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Adding infinite and finite numbers: why doesn't 0=1?
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Okay, so,
$$infty + 1 = infty$$
subtract infinity from both sides.
$$1=0$$
At first I thought, duh, $infty neq infty+1$, but, now, I'm just more confused because my brother rephrased it in terms of geometry, and it seems to hold there i.e., if you have a ray of infinite (unbounded) length, and then you start a parallel ray one unit behind it, how long is the new ray? I want to say infinite, but then, if you subtract the length of the ray beside it, then the result is the same as in the first problem.
Is there any way someone could explain why this doesn't work?
infinity fake-proofs paradoxes
edited Jul 17 at 1:54
smci
348211
asked Jul 16 at 19:22
lware
234
add a comment |Â
up vote
0
down vote
favorite
Okay, so,
$$infty + 1 = infty$$
subtract infinity from both sides.
$$1=0$$
At first I thought, duh, $infty neq infty+1$, but, now, I'm just more confused because my brother rephrased it in terms of geometry, and it seems to hold there i.e., if you have a ray of infinite (unbounded) length, and then you start a parallel ray one unit behind it, how long is the new ray? I want to say infinite, but then, if you subtract the length of the ray beside it, then the result is the same as in the first problem.
Is there any way someone could explain why this doesn't work?
infinity fake-proofs paradoxes
edited Jul 17 at 1:54
smci
348211
asked Jul 16 at 19:22
lware
234
13
Infinity is not a number, hence you cannot apply normal arithmetic to it. You could argue that $lim_x to infty x=lim_xto infty x+1$ but that holds a different meaning.
– Sorfosh
Jul 16 at 19:24
You can't subtract infinity from both sides because it is not a number
– gd1035
Jul 16 at 19:25
9
Some people work with $infty$ as a "number", but even those won't even dare to subtract infinity from infinity..
– Shashi
Jul 16 at 19:45
1
Many answers have mentioned that "infinity is not a number," but not many have tried to explain why. If you are the curious type, I recommend watching the VSauce video How to Count Past Infinity which is the best layman's introduction to how infinities work that I know of. It does a great job of capturing some of the wierdnesses that appear around infinities and how we try to conquer them.
– Cort Ammon
Jul 17 at 0:48
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Okay, so,
$$infty + 1 = infty$$
subtract infinity from both sides.
$$1=0$$
At first I thought, duh, $infty neq infty+1$, but, now, I'm just more confused because my brother rephrased it in terms of geometry, and it seems to hold there i.e., if you have a ray of infinite (unbounded) length, and then you start a parallel ray one unit behind it, how long is the new ray? I want to say infinite, but then, if you subtract the length of the ray beside it, then the result is the same as in the first problem.
Is there any way someone could explain why this doesn't work?
infinity fake-proofs paradoxes
edited Jul 17 at 1:54
smci
348211
asked Jul 16 at 19:22
lware
234
Okay, so,
$$infty + 1 = infty$$
subtract infinity from both sides.
$$1=0$$
At first I thought, duh, $infty neq infty+1$, but, now, I'm just more confused because my brother rephrased it in terms of geometry, and it seems to hold there i.e., if you have a ray of infinite (unbounded) length, and then you start a parallel ray one unit behind it, how long is the new ray? I want to say infinite, but then, if you subtract the length of the ray beside it, then the result is the same as in the first problem.
Is there any way someone could explain why this doesn't work?
infinity fake-proofs paradoxes
edited Jul 17 at 1:54
smci
348211
asked Jul 16 at 19:22
lware
234
edited Jul 17 at 1:54
smci
348211
edited Jul 17 at 1:54
smci
348211
edited Jul 17 at 1:54
smci
348211
348211
asked Jul 16 at 19:22
lware
234
asked Jul 16 at 19:22
lware
234
asked Jul 16 at 19:22
lware
234
234
13
Infinity is not a number, hence you cannot apply normal arithmetic to it. You could argue that $lim_x to infty x=lim_xto infty x+1$ but that holds a different meaning.
– Sorfosh
Jul 16 at 19:24
You can't subtract infinity from both sides because it is not a number
– gd1035
Jul 16 at 19:25
9
Some people work with $infty$ as a "number", but even those won't even dare to subtract infinity from infinity..
