What is a binary expansion of a real number?
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Related to this question I asked.
I want to know what is exactly meant by a binary expansion of a number, real or natural.
Can someone show me via example, how would you binary expand a real number like 1.575? And can someone show me how 535 would be broken up?
My next question is why does this work? Why do binary expansions work? Why can't I get the same thing out of a power of 3 or 4 or 5 for that matter?
Why does powers of 2 work?
sequences-and-series power-series binary
asked Aug 20 at 8:46
ng.newbie
1998
add a comment |Â
up vote
0
down vote
favorite
Related to this question I asked.
I want to know what is exactly meant by a binary expansion of a number, real or natural.
Can someone show me via example, how would you binary expand a real number like 1.575? And can someone show me how 535 would be broken up?
My next question is why does this work? Why do binary expansions work? Why can't I get the same thing out of a power of 3 or 4 or 5 for that matter?
Why does powers of 2 work?
sequences-and-series power-series binary
asked Aug 20 at 8:46
ng.newbie
1998
2
Of course there are base 3, 4, or 5 expansions.
– Martin R
Aug 20 at 8:50
@MartinR You mean I can get any number expanded in the base of 3 or 4? What about 2? How can I get 2 from an expansion of 3 or 4? This is really confusing
– ng.newbie
Aug 20 at 8:51
Your question is unclear (to me). Perhaps this: A "base-b" expansion ($b ge 2$) is a sum of powers of $b$, multiplied with “digits†in the range $0..b-1$. As an example, the base-2 expansion of $11$ is $2^3 + 2^1 + 2^0$, and the base-3 expansion of $11$ is $3^2 + 2 cdot 3^0$. – Base 2 is only “special†in the sense that digits are only zero or one, so every integer can be written as a sum of (different) powers of 2.
– Martin R
Aug 20 at 9:07
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Related to this question I asked.
I want to know what is exactly meant by a binary expansion of a number, real or natural.
Can someone show me via example, how would you binary expand a real number like 1.575? And can someone show me how 535 would be broken up?
My next question is why does this work? Why do binary expansions work? Why can't I get the same thing out of a power of 3 or 4 or 5 for that matter?
Why does powers of 2 work?
sequences-and-series power-series binary
asked Aug 20 at 8:46
ng.newbie
1998
Related to this question I asked.
I want to know what is exactly meant by a binary expansion of a number, real or natural.
Can someone show me via example, how would you binary expand a real number like 1.575? And can someone show me how 535 would be broken up?
My next question is why does this work? Why do binary expansions work? Why can't I get the same thing out of a power of 3 or 4 or 5 for that matter?
Why does powers of 2 work?
sequences-and-series power-series binary
asked Aug 20 at 8:46
ng.newbie
1998
asked Aug 20 at 8:46
ng.newbie
1998
asked Aug 20 at 8:46
ng.newbie
1998
asked Aug 20 at 8:46
ng.newbie
1998
1998
2
Of course there are base 3, 4, or 5 expansions.
– Martin R
Aug 20 at 8:50
@MartinR You mean I can get any number expanded in the base of 3 or 4? What about 2? How can I get 2 from an expansion of 3 or 4? This is really confusing
– ng.newbie
Aug 20 at 8:51
Your question is unclear (to me). Perhaps this: A "base-b" expansion ($b ge 2$) is a sum of powers of $b$, multiplied with “digits†in the range $0..b-1$. As an example, the base-2 expansion of $11$ is $2^3 + 2^1 + 2^0$, and the base-3 expansion of $11$ is $3^2 + 2 cdot 3^0$. – Base 2 is only “special†in the sense that digits are only zero or one, so every integer can be written as a sum of (different) powers of 2.
– Martin R
Aug 20 at 9:07
add a comment |Â
2
Of course there are base 3, 4, or 5 expansions.
– Martin R
Aug 20 at 8:50
@MartinR You mean I can get any number expanded in the base of 3 or 4? What about 2? How can I get 2 from an expansion of 3 or 4? This is really confusing
– ng.newbie
Aug 20 at 8:51
Your question is unclear (to me). Perhaps this: A "base-b" expansion ($b ge 2$) is a sum of powers of $b$, multiplied with “digits†in the range $0..b-1$. As an example, the base-2 expansion of $11$ is $2^3 + 2^1 + 2^0$, and the base-3 expansion of $11$ is $3^2 + 2 cdot 3^0$. – Base 2 is only “special†in the sense that digits are only zero or one, so every integer can be written as a sum of (different) powers of 2.
