Infinite non-periodic binary fraction
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I have an infinite non-periodic binary fraction. For example:
$frac_1=0.101111011100110010001001010010000001001...$
Is it always true that $1-frac_1$ = non-periodic binary fraction?
fractions
asked Nov 19 at 19:50
bvl
72
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up vote
0
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favorite
I have an infinite non-periodic binary fraction. For example:
$frac_1=0.101111011100110010001001010010000001001...$
Is it always true that $1-frac_1$ = non-periodic binary fraction?
fractions
asked Nov 19 at 19:50
bvl
72
By non-periodic, I suppose you mean not "ultimately periodic". The answer is quite obviously yes. Notice that $1-frac_1$ has the same binary expansion where you switch $0$ and $1$.
– Jean-Claude Arbaut
Nov 19 at 19:54
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have an infinite non-periodic binary fraction. For example:
$frac_1=0.101111011100110010001001010010000001001...$
Is it always true that $1-frac_1$ = non-periodic binary fraction?
fractions
asked Nov 19 at 19:50
bvl
72
I have an infinite non-periodic binary fraction. For example:
$frac_1=0.101111011100110010001001010010000001001...$
Is it always true that $1-frac_1$ = non-periodic binary fraction?
fractions
fractions
asked Nov 19 at 19:50
bvl
72
asked Nov 19 at 19:50
bvl
72
asked Nov 19 at 19:50
bvl
72
asked Nov 19 at 19:50
bvl
72
asked Nov 19 at 19:50
bvl
72
72
By non-periodic, I suppose you mean not "ultimately periodic". The answer is quite obviously yes. Notice that $1-frac_1$ has the same binary expansion where you switch $0$ and $1$.
– Jean-Claude Arbaut
Nov 19 at 19:54
add a comment |
By non-periodic, I suppose you mean not "ultimately periodic". The answer is quite obviously yes. Notice that $1-frac_1$ has the same binary expansion where you switch $0$ and $1$.
– Jean-Claude Arbaut
Nov 19 at 19:54
By non-periodic, I suppose you mean not "ultimately periodic". The answer is quite obviously yes. Notice that $1-frac_1$ has the same binary expansion where you switch $0$ and $1$.
– Jean-Claude Arbaut
Nov 19 at 19:54
By non-periodic, I suppose you mean not "ultimately periodic". The answer is quite obviously yes. Notice that $1-frac_1$ has the same binary expansion where you switch $0$ and $1$.
– Jean-Claude Arbaut
Nov 19 at 19:54
add a comment |
1 Answer
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Well, yes. This is actually quite obvious.
If a fraction repeats (is periodic) then it is a fraction. If it's not, then it's not a fraction (that is, it's irrational).
Clearly, a number $q$ is a fraction if and only if $1-q$ is. Hence, if the digits of some number $q$ do not repeat, neither will the digits of $1-q$.
In fact, as pointed out in the comments, it's even easier to see it when you realize $1-q$ (if it's binary) is just $q$ with the digits flipped (changing $0$s into $1$s and vice versa). So, if $q$ doesn't repeat, neither will $1-q$.
answered Nov 19 at 19:59
vrugtehagel
10.7k1549
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
Well, yes. This is actually quite obvious.
If a fraction repeats (is periodic) then it is a fraction. If it's not, then it's not a fraction (that is, it's irrational).
Clearly, a number $q$ is a fraction if and only if $1-q$ is. Hence, if the digits of some number $q$ do not repeat, neither will the digits of $1-q$.
In fact, as pointed out in the comments, it's even easier to see it when you realize $1-q$ (if it's binary) is just $q$ with the digits flipped (changing $0$s into $1$s and vice versa). So, if $q$ doesn't repeat, neither will $1-q$.
answered Nov 19 at 19:59
vrugtehagel
10.7k1549
add a comment |
up vote
0
down vote
accepted
Well, yes. This is actually quite obvious.
If a fraction repeats (is periodic) then it is a fraction. If it's not, then it's not a fraction (that is, it's irrational).
Clearly, a number $q$ is a fraction if and only if $1-q$ is. Hence, if the digits of some number $q$ do not repeat, neither will the digits of $1-q$.
In fact, as pointed out in the comments, it's even easier to see it when you realize $1-q$ (if it's binary) is just $q$ with the digits flipped (changing $0$s into $1$s and vice versa). So, if $q$ doesn't repeat, neither will $1-q$.
answered Nov 19 at 19:59
vrugtehagel
10.7k1549
add a comment |
up vote
0
down vote
accepted
up vote
0
down vote
accepted
Well, yes. This is actually quite obvious.
If a fraction repeats (is periodic) then it is a fraction. If it's not, then it's not a fraction (that is, it's irrational).
Clearly, a number $q$ is a fraction if and only if $1-q$ is. Hence, if the digits of some number $q$ do not repeat, neither will the digits of $1-q$.
In fact, as pointed out in the comments, it's even easier to see it when you realize $1-q$ (if it's binary) is just $q$ with the digits flipped (changing $0$s into $1$s and vice versa). So, if $q$ doesn't repeat, neither will $1-q$.
answered Nov 19 at 19:59
vrugtehagel
10.7k1549
Well, yes. This is actually quite obvious.
If a fraction repeats (is periodic) then it is a fraction. If it's not, then it's not a fraction (that is, it's irrational).
Clearly, a number $q$ is a fraction if and only if $1-q$ is. Hence, if the digits of some number $q$ do not repeat, neither will the digits of $1-q$.
In fact, as pointed out in the comments, it's even easier to see it when you realize $1-q$ (if it's binary) is just $q$ with the digits flipped (changing $0$s into $1$s and vice versa). So, if $q$ doesn't repeat, neither will $1-q$.
answered Nov 19 at 19:59
vrugtehagel
10.7k1549
answered Nov 19 at 19:59
vrugtehagel
10.7k1549
answered Nov 19 at 19:59
vrugtehagel
10.7k1549
answered Nov 19 at 19:59
vrugtehagel
10.7k1549
10.7k1549
add a comment |
add a comment |
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By non-periodic, I suppose you mean not "ultimately periodic". The answer is quite obviously yes. Notice that $1-frac_1$ has the same binary expansion where you switch $0$ and $1$.
– Jean-Claude Arbaut
Nov 19 at 19:54