Difference between revisions of "Normal number"
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756023.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756023.png" /></td> </tr></table> | ||
− | The concept of a normal number was introduced for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756024.png" /> by E. Borel (see | + | The concept of a normal number was introduced for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756024.png" /> by E. Borel (see {{Cite|B}}, {{Cite|B2}}, p. 197). He called a real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756025.png" /> weakly normal to the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756027.png" /> if |
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756028.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756028.png" /></td> </tr></table> | ||
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756036.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756036.png" /></td> </tr></table> | ||
− | for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756037.png" /> and any given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756038.png" />-tuple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756039.png" />. Later it was proved (see | + | for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756037.png" /> and any given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756038.png" />-tuple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756039.png" />. Later it was proved (see {{Cite|Pi}}, {{Cite|NZ}}, and also {{Cite|Po}}) that the last relation is equivalent to Borel's definition of a normal number. |
− | A number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756040.png" /> is called absolutely normal if it is normal with respect to every base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756041.png" />. 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 | + | A number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756040.png" /> is called absolutely normal if it is normal with respect to every base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756041.png" />. 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 {{Cite|C}}. Earlier (see {{Cite|S}}, {{Cite|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 {{Cite|Po}}. |
[[Uniform distribution|Uniform distribution]] of the fractional parts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756043.png" /> on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756044.png" /> is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756045.png" /> being normal. | [[Uniform distribution|Uniform distribution]] of the fractional parts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756043.png" /> on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756044.png" /> is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756045.png" /> being normal. | ||
====References==== | ====References==== | ||
− | + | {| | |
− | + | |valign="top"|{{Ref|B}}|| E. Borel, "Les probabilités dénombrables et leurs applications arithmétiques" ''Rend. Circ. Math. Palermo'' , '''27''' (1909) pp. 247–271 {{MR|}} {{ZBL|40.0283.01}} | |
− | + | |- | |
+ | |valign="top"|{{Ref|B2}}|| E. Borel, "Leçons sur la théorie des fonctions" , Gauthier-Villars (1928) {{MR|0033328}} {{ZBL|54.0327.02}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Pi}}|| S. Pillai, "On normal numbers" ''Proc. Indian Acad. Sci. Sect. A'' , '''12''' (1940) pp. 179–184 {{MR|0002324}} {{ZBL|0025.30802}} {{ZBL|66.1212.02}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|NZ}}|| I. Niven, H. Zuckerman, "On the definition of normal numbers" ''Pacific J. Math.'' , '''1''' (1951) pp. 103–109 {{MR|0044560}} {{ZBL|0042.26902}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|C}}|| D.G. Champernowne, "The construction of decimals normal in the scale of ten" ''J. London Math. Soc.'' , '''8''' (1933) pp. 254–260 {{MR|}} {{ZBL|0007.33701}} {{ZBL|59.0214.01}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|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 {{MR|0073664}} {{MR|0055398}} {{MR|0021058}} {{MR|1550055}} {{ZBL|}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|L}}|| H. Lebesgue, "Sur certaines démonstrations d'existence" ''Bull. Soc. Math. France'' , '''45''' (1917) pp. 132–144 {{MR|1504765}} {{ZBL|}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Po}}|| A.G. Postnikov, "Arithmetic modelling of random processes" ''Trudy Mat. Inst. Steklov.'' , '''57''' (1960) (In Russian) | ||
+ | |} | ||
====Comments==== | ====Comments==== | ||
− | Almost-all numbers are normal with respect to every base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756046.png" /> (see e.g. Theorem 8.11 in | + | Almost-all numbers are normal with respect to every base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756046.png" /> (see e.g. Theorem 8.11 in {{Cite|N}}). It is not known whether familiar numbers like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756047.png" /> are normal or not. Normal numbers are potentially interesting in the context of random number generators. A normal number to a base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756048.png" /> is necessarily irrational. The weakly-normal number (to base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756049.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756050.png" /> is of course rational. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756051.png" />, obtained as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756052.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756053.png" /> stands for the group of digits representing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756054.png" /> to base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756055.png" />, is normal to base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067560/n06756056.png" /> {{Cite|C}}. The same recipe works to obtain normal numbers to any given base. |
====References==== | ====References==== | ||
− | + | {| | |
+ | |valign="top"|{{Ref|N}}|| I. Niven, "Irrational numbers" , Math. Assoc. Amer. (1956) {{MR|1570844}} {{MR|0080123}} {{ZBL|0070.27101}} | ||
+ | |} |
Revision as of 18:06, 26 May 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 [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 (see e.g. Theorem 8.11 in [N]). 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
[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