– Martin R
Aug 20 at 9:07
2
2
Of course there are base 3, 4, or 5 expansions.
– Martin R
Aug 20 at 8:50
Of course there are base 3, 4, or 5 expansions.
– Martin R
Aug 20 at 8:50
@MartinR You mean I can get any number expanded in the base of 3 or 4? What about 2? How can I get 2 from an expansion of 3 or 4? This is really confusing
– ng.newbie
Aug 20 at 8:51
@MartinR You mean I can get any number expanded in the base of 3 or 4? What about 2? How can I get 2 from an expansion of 3 or 4? This is really confusing
– ng.newbie
Aug 20 at 8:51
Your question is unclear (to me). Perhaps this: A "base-b" expansion ($b ge 2$) is a sum of powers of $b$, multiplied with “digits†in the range $0..b-1$. As an example, the base-2 expansion of $11$ is $2^3 + 2^1 + 2^0$, and the base-3 expansion of $11$ is $3^2 + 2 cdot 3^0$. – Base 2 is only “special†in the sense that digits are only zero or one, so every integer can be written as a sum of (different) powers of 2.
– Martin R
Aug 20 at 9:07
Your question is unclear (to me). Perhaps this: A "base-b" expansion ($b ge 2$) is a sum of powers of $b$, multiplied with “digits†in the range $0..b-1$. As an example, the base-2 expansion of $11$ is $2^3 + 2^1 + 2^0$, and the base-3 expansion of $11$ is $3^2 + 2 cdot 3^0$. – Base 2 is only “special†in the sense that digits are only zero or one, so every integer can be written as a sum of (different) powers of 2.
– Martin R
Aug 20 at 9:07
add a comment |Â
1 Answer
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For any base $bgeq 2$ (of course, $binmathbb N$) and any real number $x$, you can write the number in base $b$ as
$$x = a_na_n-1dots a_0.b_1b_2b_3dots$$
where this means that $$x=a_0 + a_1cdot b + a_2cdot b^2 + cdots + a_nb^n + b_1b^-1 + b_2b^-2 + cdots$$ and all numbers $a_i$ and $b_i$ are elements of $0,1,dots, b-1$
this can be done no matter what number $b$ you choose as the base. For example, in base $10$, the number $16$ is written as $16$ because $16=1cdot 10 + 6cdot 10^0$. In base $2$, the number is written as $10000$, because $$16=0cdot 2^0 + 0cdot 2^1 + 0cdot 2^2 + 0cdot 2^3 + 1cdot 2^4.$$
This is also true for numbers that are not integers, for example, in base $10$, you write $frac14$ as $0.25$ because $$frac14 = 2cdot 10^-1 + 5cdot 10^-2$$ while in base $2$, it's written as $0.01$ because $$frac14 = 0cdot 2^-1 + 1cdot 2^-2.$$
However, for non-integers, the expansions can sometimes be (more like usually is:) infinite. For example, the number $frac13$ is written as $0.3333333dots$ in base $10$, but in base $3$, it's simply $0.1$.
Naturally, for bases higher than $10$, we don't have enough symbols to write all the numbers, but we can make more up. For example, the base $16$ system uses the letters $a,b,c,d,e,f$ to represent the numbers $10,11,12,13,14,15$, meaning that $31$ in base $16$ is written as $$1f$$ because $$31 =1cdot 16 + fcdot 16^0.$$
Similarly, in that base, $$frac1416 = 0.e$$
answered Aug 20 at 9:03
5xum
82.4k383147
Can you show me the break up of 16 in base 5?
– ng.newbie
Aug 20 at 9:26
@ng.newbie sure. $15=3cdot 5^1 +1cdot5^0$ so in base $5$, the number $15$ is written as $31$
– 5xum
Aug 20 at 9:29
Okay so I have to get the representation of that number in that particular base then I can convert it. For example I first have to get the binary representation of 16 before I can expand it in base 2. Correct?
– ng.newbie
Aug 20 at 9:30
Really good and clear answer. Just for a bit of nitpicking: in your first exemple, I think it would have been clearer if you wrote the power of 2 in decreasing order, so the 10000 clearly shows himself
– F.Carette
Aug 20 at 9:32
@ng.newbie What's the difference from a binary representation and an expansion in that base?
