fractional parts ,
on the interval
is equivalent to
being normal.
References
[B] | E. Borel, "Les probabilités dénombrables et leurs applications arithmétiques" Rend. Circ. Math. Palermo , 27 (1909) pp. 247–271 Zbl 40.0283.01 |
[B2] | E. Borel, "Leçons sur la théorie des fonctions" , Gauthier-Villars (1928) MR0033328 Zbl 54.0327.02 |
[Pi] | S. Pillai, "On normal numbers" Proc. Indian Acad. Sci. Sect. A , 12 (1940) pp. 179–184 MR0002324 Zbl 0025.30802 Zbl 66.1212.02 |
[NZ] | I. Niven, H. Zuckerman, "On the definition of normal numbers" Pacific J. Math. , 1 (1951) pp. 103–109 MR0044560 Zbl 0042.26902 |
[C] | D.G. Champernowne, "The construction of decimals normal in the scale of ten" J. London Math. Soc. , 8 (1933) pp. 254–260 Zbl 0007.33701 Zbl 59.0214.01 |
[S] | W. Sierpiński, "Démonstration élémentaire d'un théorème de M. Borel sur les nombres absolument normaux et détermination effective d'un tel nombre" Bull. Soc. Math. France , 45 (1917) pp. 127–132 MR0073664 MR0055398 MR0021058 MR1550055 |
[L] | H. Lebesgue, "Sur certaines démonstrations d'existence" Bull. Soc. Math. France , 45 (1917) pp. 132–144 MR1504765 |
[Po] | A.G. Postnikov, "Arithmetic modelling of random processes" Trudy Mat. Inst. Steklov. , 57 (1960) (In Russian) |
Comments
Almost-all numbers are normal with respect to every base $g$ (see e.g. Theorem 8.11 in [N]). It is not known whether familiar numbers like $\sqrt2,\,e,',\pi$ are normal or not. Normal numbers are potentially interesting in the context of random number generators. A normal number to a base $g$ is necessarily irrational. The weakly-normal number (to base $10$) $0\cdot12345678901234567890\ldots$ is of course rational. The number $x = 0\cdot1234567891011121314\ldots, obtained as $x = 0 \cdot \alpha_1 \alpha_2 \alpha_3 \ldots$ where $\alpha_i$ stands for the group of digits representing $i$ to base $10$, is normal to base $10$ [C]. The same recipe works to obtain normal numbers to any given base.
References
[N] | I. Niven, "Irrational numbers" , Math. Assoc. Amer. (1956) MR1570844 MR0080123 Zbl 0070.27101 |
Normal number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_number&oldid=26888