What are binary numbers that are equal to their length?
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What are binary numbers that are equal to their length?
I am trying to find numbers where their value is equal to the length of their binary representation. They cannot start with zero.
For example:
1 = 1 and has a length of 1
10 = 2 and has a length of 2
Additional Thoughts:
Is there a name for numbers like this (maybe in different bases)?
I'm not sure if there are any more out there that exist.
Technically zero as "" is also true. It is nothing and has no numbers in it's representation, however, you can't write it. (so it's not real?)
number-theory binary
asked Nov 15 at 19:10
Tgwizman
1012
|
show 2 more comments
up vote
0
down vote
favorite
What are binary numbers that are equal to their length?
I am trying to find numbers where their value is equal to the length of their binary representation. They cannot start with zero.
For example:
1 = 1 and has a length of 1
10 = 2 and has a length of 2
Additional Thoughts:
Is there a name for numbers like this (maybe in different bases)?
I'm not sure if there are any more out there that exist.
Technically zero as "" is also true. It is nothing and has no numbers in it's representation, however, you can't write it. (so it's not real?)
number-theory binary
asked Nov 15 at 19:10
Tgwizman
1012
I guess in any base, 1 is always going to work.
– Tgwizman
Nov 15 at 19:10
4
If the number has length $n$, then it is at least $2^{n-1}$. So you need $2^{n-1}≤n$. Therefore...
– lulu
Nov 15 at 19:13
Well $0$ does have numbers in its representation; $0$. And yes $0$ is real and yes you can write it and in a way, it's not nothing. Just because "zero apples" means you have no apples, doesn't mean zero is nothing. If you have zero of something, then you have nothing. Zero itself is not nothing.
– vrugtehagel
Nov 15 at 19:17
Yes, there is a name for such numbers; it is $1$.
– Servaes
Nov 15 at 19:26
2
It is true for all numbers in the unary number system.
– Jens
Nov 15 at 19:32
|
show 2 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
What are binary numbers that are equal to their length?
I am trying to find numbers where their value is equal to the length of their binary representation. They cannot start with zero.
For example:
1 = 1 and has a length of 1
10 = 2 and has a length of 2
Additional Thoughts:
Is there a name for numbers like this (maybe in different bases)?
I'm not sure if there are any more out there that exist.
Technically zero as "" is also true. It is nothing and has no numbers in it's representation, however, you can't write it. (so it's not real?)
number-theory binary
asked Nov 15 at 19:10
Tgwizman
1012
What are binary numbers that are equal to their length?
I am trying to find numbers where their value is equal to the length of their binary representation. They cannot start with zero.
For example:
1 = 1 and has a length of 1
10 = 2 and has a length of 2
Additional Thoughts:
Is there a name for numbers like this (maybe in different bases)?
I'm not sure if there are any more out there that exist.
Technically zero as "" is also true. It is nothing and has no numbers in it's representation, however, you can't write it. (so it's not real?)
number-theory binary
number-theory binary
asked Nov 15 at 19:10
Tgwizman
1012
asked Nov 15 at 19:10
Tgwizman
1012
asked Nov 15 at 19:10
Tgwizman
1012
asked Nov 15 at 19:10
Tgwizman
1012
asked Nov 15 at 19:10
Tgwizman
1012
1012
I guess in any base, 1 is always going to work.
– Tgwizman
Nov 15 at 19:10
4
If the number has length $n$, then it is at least $2^{n-1}$. So you need $2^{n-1}≤n$. Therefore...
– lulu
Nov 15 at 19:13
Well $0$ does have numbers in its representation; $0$. And yes $0$ is real and yes you can write it and in a way, it's not nothing. Just because "zero apples" means you have no apples, doesn't mean zero is nothing. If you have zero of something, then you have nothing. Zero itself is not nothing.
– vrugtehagel
Nov 15 at 19:17
Yes, there is a name for such numbers; it is $1$.
– Servaes
Nov 15 at 19:26
2
It is true for all numbers in the unary number system.
– Jens
Nov 15 at 19:32
|
show 2 more comments
I guess in any base, 1 is always going to work.
– Tgwizman
Nov 15 at 19:10
4
If the number has length $n$, then it is at least $2^{n-1}$. So you need $2^{n-1}≤n$. Therefore...
– lulu
Nov 15 at 19:13
Well $0$ does have numbers in its representation; $0$. And yes $0$ is real and yes you can write it and in a way, it's not nothing. Just because "zero apples" means you have no apples, doesn't mean zero is nothing. If you have zero of something, then you have nothing. Zero itself is not nothing.
– vrugtehagel
Nov 15 at 19:17
Yes, there is a name for such numbers; it is $1$.
– Servaes
Nov 15 at 19:26
2
It is true for all numbers in the unary number system.
– Jens
Nov 15 at 19:32
I guess in any base, 1 is always going to work.
– Tgwizman
Nov 15 at 19:10
I guess in any base, 1 is always going to work.
– Tgwizman
Nov 15 at 19:10
4
4
If the number has length $n$, then it is at least $2^{n-1}$. So you need $2^{n-1}≤n$. Therefore...
– lulu
Nov 15 at 19:13
If the number has length $n$, then it is at least $2^{n-1}$. So you need $2^{n-1}≤n$. Therefore...
– lulu
Nov 15 at 19:13
Well $0$ does have numbers in its representation; $0$. And yes $0$ is real and yes you can write it and in a way, it's not nothing. Just because "zero apples" means you have no apples, doesn't mean zero is nothing. If you have zero of something, then you have nothing. Zero itself is not nothing.
– vrugtehagel
Nov 15 at 19:17
Well $0$ does have numbers in its representation; $0$. And yes $0$ is real and yes you can write it and in a way, it's not nothing. Just because "zero apples" means you have no apples, doesn't mean zero is nothing. If you have zero of something, then you have nothing. Zero itself is not nothing.
– vrugtehagel
Nov 15 at 19:17
Yes, there is a name for such numbers; it is $1$.
– Servaes
Nov 15 at 19:26
Yes, there is a name for such numbers; it is $1$.
– Servaes
Nov 15 at 19:26
2
2
It is true for all numbers in the unary number system.
– Jens
Nov 15 at 19:32
It is true for all numbers in the unary number system.
– Jens
Nov 15 at 19:32
|
show 2 more comments
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I guess in any base, 1 is always going to work.
– Tgwizman
Nov 15 at 19:10
4
If the number has length $n$, then it is at least $2^{n-1}$. So you need $2^{n-1}≤n$. Therefore...
– lulu
Nov 15 at 19:13
Well $0$ does have numbers in its representation; $0$. And yes $0$ is real and yes you can write it and in a way, it's not nothing. Just because "zero apples" means you have no apples, doesn't mean zero is nothing. If you have zero of something, then you have nothing. Zero itself is not nothing.
– vrugtehagel
Nov 15 at 19:17
Yes, there is a name for such numbers; it is $1$.
– Servaes
Nov 15 at 19:26
2
It is true for all numbers in the unary number system.
– Jens
Nov 15 at 19:32