– 5xum
Aug 20 at 10:04
 |Â
show 8 more comments
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
For any base $bgeq 2$ (of course, $binmathbb N$) and any real number $x$, you can write the number in base $b$ as
$$x = a_na_n-1dots a_0.b_1b_2b_3dots$$
where this means that $$x=a_0 + a_1cdot b + a_2cdot b^2 + cdots + a_nb^n + b_1b^-1 + b_2b^-2 + cdots$$ and all numbers $a_i$ and $b_i$ are elements of $0,1,dots, b-1$
this can be done no matter what number $b$ you choose as the base. For example, in base $10$, the number $16$ is written as $16$ because $16=1cdot 10 + 6cdot 10^0$. In base $2$, the number is written as $10000$, because $$16=0cdot 2^0 + 0cdot 2^1 + 0cdot 2^2 + 0cdot 2^3 + 1cdot 2^4.$$
This is also true for numbers that are not integers, for example, in base $10$, you write $frac14$ as $0.25$ because $$frac14 = 2cdot 10^-1 + 5cdot 10^-2$$ while in base $2$, it's written as $0.01$ because $$frac14 = 0cdot 2^-1 + 1cdot 2^-2.$$
However, for non-integers, the expansions can sometimes be (more like usually is:) infinite. For example, the number $frac13$ is written as $0.3333333dots$ in base $10$, but in base $3$, it's simply $0.1$.
Naturally, for bases higher than $10$, we don't have enough symbols to write all the numbers, but we can make more up. For example, the base $16$ system uses the letters $a,b,c,d,e,f$ to represent the numbers $10,11,12,13,14,15$, meaning that $31$ in base $16$ is written as $$1f$$ because $$31 =1cdot 16 + fcdot 16^0.$$
Similarly, in that base, $$frac1416 = 0.e$$
answered Aug 20 at 9:03
5xum
82.4k383147
Can you show me the break up of 16 in base 5?
– ng.newbie
Aug 20 at 9:26
@ng.newbie sure. $15=3cdot 5^1 +1cdot5^0$ so in base $5$, the number $15$ is written as $31$
– 5xum
Aug 20 at 9:29
Okay so I have to get the representation of that number in that particular base then I can convert it. For example I first have to get the binary representation of 16 before I can expand it in base 2. Correct?
– ng.newbie
Aug 20 at 9:30
Really good and clear answer. Just for a bit of nitpicking: in your first exemple, I think it would have been clearer if you wrote the power of 2 in decreasing order, so the 10000 clearly shows himself
– F.Carette
Aug 20 at 9:32
@ng.newbie What's the difference from a binary representation and an expansion in that base?
– 5xum
Aug 20 at 10:04
 |Â
show 8 more comments
up vote
2
down vote
accepted
For any base $bgeq 2$ (of course, $binmathbb N$) and any real number $x$, you can write the number in base $b$ as
$$x = a_na_n-1dots a_0.b_1b_2b_3dots$$
where this means that $$x=a_0 + a_1cdot b + a_2cdot b^2 + cdots + a_nb^n + b_1b^-1 + b_2b^-2 + cdots$$ and all numbers $a_i$ and $b_i$ are elements of $0,1,dots, b-1$
this can be done no matter what number $b$ you choose as the base. For example, in base $10$, the number $16$ is written as $16$ because $16=1cdot 10 + 6cdot 10^0$. In base $2$, the number is written as $10000$, because $$16=0cdot 2^0 + 0cdot 2^1 + 0cdot 2^2 + 0cdot 2^3 + 1cdot 2^4.$$
This is also true for numbers that are not integers, for example, in base $10$, you write $frac14$ as $0.25$ because $$frac14 = 2cdot 10^-1 + 5cdot 10^-2$$ while in base $2$, it's written as $0.01$ because $$frac14 = 0cdot 2^-1 + 1cdot 2^-2.$$
However, for non-integers, the expansions can sometimes be (more like usually is:) infinite. For example, the number $frac13$ is written as $0.3333333dots$ in base $10$, but in base $3$, it's simply $0.1$.
Naturally, for bases higher than $10$, we don't have enough symbols to write all the numbers, but we can make more up. For example, the base $16$ system uses the letters $a,b,c,d,e,f$ to represent the numbers $10,11,12,13,14,15$, meaning that $31$ in base $16$ is written as $$1f$$ because $$31 =1cdot 16 + fcdot 16^0.$$
Similarly, in that base, $$frac1416 = 0.e$$
answered Aug 20 at 9:03
5xum
82.4k383147
Can you show me the break up of 16 in base 5?
