Difference between revisions of "Normal number"
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A real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n0675601.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n0675602.png" />, having the following property: For every natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n0675603.png" />, any given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n0675604.png" />-tuple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n0675605.png" /> consisting of the symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n0675606.png" /> appears with asymptotic frequency <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n0675607.png" /> in the sequence | A real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n0675601.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n0675602.png" />, having the following property: For every natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n0675603.png" />, any given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n0675604.png" />-tuple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n0675605.png" /> consisting of the symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n0675606.png" /> appears with asymptotic frequency <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n0675607.png" /> in the sequence | ||
Revision as of 19:06, 22 March 2012
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 [1], [2], 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 [3], [4], and also [8]) 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 [5]. Earlier (see [6], [7]) 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 [8].
Uniform distribution of the fractional parts ,
on the interval
is equivalent to
being normal.
References
[1] | E. Borel, "Les probabilités dénombrables et leurs applications arithmétiques" Rend. Circ. Math. Palermo , 27 (1909) pp. 247–271 |
[2] | E. Borel, "Leçons sur la théorie des fonctions" , Gauthier-Villars (1928) |
[3] | S. Pillai, "On normal numbers" Proc. Indian Acad. Sci. Sect. A , 12 (1940) pp. 179–184 |
[4] | I. Niven, H. Zuckerman, "On the definition of normal numbers" Pacific J. Math. , 1 (1951) pp. 103–109 |
[5] | D.G. Champernowne, "The construction of decimals normal in the scale of ten" J. London Math. Soc. , 8 (1933) pp. 254–260 |
[6] | 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 |
[7] | H. Lebesgue, "Sur certaines démonstrations d'existence" Bull. Soc. Math. France , 45 (1917) pp. 132–144 |
[8] | 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 (see e.g. Theorem 8.11 in [a1]). It is not known whether familiar numbers like
are normal or not. Normal numbers are potentially interesting in the context of random number generators. A normal number to a base
is necessarily irrational. The weakly-normal number (to base
)
is of course rational. The number
, obtained as
where
stands for the group of digits representing
to base
, is normal to base
[5]. The same recipe works to obtain normal numbers to any given base.
References
[a1] | I. Niven, "Irrational numbers" , Math. Assoc. Amer. (1956) |
Normal number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_number&oldid=18704