– Shashi
Jul 16 at 19:45
1
Many answers have mentioned that "infinity is not a number," but not many have tried to explain why. If you are the curious type, I recommend watching the VSauce video How to Count Past Infinity which is the best layman's introduction to how infinities work that I know of. It does a great job of capturing some of the wierdnesses that appear around infinities and how we try to conquer them.
– Cort Ammon
Jul 17 at 0:48
add a comment |Â
13
Infinity is not a number, hence you cannot apply normal arithmetic to it. You could argue that $lim_x to infty x=lim_xto infty x+1$ but that holds a different meaning.
– Sorfosh
Jul 16 at 19:24
You can't subtract infinity from both sides because it is not a number
– gd1035
Jul 16 at 19:25
9
Some people work with $infty$ as a "number", but even those won't even dare to subtract infinity from infinity..
– Shashi
Jul 16 at 19:45
1
Many answers have mentioned that "infinity is not a number," but not many have tried to explain why. If you are the curious type, I recommend watching the VSauce video How to Count Past Infinity which is the best layman's introduction to how infinities work that I know of. It does a great job of capturing some of the wierdnesses that appear around infinities and how we try to conquer them.
– Cort Ammon
Jul 17 at 0:48
13
13
Infinity is not a number, hence you cannot apply normal arithmetic to it. You could argue that $lim_x to infty x=lim_xto infty x+1$ but that holds a different meaning.
– Sorfosh
Jul 16 at 19:24
Infinity is not a number, hence you cannot apply normal arithmetic to it. You could argue that $lim_x to infty x=lim_xto infty x+1$ but that holds a different meaning.
– Sorfosh
Jul 16 at 19:24
You can't subtract infinity from both sides because it is not a number
– gd1035
Jul 16 at 19:25
You can't subtract infinity from both sides because it is not a number
– gd1035
Jul 16 at 19:25
9
9
Some people work with $infty$ as a "number", but even those won't even dare to subtract infinity from infinity..
– Shashi
Jul 16 at 19:45
Some people work with $infty$ as a "number", but even those won't even dare to subtract infinity from infinity..
– Shashi
Jul 16 at 19:45
1
1
Many answers have mentioned that "infinity is not a number," but not many have tried to explain why. If you are the curious type, I recommend watching the VSauce video How to Count Past Infinity which is the best layman's introduction to how infinities work that I know of. It does a great job of capturing some of the wierdnesses that appear around infinities and how we try to conquer them.
– Cort Ammon
Jul 17 at 0:48
Many answers have mentioned that "infinity is not a number," but not many have tried to explain why. If you are the curious type, I recommend watching the VSauce video How to Count Past Infinity which is the best layman's introduction to how infinities work that I know of. It does a great job of capturing some of the wierdnesses that appear around infinities and how we try to conquer them.
– Cort Ammon
Jul 17 at 0:48
add a comment |Â
4 Answers
4
active
oldest
votes
up vote
5
down vote
accepted
When we extend arithmetic to include $pm infty$, the arithmetic operations (and various other functions) are defined by continuous extension.
This means that we leave $infty - infty$ undefined (much like how we leave $1/0$ defined).
In more detail, if $x$ and $y$ are extended real numbers, then we define subtraction in the maximal way so that, whenever subtraction is defined, we have
$$ left( lim_n x_n right) - left( lim_n y_n right)
= lim_n left( x_n - y_n right) $$
This means that $x-y$ is only defined if the right hand side has the same value for every pair of sequences $x_n$ and $y_n$ that converge to $x$ and $y$ respectively.
It's easy to find two sequences that converge to different limits:
- If $x_n = y_n = n$, the right hand side is zero
- If $x_n = n+1$ and $y_n = n$, the right hand side is one
Thus, we don't define $infty - infty$.
answered Jul 16 at 23:25
Hurkyl
108k9112253
add a comment |Â
up vote
6
down vote
The whole idea of $infty = infty + 1$ is not properly defined, as is often the case for arithmetic and even geometry involving infinity. Length is defined to be a real number and, as has already been pointed out, infinity is not a real number. So the idea that there are rays of infinite length is not properly defined.
There are instances outside usual geometry and arithmetic where infinity (or rather infinities) are defined and equations with them can be created and make sense. I would direct you to learning about ordinal numbers (https://en.wikipedia.org/wiki/Ordinal_number#Arithmetic_of_ordinals) and cardinal numbers (https://en.wikipedia.org/wiki/Cardinal_number#Cardinal_arithmetic) if you're looking for arithmetic involving the infinite.