– ng.newbie
Aug 20 at 9:26
@ng.newbie sure. $15=3cdot 5^1 +1cdot5^0$ so in base $5$, the number $15$ is written as $31$
– 5xum
Aug 20 at 9:29
Okay so I have to get the representation of that number in that particular base then I can convert it. For example I first have to get the binary representation of 16 before I can expand it in base 2. Correct?
– ng.newbie
Aug 20 at 9:30
Really good and clear answer. Just for a bit of nitpicking: in your first exemple, I think it would have been clearer if you wrote the power of 2 in decreasing order, so the 10000 clearly shows himself
– F.Carette
Aug 20 at 9:32
@ng.newbie What's the difference from a binary representation and an expansion in that base?
– 5xum
Aug 20 at 10:04
 |Â
show 8 more comments
up vote
2
down vote
accepted
up vote
2
down vote
accepted
For any base $bgeq 2$ (of course, $binmathbb N$) and any real number $x$, you can write the number in base $b$ as
$$x = a_na_n-1dots a_0.b_1b_2b_3dots$$
where this means that $$x=a_0 + a_1cdot b + a_2cdot b^2 + cdots + a_nb^n + b_1b^-1 + b_2b^-2 + cdots$$ and all numbers $a_i$ and $b_i$ are elements of $0,1,dots, b-1$
this can be done no matter what number $b$ you choose as the base. For example, in base $10$, the number $16$ is written as $16$ because $16=1cdot 10 + 6cdot 10^0$. In base $2$, the number is written as $10000$, because $$16=0cdot 2^0 + 0cdot 2^1 + 0cdot 2^2 + 0cdot 2^3 + 1cdot 2^4.$$
This is also true for numbers that are not integers, for example, in base $10$, you write $frac14$ as $0.25$ because $$frac14 = 2cdot 10^-1 + 5cdot 10^-2$$ while in base $2$, it's written as $0.01$ because $$frac14 = 0cdot 2^-1 + 1cdot 2^-2.$$
However, for non-integers, the expansions can sometimes be (more like usually is:) infinite. For example, the number $frac13$ is written as $0.3333333dots$ in base $10$, but in base $3$, it's simply $0.1$.
Naturally, for bases higher than $10$, we don't have enough symbols to write all the numbers, but we can make more up. For example, the base $16$ system uses the letters $a,b,c,d,e,f$ to represent the numbers $10,11,12,13,14,15$, meaning that $31$ in base $16$ is written as $$1f$$ because $$31 =1cdot 16 + fcdot 16^0.$$
Similarly, in that base, $$frac1416 = 0.e$$
answered Aug 20 at 9:03
5xum
82.4k383147
For any base $bgeq 2$ (of course, $binmathbb N$) and any real number $x$, you can write the number in base $b$ as
$$x = a_na_n-1dots a_0.b_1b_2b_3dots$$
where this means that $$x=a_0 + a_1cdot b + a_2cdot b^2 + cdots + a_nb^n + b_1b^-1 + b_2b^-2 + cdots$$ and all numbers $a_i$ and $b_i$ are elements of $0,1,dots, b-1$
this can be done no matter what number $b$ you choose as the base. For example, in base $10$, the number $16$ is written as $16$ because $16=1cdot 10 + 6cdot 10^0$. In base $2$, the number is written as $10000$, because $$16=0cdot 2^0 + 0cdot 2^1 + 0cdot 2^2 + 0cdot 2^3 + 1cdot 2^4.$$
This is also true for numbers that are not integers, for example, in base $10$, you write $frac14$ as $0.25$ because $$frac14 = 2cdot 10^-1 + 5cdot 10^-2$$ while in base $2$, it's written as $0.01$ because $$frac14 = 0cdot 2^-1 + 1cdot 2^-2.$$
However, for non-integers, the expansions can sometimes be (more like usually is:) infinite. For example, the number $frac13$ is written as $0.3333333dots$ in base $10$, but in base $3$, it's simply $0.1$.
Naturally, for bases higher than $10$, we don't have enough symbols to write all the numbers, but we can make more up. For example, the base $16$ system uses the letters $a,b,c,d,e,f$ to represent the numbers $10,11,12,13,14,15$, meaning that $31$ in base $16$ is written as $$1f$$ because $$31 =1cdot 16 + fcdot 16^0.$$
Similarly, in that base, $$frac1416 = 0.e$$
answered Aug 20 at 9:03
5xum
82.4k383147
edited Aug 20 at 9:09
answered Aug 20 at 9:03
5xum
82.4k383147
answered Aug 20 at 9:03
5xum
82.4k383147
answered Aug 20 at 9:03
5xum
82.4k383147
82.4k383147
Can you show me the break up of 16 in base 5?
