A number is normal base b iff it is simply normal in bases $b^k$












1












$begingroup$


I am trying to prove that a number is normal base b $iff$ it is simply normal in all bases $b^k$ for every integer $k geq 1$.



I'm a little confused on this because if for example we take a number that is normal is base 3 how would that be simply normal in base 9 as it would not have any digits greater than 3.



These are my definitions for normal and simply normal:



We say a number $x$ in decimal expansion form base-$b$ is simply normal base-$b$ if
$lim_{n to infty} frac{N_n^b(x;{w})}{n} = frac{1}{b},$
$forall w in {0,1,2,...b-1}$.



A number $x$ in decimal expansion form base $b$ is normal base-$b$ if for any arbitrary finite string (or word) $w$ with letters from the alphabet ${0,1,2,...,b-1}$
$ lim_{n to infty}frac{N_n^b(x;w)}{n} = frac{1}{b^{|w|}}$
where $|w|$ denotes the lengh of the word.



If somebody could help me get started on how to prove this if and only if statement that would be great.



Update: I have figured out the forward direction, I am still confused on the reverse.










share|cite|improve this question











$endgroup$












  • $begingroup$
    In base 9 it would have digits 0--8. That is, block it off in subparts of length 2 and convert each to a number from 1 to 8. Example 122122=(12)(21)(22)=(5)(7)(8) base 9. Could you put definitions of normal and simply normal in question? (not as comm4ent or link)
    $endgroup$
    – coffeemath
    Dec 4 '18 at 2:45








  • 1




    $begingroup$
    @coffeemath done
    $endgroup$
    – Sasha
    Dec 4 '18 at 3:00










  • $begingroup$
    This is not set-theory. Please do not add the tag back.
    $endgroup$
    – Andrés E. Caicedo
    Dec 4 '18 at 5:50


















1












$begingroup$


I am trying to prove that a number is normal base b $iff$ it is simply normal in all bases $b^k$ for every integer $k geq 1$.



I'm a little confused on this because if for example we take a number that is normal is base 3 how would that be simply normal in base 9 as it would not have any digits greater than 3.



These are my definitions for normal and simply normal:



We say a number $x$ in decimal expansion form base-$b$ is simply normal base-$b$ if
$lim_{n to infty} frac{N_n^b(x;{w})}{n} = frac{1}{b},$
$forall w in {0,1,2,...b-1}$.



A number $x$ in decimal expansion form base $b$ is normal base-$b$ if for any arbitrary finite string (or word) $w$ with letters from the alphabet ${0,1,2,...,b-1}$
$ lim_{n to infty}frac{N_n^b(x;w)}{n} = frac{1}{b^{|w|}}$
where $|w|$ denotes the lengh of the word.



If somebody could help me get started on how to prove this if and only if statement that would be great.



Update: I have figured out the forward direction, I am still confused on the reverse.










share|cite|improve this question











$endgroup$












  • $begingroup$
    In base 9 it would have digits 0--8. That is, block it off in subparts of length 2 and convert each to a number from 1 to 8. Example 122122=(12)(21)(22)=(5)(7)(8) base 9. Could you put definitions of normal and simply normal in question? (not as comm4ent or link)
    $endgroup$
    – coffeemath
    Dec 4 '18 at 2:45








  • 1




    $begingroup$
    @coffeemath done
    $endgroup$
    – Sasha
    Dec 4 '18 at 3:00










  • $begingroup$
    This is not set-theory. Please do not add the tag back.
    $endgroup$
    – Andrés E. Caicedo
    Dec 4 '18 at 5:50
















1












1








1


0



$begingroup$


I am trying to prove that a number is normal base b $iff$ it is simply normal in all bases $b^k$ for every integer $k geq 1$.



I'm a little confused on this because if for example we take a number that is normal is base 3 how would that be simply normal in base 9 as it would not have any digits greater than 3.



These are my definitions for normal and simply normal:



We say a number $x$ in decimal expansion form base-$b$ is simply normal base-$b$ if
$lim_{n to infty} frac{N_n^b(x;{w})}{n} = frac{1}{b},$
$forall w in {0,1,2,...b-1}$.



