Properties of Normal Numbers
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I found these two properties of normal numbers on the Wikipedia page for "normal numbers", but I can't find any other source for them. I was wondering if they were in fact true, and if somebody could give me a hint on how to prove them.
- Any number is the product of two absolutely normal numbers.
- If $x$ is normal in base $b$, and $q neq 0$ is a rational number, then $ xcdot q$ is normal in base $b$.
I was thinking that 1 could sort of follow from Borel's proof that almost every real number is normal. However I'm not sure about the second.
number-theory lebesgue-measure
edited Dec 6 '18 at 20:17
Blue
47.8k870152
asked Dec 6 '18 at 20:09
SashaSasha
537
$endgroup$
add a comment |
$begingroup$
I found these two properties of normal numbers on the Wikipedia page for "normal numbers", but I can't find any other source for them. I was wondering if they were in fact true, and if somebody could give me a hint on how to prove them.
- Any number is the product of two absolutely normal numbers.
- If $x$ is normal in base $b$, and $q neq 0$ is a rational number, then $ xcdot q$ is normal in base $b$.
I was thinking that 1 could sort of follow from Borel's proof that almost every real number is normal. However I'm not sure about the second.
number-theory lebesgue-measure
edited Dec 6 '18 at 20:17
Blue
47.8k870152
asked Dec 6 '18 at 20:09
SashaSasha
537
$endgroup$
add a comment |
$begingroup$
I found these two properties of normal numbers on the Wikipedia page for "normal numbers", but I can't find any other source for them. I was wondering if they were in fact true, and if somebody could give me a hint on how to prove them.
- Any number is the product of two absolutely normal numbers.
- If $x$ is normal in base $b$, and $q neq 0$ is a rational number, then $ xcdot q$ is normal in base $b$.
I was thinking that 1 could sort of follow from Borel's proof that almost every real number is normal. However I'm not sure about the second.
number-theory lebesgue-measure
edited Dec 6 '18 at 20:17
Blue
47.8k870152
asked Dec 6 '18 at 20:09
SashaSasha
537
$endgroup$
I found these two properties of normal numbers on the Wikipedia page for "normal numbers", but I can't find any other source for them. I was wondering if they were in fact true, and if somebody could give me a hint on how to prove them.
- Any number is the product of two absolutely normal numbers.
- If $x$ is normal in base $b$, and $q neq 0$ is a rational number, then $ xcdot q$ is normal in base $b$.
I was thinking that 1 could sort of follow from Borel's proof that almost every real number is normal. However I'm not sure about the second.
number-theory lebesgue-measure
number-theory lebesgue-measure
edited Dec 6 '18 at 20:17
Blue
47.8k870152
asked Dec 6 '18 at 20:09
SashaSasha
537
edited Dec 6 '18 at 20:17
Blue
47.8k870152
asked Dec 6 '18 at 20:09
SashaSasha
537
edited Dec 6 '18 at 20:17
Blue
47.8k870152
edited Dec 6 '18 at 20:17
Blue
47.8k870152
edited Dec 6 '18 at 20:17
Blue
47.8k870152
47.8k870152
asked Dec 6 '18 at 20:09
SashaSasha
537
asked Dec 6 '18 at 20:09
SashaSasha
537
asked Dec 6 '18 at 20:09
SashaSasha
537
537
add a comment |
add a comment |
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