For example, $omega$ is defined as the first infinite ordinal and arithmetic with it can be kind of interesting. For example, $$omega + 1 neq omega$$ but $$1 + omega = omega$$ So some of the more comfortable and familiar properties like commutativity of numbers breaks down with infinity. And as such, thought experiments like mental pictures involving lines and rays, or sets and orderings need to have their terms carefully defined in order to have any real meaning.
answered Jul 16 at 19:35
Allen O'Hara
1086
add a comment |Â
up vote
6
down vote
Hint:
Here is a quiz: as $infty+1=infty$, the two sides are interchangeable.
So what is meant by $infty-infty$ ?
- a: $infty-infty$,
- b: $infty+1-infty$,
- c: $infty-(infty+1)$,
- d: $infty+1-(infty+1)$,
- e: none of these,
- f: all of these.
answered Jul 16 at 19:33
Yves Daoust
111k665204
I can't see how this is a helpful answer to OP. It seems to be a comment, since it does not answer nothing, although provides interesting insights.
– rafa11111
Jul 16 at 19:41
I understand your frustration with this often-asked question. That said, it's hard to see how this could lead to the OP realizing "I see - the usual rules of arithmetic can't be consistent if infinity is a number. Now I understand."
– Ethan Bolker
Jul 16 at 20:02
@EthanBolker: no, this is not my message. The message, though not made explicit is "how could you deal with something that has several values that are at the same time different and equal" ? Infinity doesn't have a "memory" of what was added to it.
– Yves Daoust
Jul 16 at 20:07
@rafa11111: an answer need not be written in the affirmative mode.
– Yves Daoust
Jul 16 at 20:09
7
I like this answer. Makes you realize why infinity cannot be a number, instead of just telling you that it is not. Proof by contradiction of sorts.
– Sorfosh
Jul 16 at 20:57
 |Â
show 1 more comment
up vote
2
down vote
$infty$ is not a number. On the real number line, $mathbbR$ we only have numbers. The extended real numbers, $overlinemathbbR:=mathbbR cup-infty;infty$ that has a property that $forall x in mathbbR$, $infty + x =infty$ so you couldn't subtract it. This also holds for other fields rather than $mathbbR$.
answered Jul 16 at 19:31
Mario 04
6013
add a comment |Â
4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
5
down vote
accepted
When we extend arithmetic to include $pm infty$, the arithmetic operations (and various other functions) are defined by continuous extension.
This means that we leave $infty - infty$ undefined (much like how we leave $1/0$ defined).
In more detail, if $x$ and $y$ are extended real numbers, then we define subtraction in the maximal way so that, whenever subtraction is defined, we have
$$ left( lim_n x_n right) - left( lim_n y_n right)
= lim_n left( x_n - y_n right) $$
This means that $x-y$ is only defined if the right hand side has the same value for every pair of sequences $x_n$ and $y_n$ that converge to $x$ and $y$ respectively.
It's easy to find two sequences that converge to different limits:
- If $x_n = y_n = n$, the right hand side is zero
- If $x_n = n+1$ and $y_n = n$, the right hand side is one
Thus, we don't define $infty - infty$.
answered Jul 16 at 23:25
Hurkyl
108k9112253
add a comment |Â
up vote
5
down vote
accepted
When we extend arithmetic to include $pm infty$, the arithmetic operations (and various other functions) are defined by continuous extension.
This means that we leave $infty - infty$ undefined (much like how we leave $1/0$ defined).
In more detail, if $x$ and $y$ are extended real numbers, then we define subtraction in the maximal way so that, whenever subtraction is defined, we have
$$ left( lim_n x_n right) - left( lim_n y_n right)
= lim_n left( x_n - y_n right) $$
This means that $x-y$ is only defined if the right hand side has the same value for every pair of sequences $x_n$ and $y_n$ that converge to $x$ and $y$ respectively.
It's easy to find two sequences that converge to different limits:
- If $x_n = y_n = n$, the right hand side is zero
- If $x_n = n+1$ and $y_n = n$, the right hand side is one
Thus, we don't define $infty - infty$.
answered Jul 16 at 23:25
Hurkyl
108k9112253
add a comment |Â
up vote
5
down vote
accepted
up vote
5
down vote
accepted
When we extend arithmetic to include $pm infty$, the arithmetic operations (and various other functions) are defined by continuous extension.