A number $x$ in decimal expansion form base $b$ is normal base-$b$ if for any arbitrary finite string (or word) $w$ with letters from the alphabet ${0,1,2,...,b-1}$
$ lim_{n to infty}frac{N_n^b(x;w)}{n} = frac{1}{b^{|w|}}$
where $|w|$ denotes the lengh of the word.



If somebody could help me get started on how to prove this if and only if statement that would be great.



Update: I have figured out the forward direction, I am still confused on the reverse.










share|cite|improve this question











$endgroup$




I am trying to prove that a number is normal base b $iff$ it is simply normal in all bases $b^k$ for every integer $k geq 1$.



I'm a little confused on this because if for example we take a number that is normal is base 3 how would that be simply normal in base 9 as it would not have any digits greater than 3.



These are my definitions for normal and simply normal:



We say a number $x$ in decimal expansion form base-$b$ is simply normal base-$b$ if
$lim_{n to infty} frac{N_n^b(x;{w})}{n} = frac{1}{b},$
$forall w in {0,1,2,...b-1}$.



A number $x$ in decimal expansion form base $b$ is normal base-$b$ if for any arbitrary finite string (or word) $w$ with letters from the alphabet ${0,1,2,...,b-1}$
$ lim_{n to infty}frac{N_n^b(x;w)}{n} = frac{1}{b^{|w|}}$
where $|w|$ denotes the lengh of the word.



If somebody could help me get started on how to prove this if and only if statement that would be great.



Update: I have figured out the forward direction, I am still confused on the reverse.







number-theory






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 4 '18 at 5:50









Andrés E. Caicedo

65.1k8158247




65.1k8158247










asked Dec 4 '18 at 2:29









SashaSasha

537




537












  • $begingroup$
    In base 9 it would have digits 0--8. That is, block it off in subparts of length 2 and convert each to a number from 1 to 8. Example 122122=(12)(21)(22)=(5)(7)(8) base 9. Could you put definitions of normal and simply normal in question? (not as comm4ent or link)
    $endgroup$
    – coffeemath
    Dec 4 '18 at 2:45








  • 1




    $begingroup$
    @coffeemath done
    $endgroup$
    – Sasha
    Dec 4 '18 at 3:00










  • $begingroup$
    This is not set-theory. Please do not add the tag back.
    $endgroup$
    – Andrés E. Caicedo
    Dec 4 '18 at 5:50




















  • $begingroup$
    In base 9 it would have digits 0--8. That is, block it off in subparts of length 2 and convert each to a number from 1 to 8. Example 122122=(12)(21)(22)=(5)(7)(8) base 9. Could you put definitions of normal and simply normal in question? (not as comm4ent or link)
    $endgroup$
    – coffeemath
    Dec 4 '18 at 2:45








  • 1




    $begingroup$
    @coffeemath done
    $endgroup$
    – Sasha
    Dec 4 '18 at 3:00










  • $begingroup$
    This is not set-theory. Please do not add the tag back.
    $endgroup$
    – Andrés E. Caicedo
    Dec 4 '18 at 5:50


















$begingroup$
In base 9 it would have digits 0--8. That is, block it off in subparts of length 2 and convert each to a number from 1 to 8. Example 122122=(12)(21)(22)=(5)(7)(8) base 9. Could you put definitions of normal and simply normal in question? (not as comm4ent or link)
$endgroup$
– coffeemath
Dec 4 '18 at 2:45






$begingroup$
In base 9 it would have digits 0--8. That is, block it off in subparts of length 2 and convert each to a number from 1 to 8. Example 122122=(12)(21)(22)=(5)(7)(8) base 9. Could you put definitions of normal and simply normal in question? (not as comm4ent or link)
$endgroup$
– coffeemath
Dec 4 '18 at 2:45






1




1




$begingroup$
@coffeemath done
$endgroup$
– Sasha
Dec 4 '18 at 3:00




$begingroup$
@coffeemath done
$endgroup$
– Sasha
Dec 4 '18 at 3:00












$begingroup$
This is not set-theory. Please do not add the tag back.
$endgroup$
– Andrés E. Caicedo
Dec 4 '18 at 5:50






$begingroup$
This is not set-theory. Please do not add the tag back.
$endgroup$
– Andrés E. Caicedo
Dec 4 '18 at 5:50












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