– ng.newbie
Aug 20 at 9:26
@ng.newbie sure. $15=3cdot 5^1 +1cdot5^0$ so in base $5$, the number $15$ is written as $31$
– 5xum
Aug 20 at 9:29
Okay so I have to get the representation of that number in that particular base then I can convert it. For example I first have to get the binary representation of 16 before I can expand it in base 2. Correct?
– ng.newbie
Aug 20 at 9:30
Really good and clear answer. Just for a bit of nitpicking: in your first exemple, I think it would have been clearer if you wrote the power of 2 in decreasing order, so the 10000 clearly shows himself
– F.Carette
Aug 20 at 9:32
@ng.newbie What's the difference from a binary representation and an expansion in that base?
– 5xum
Aug 20 at 10:04
 |Â
show 8 more comments
Can you show me the break up of 16 in base 5?
– ng.newbie
Aug 20 at 9:26
@ng.newbie sure. $15=3cdot 5^1 +1cdot5^0$ so in base $5$, the number $15$ is written as $31$
– 5xum
Aug 20 at 9:29
Okay so I have to get the representation of that number in that particular base then I can convert it. For example I first have to get the binary representation of 16 before I can expand it in base 2. Correct?
– ng.newbie
Aug 20 at 9:30
Really good and clear answer. Just for a bit of nitpicking: in your first exemple, I think it would have been clearer if you wrote the power of 2 in decreasing order, so the 10000 clearly shows himself
– F.Carette
Aug 20 at 9:32
@ng.newbie What's the difference from a binary representation and an expansion in that base?
– 5xum
Aug 20 at 10:04
Can you show me the break up of 16 in base 5?
– ng.newbie
Aug 20 at 9:26
Can you show me the break up of 16 in base 5?
– ng.newbie
Aug 20 at 9:26
@ng.newbie sure. $15=3cdot 5^1 +1cdot5^0$ so in base $5$, the number $15$ is written as $31$
– 5xum
Aug 20 at 9:29
@ng.newbie sure. $15=3cdot 5^1 +1cdot5^0$ so in base $5$, the number $15$ is written as $31$
– 5xum
Aug 20 at 9:29
Okay so I have to get the representation of that number in that particular base then I can convert it. For example I first have to get the binary representation of 16 before I can expand it in base 2. Correct?
– ng.newbie
Aug 20 at 9:30
Okay so I have to get the representation of that number in that particular base then I can convert it. For example I first have to get the binary representation of 16 before I can expand it in base 2. Correct?
– ng.newbie
Aug 20 at 9:30
Really good and clear answer. Just for a bit of nitpicking: in your first exemple, I think it would have been clearer if you wrote the power of 2 in decreasing order, so the 10000 clearly shows himself
– F.Carette
Aug 20 at 9:32
Really good and clear answer. Just for a bit of nitpicking: in your first exemple, I think it would have been clearer if you wrote the power of 2 in decreasing order, so the 10000 clearly shows himself
– F.Carette
Aug 20 at 9:32
@ng.newbie What's the difference from a binary representation and an expansion in that base?
– 5xum
Aug 20 at 10:04
@ng.newbie What's the difference from a binary representation and an expansion in that base?
– 5xum
Aug 20 at 10:04
 |Â
show 8 more comments
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2
Of course there are base 3, 4, or 5 expansions.
– Martin R
Aug 20 at 8:50
@MartinR You mean I can get any number expanded in the base of 3 or 4? What about 2? How can I get 2 from an expansion of 3 or 4? This is really confusing
– ng.newbie
Aug 20 at 8:51
Your question is unclear (to me). Perhaps this: A "base-b" expansion ($b ge 2$) is a sum of powers of $b$, multiplied with “digits†in the range $0..b-1$. As an example, the base-2 expansion of $11$ is $2^3 + 2^1 + 2^0$, and the base-3 expansion of $11$ is $3^2 + 2 cdot 3^0$. – Base 2 is only “special†in the sense that digits are only zero or one, so every integer can be written as a sum of (different) powers of 2.
– Martin R
Aug 20 at 9:07