This means that we leave $infty - infty$ undefined (much like how we leave $1/0$ defined).
In more detail, if $x$ and $y$ are extended real numbers, then we define subtraction in the maximal way so that, whenever subtraction is defined, we have
$$ left( lim_n x_n right) - left( lim_n y_n right)
= lim_n left( x_n - y_n right) $$
This means that $x-y$ is only defined if the right hand side has the same value for every pair of sequences $x_n$ and $y_n$ that converge to $x$ and $y$ respectively.
It's easy to find two sequences that converge to different limits:
- If $x_n = y_n = n$, the right hand side is zero
- If $x_n = n+1$ and $y_n = n$, the right hand side is one
Thus, we don't define $infty - infty$.
answered Jul 16 at 23:25
Hurkyl
108k9112253
When we extend arithmetic to include $pm infty$, the arithmetic operations (and various other functions) are defined by continuous extension.
This means that we leave $infty - infty$ undefined (much like how we leave $1/0$ defined).
In more detail, if $x$ and $y$ are extended real numbers, then we define subtraction in the maximal way so that, whenever subtraction is defined, we have
$$ left( lim_n x_n right) - left( lim_n y_n right)
= lim_n left( x_n - y_n right) $$
This means that $x-y$ is only defined if the right hand side has the same value for every pair of sequences $x_n$ and $y_n$ that converge to $x$ and $y$ respectively.
It's easy to find two sequences that converge to different limits:
- If $x_n = y_n = n$, the right hand side is zero
- If $x_n = n+1$ and $y_n = n$, the right hand side is one
Thus, we don't define $infty - infty$.
answered Jul 16 at 23:25
Hurkyl
108k9112253
answered Jul 16 at 23:25
Hurkyl
108k9112253
answered Jul 16 at 23:25
Hurkyl
108k9112253
answered Jul 16 at 23:25
Hurkyl
108k9112253
108k9112253
add a comment |Â
add a comment |Â
up vote
6
down vote
The whole idea of $infty = infty + 1$ is not properly defined, as is often the case for arithmetic and even geometry involving infinity. Length is defined to be a real number and, as has already been pointed out, infinity is not a real number. So the idea that there are rays of infinite length is not properly defined.
There are instances outside usual geometry and arithmetic where infinity (or rather infinities) are defined and equations with them can be created and make sense. I would direct you to learning about ordinal numbers (https://en.wikipedia.org/wiki/Ordinal_number#Arithmetic_of_ordinals) and cardinal numbers (https://en.wikipedia.org/wiki/Cardinal_number#Cardinal_arithmetic) if you're looking for arithmetic involving the infinite.
For example, $omega$ is defined as the first infinite ordinal and arithmetic with it can be kind of interesting. For example, $$omega + 1 neq omega$$ but $$1 + omega = omega$$ So some of the more comfortable and familiar properties like commutativity of numbers breaks down with infinity. And as such, thought experiments like mental pictures involving lines and rays, or sets and orderings need to have their terms carefully defined in order to have any real meaning.
answered Jul 16 at 19:35
Allen O'Hara
1086
add a comment |Â
up vote
6
down vote
The whole idea of $infty = infty + 1$ is not properly defined, as is often the case for arithmetic and even geometry involving infinity. Length is defined to be a real number and, as has already been pointed out, infinity is not a real number. So the idea that there are rays of infinite length is not properly defined.
There are instances outside usual geometry and arithmetic where infinity (or rather infinities) are defined and equations with them can be created and make sense. I would direct you to learning about ordinal numbers (https://en.wikipedia.org/wiki/Ordinal_number#Arithmetic_of_ordinals) and cardinal numbers (https://en.wikipedia.org/wiki/Cardinal_number#Cardinal_arithmetic) if you're looking for arithmetic involving the infinite.
