A number is normal base b iff it is simply normal in bases $b^k$
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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
edited Dec 4 '18 at 5:50
Andrés E. Caicedo
65.5k8159250
asked Dec 4 '18 at 2:29
SashaSasha
658
$endgroup$
add a comment |
$begingroup$
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.
number-theory
edited Dec 4 '18 at 5:50
Andrés E. Caicedo
65.5k8159250
asked Dec 4 '18 at 2:29
SashaSasha
658
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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.
number-theory
edited Dec 4 '18 at 5:50
Andrés E. Caicedo
65.5k8159250
asked Dec 4 '18 at 2:29
SashaSasha
658
$endgroup$
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
number-theory
edited Dec 4 '18 at 5:50
Andrés E. Caicedo
65.5k8159250
asked Dec 4 '18 at 2:29
SashaSasha
658
edited Dec 4 '18 at 5:50
Andrés E. Caicedo
65.5k8159250
asked Dec 4 '18 at 2:29
SashaSasha
658
edited Dec 4 '18 at 5:50
Andrés E. Caicedo
65.5k8159250
edited Dec 4 '18 at 5:50
Andrés E. Caicedo
65.5k8159250
edited Dec 4 '18 at 5:50
Andrés E. Caicedo
65.5k8159250
65.5k8159250
asked Dec 4 '18 at 2:29
SashaSasha
658
asked Dec 4 '18 at 2:29
SashaSasha
658
asked Dec 4 '18 at 2:29
SashaSasha
658
658
$begingroup$
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
add a comment |
$begingroup$
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
add a comment |
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$begingroup$
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