For example, $omega$ is defined as the first infinite ordinal and arithmetic with it can be kind of interesting. For example, $$omega + 1 neq omega$$ but $$1 + omega = omega$$ So some of the more comfortable and familiar properties like commutativity of numbers breaks down with infinity. And as such, thought experiments like mental pictures involving lines and rays, or sets and orderings need to have their terms carefully defined in order to have any real meaning.
answered Jul 16 at 19:35
Allen O'Hara
1086
add a comment |Â
up vote
6
down vote
up vote
6
down vote
The whole idea of $infty = infty + 1$ is not properly defined, as is often the case for arithmetic and even geometry involving infinity. Length is defined to be a real number and, as has already been pointed out, infinity is not a real number. So the idea that there are rays of infinite length is not properly defined.
There are instances outside usual geometry and arithmetic where infinity (or rather infinities) are defined and equations with them can be created and make sense. I would direct you to learning about ordinal numbers (https://en.wikipedia.org/wiki/Ordinal_number#Arithmetic_of_ordinals) and cardinal numbers (https://en.wikipedia.org/wiki/Cardinal_number#Cardinal_arithmetic) if you're looking for arithmetic involving the infinite.
For example, $omega$ is defined as the first infinite ordinal and arithmetic with it can be kind of interesting. For example, $$omega + 1 neq omega$$ but $$1 + omega = omega$$ So some of the more comfortable and familiar properties like commutativity of numbers breaks down with infinity. And as such, thought experiments like mental pictures involving lines and rays, or sets and orderings need to have their terms carefully defined in order to have any real meaning.
answered Jul 16 at 19:35
Allen O'Hara
1086
The whole idea of $infty = infty + 1$ is not properly defined, as is often the case for arithmetic and even geometry involving infinity. Length is defined to be a real number and, as has already been pointed out, infinity is not a real number. So the idea that there are rays of infinite length is not properly defined.
There are instances outside usual geometry and arithmetic where infinity (or rather infinities) are defined and equations with them can be created and make sense. I would direct you to learning about ordinal numbers (https://en.wikipedia.org/wiki/Ordinal_number#Arithmetic_of_ordinals) and cardinal numbers (https://en.wikipedia.org/wiki/Cardinal_number#Cardinal_arithmetic) if you're looking for arithmetic involving the infinite.
For example, $omega$ is defined as the first infinite ordinal and arithmetic with it can be kind of interesting. For example, $$omega + 1 neq omega$$ but $$1 + omega = omega$$ So some of the more comfortable and familiar properties like commutativity of numbers breaks down with infinity. And as such, thought experiments like mental pictures involving lines and rays, or sets and orderings need to have their terms carefully defined in order to have any real meaning.
answered Jul 16 at 19:35
Allen O'Hara
1086
answered Jul 16 at 19:35
Allen O'Hara
1086
answered Jul 16 at 19:35
Allen O'Hara
1086
answered Jul 16 at 19:35
Allen O'Hara
1086
1086
add a comment |Â
add a comment |Â
up vote
6
down vote
Hint:
Here is a quiz: as $infty+1=infty$, the two sides are interchangeable.
So what is meant by $infty-infty$ ?
- a: $infty-infty$,
- b: $infty+1-infty$,
- c: $infty-(infty+1)$,
- d: $infty+1-(infty+1)$,
- e: none of these,
- f: all of these.
answered Jul 16 at 19:33
Yves Daoust
111k665204
I can't see how this is a helpful answer to OP. It seems to be a comment, since it does not answer nothing, although provides interesting insights.
– rafa11111
Jul 16 at 19:41
I understand your frustration with this often-asked question. That said, it's hard to see how this could lead to the OP realizing "I see - the usual rules of arithmetic can't be consistent if infinity is a number. Now I understand."
– Ethan Bolker
Jul 16 at 20:02
@EthanBolker: no, this is not my message. The message, though not made explicit is "how could you deal with something that has several values that are at the same time different and equal" ? Infinity doesn't have a "memory" of what was added to it.
– Yves Daoust
Jul 16 at 20:07
@rafa11111: an answer need not be written in the affirmative mode.
– Yves Daoust
Jul 16 at 20:09
7
I like this answer. Makes you realize why infinity cannot be a number, instead of just telling you that it is not. Proof by contradiction of sorts.
– Sorfosh
Jul 16 at 20:57
 |Â
show 1 more comment
up vote
6
down vote
Hint:
Here is a quiz: as $infty+1=infty$, the two sides are interchangeable.
So what is meant by $infty-infty$ ?
- a: $infty-infty$,
- b: $infty+1-infty$,
- c: $infty-(infty+1)$,
- d: $infty+1-(infty+1)$,
- e: none of these,
- f: all of these.
answered Jul 16 at 19:33
Yves Daoust
111k665204
I can't see how this is a helpful answer to OP. It seems to be a comment, since it does not answer nothing, although provides interesting insights.
– rafa11111
Jul 16 at 19:41
I understand your frustration with this often-asked question. That said, it's hard to see how this could lead to the OP realizing "I see - the usual rules of arithmetic can't be consistent if infinity is a number. Now I understand."
– Ethan Bolker
Jul 16 at 20:02
@EthanBolker: no, this is not my message. The message, though not made explicit is "how could you deal with something that has several values that are at the same time different and equal" ? Infinity doesn't have a "memory" of what was added to it.
– Yves Daoust
Jul 16 at 20:07
@rafa11111: an answer need not be written in the affirmative mode.
– Yves Daoust
Jul 16 at 20:09
7
I like this answer. Makes you realize why infinity cannot be a number, instead of just telling you that it is not. Proof by contradiction of sorts.
– Sorfosh
Jul 16 at 20:57
 |Â
show 1 more comment
up vote
6
down vote
up vote
6
down vote
Hint:
Here is a quiz: as $infty+1=infty$, the two sides are interchangeable.
So what is meant by $infty-infty$ ?
- a: $infty-infty$,
- b: $infty+1-infty$,
- c: $infty-(infty+1)$,
- d: $infty+1-(infty+1)$,
- e: none of these,
- f: all of these.
answered Jul 16 at 19:33
Yves Daoust
111k665204
Hint:
Here is a quiz: as $infty+1=infty$, the two sides are interchangeable.
So what is meant by $infty-infty$ ?
- a: $infty-infty$,
- b: $infty+1-infty$,
- c: $infty-(infty+1)$,
- d: $infty+1-(infty+1)$,
- e: none of these,
- f: all of these.
answered Jul 16 at 19:33
Yves Daoust
111k665204
edited Jul 16 at 20:08
answered Jul 16 at 19:33
Yves Daoust
111k665204
answered Jul 16 at 19:33
Yves Daoust
111k665204
answered Jul 16 at 19:33
Yves Daoust
111k665204
111k665204
I can't see how this is a helpful answer to OP. It seems to be a comment, since it does not answer nothing, although provides interesting insights.
– rafa11111
Jul 16 at 19:41
I understand your frustration with this often-asked question. That said, it's hard to see how this could lead to the OP realizing "I see - the usual rules of arithmetic can't be consistent if infinity is a number. Now I understand."
– Ethan Bolker
Jul 16 at 20:02
@EthanBolker: no, this is not my message. The message, though not made explicit is "how could you deal with something that has several values that are at the same time different and equal" ? Infinity doesn't have a "memory" of what was added to it.
– Yves Daoust
Jul 16 at 20:07
@rafa11111: an answer need not be written in the affirmative mode.
– Yves Daoust
Jul 16 at 20:09
7
I like this answer. Makes you realize why infinity cannot be a number, instead of just telling you that it is not. Proof by contradiction of sorts.
– Sorfosh
Jul 16 at 20:57
 |Â
show 1 more comment
I can't see how this is a helpful answer to OP. It seems to be a comment, since it does not answer nothing, although provides interesting insights.
– rafa11111
Jul 16 at 19:41
I understand your frustration with this often-asked question. That said, it's hard to see how this could lead to the OP realizing "I see - the usual rules of arithmetic can't be consistent if infinity is a number. Now I understand."
– Ethan Bolker
Jul 16 at 20:02
@EthanBolker: no, this is not my message. The message, though not made explicit is "how could you deal with something that has several values that are at the same time different and equal" ? Infinity doesn't have a "memory" of what was added to it.
– Yves Daoust
Jul 16 at 20:07
@rafa11111: an answer need not be written in the affirmative mode.
– Yves Daoust
Jul 16 at 20:09
7
I like this answer. Makes you realize why infinity cannot be a number, instead of just telling you that it is not. Proof by contradiction of sorts.
– Sorfosh
Jul 16 at 20:57
I can't see how this is a helpful answer to OP. It seems to be a comment, since it does not answer nothing, although provides interesting insights.
– rafa11111
Jul 16 at 19:41
I can't see how this is a helpful answer to OP. It seems to be a comment, since it does not answer nothing, although provides interesting insights.
– rafa11111
Jul 16 at 19:41
I understand your frustration with this often-asked question. That said, it's hard to see how this could lead to the OP realizing "I see - the usual rules of arithmetic can't be consistent if infinity is a number. Now I understand."
– Ethan Bolker
Jul 16 at 20:02
I understand your frustration with this often-asked question. That said, it's hard to see how this could lead to the OP realizing "I see - the usual rules of arithmetic can't be consistent if infinity is a number. Now I understand."
– Ethan Bolker
Jul 16 at 20:02
@EthanBolker: no, this is not my message. The message, though not made explicit is "how could you deal with something that has several values that are at the same time different and equal" ? Infinity doesn't have a "memory" of what was added to it.
– Yves Daoust
Jul 16 at 20:07
@EthanBolker: no, this is not my message. The message, though not made explicit is "how could you deal with something that has several values that are at the same time different and equal" ? Infinity doesn't have a "memory" of what was added to it.
– Yves Daoust
Jul 16 at 20:07
@rafa11111: an answer need not be written in the affirmative mode.
– Yves Daoust
Jul 16 at 20:09
@rafa11111: an answer need not be written in the affirmative mode.
– Yves Daoust
Jul 16 at 20:09
7
7
I like this answer. Makes you realize why infinity cannot be a number, instead of just telling you that it is not. Proof by contradiction of sorts.
– Sorfosh
Jul 16 at 20:57
I like this answer. Makes you realize why infinity cannot be a number, instead of just telling you that it is not. Proof by contradiction of sorts.
– Sorfosh
Jul 16 at 20:57
 |Â
show 1 more comment
up vote
2
down vote
$infty$ is not a number. On the real number line, $mathbbR$ we only have numbers. The extended real numbers, $overlinemathbbR:=mathbbR cup-infty;infty$ that has a property that $forall x in mathbbR$, $infty + x =infty$ so you couldn't subtract it. This also holds for other fields rather than $mathbbR$.
answered Jul 16 at 19:31
Mario 04
6013
add a comment |Â
up vote
2
down vote
$infty$ is not a number. On the real number line, $mathbbR$ we only have numbers. The extended real numbers, $overlinemathbbR:=mathbbR cup-infty;infty$ that has a property that $forall x in mathbbR$, $infty + x =infty$ so you couldn't subtract it. This also holds for other fields rather than $mathbbR$.
answered Jul 16 at 19:31
Mario 04
6013
add a comment |Â
up vote
2
down vote
up vote
2
down vote
$infty$ is not a number. On the real number line, $mathbbR$ we only have numbers. The extended real numbers, $overlinemathbbR:=mathbbR cup-infty;infty$ that has a property that $forall x in mathbbR$, $infty + x =infty$ so you couldn't subtract it. This also holds for other fields rather than $mathbbR$.
answered Jul 16 at 19:31
Mario 04
6013
$infty$ is not a number. On the real number line, $mathbbR$ we only have numbers. The extended real numbers, $overlinemathbbR:=mathbbR cup-infty;infty$ that has a property that $forall x in mathbbR$, $infty + x =infty$ so you couldn't subtract it. This also holds for other fields rather than $mathbbR$.
answered Jul 16 at 19:31
Mario 04
6013
answered Jul 16 at 19:31
Mario 04
6013
answered Jul 16 at 19:31
Mario 04
6013
answered Jul 16 at 19:31
Mario 04
6013
6013
add a comment |Â
add a comment |Â
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13
Infinity is not a number, hence you cannot apply normal arithmetic to it. You could argue that $lim_x to infty x=lim_xto infty x+1$ but that holds a different meaning.
– Sorfosh
Jul 16 at 19:24
You can't subtract infinity from both sides because it is not a number
– gd1035
Jul 16 at 19:25
9
Some people work with $infty$ as a "number", but even those won't even dare to subtract infinity from infinity..
– Shashi
Jul 16 at 19:45
1
Many answers have mentioned that "infinity is not a number," but not many have tried to explain why. If you are the curious type, I recommend watching the VSauce video How to Count Past Infinity which is the best layman's introduction to how infinities work that I know of. It does a great job of capturing some of the wierdnesses that appear around infinities and how we try to conquer them.
– Cort Ammon
Jul 17 at 0:48