Why do we call complex numbers “numbers” but we don’t consider 2 vectors numbers?What exactly is a number?Relationship between complex number and vectorsAre there any numbers more fundamental than Complex numbers?Why are complex numbers considered to be numbers?Does it make sense to compare complex numbers in certain circumstances?Usefulness of alternative constructions of the complex numbersComplex numbers?Why are complex numbers so magical?What can complex numbers do that linear algebra cannot?complex numbers and rotation matricesIf there is anything else introduced into equations like the complex numbers.
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Why do we call complex numbers “numbers” but we don’t consider 2 vectors numbers?
What exactly is a number?Relationship between complex number and vectorsAre there any numbers more fundamental than Complex numbers?Why are complex numbers considered to be numbers?Does it make sense to compare complex numbers in certain circumstances?Usefulness of alternative constructions of the complex numbersComplex numbers?Why are complex numbers so magical?What can complex numbers do that linear algebra cannot?complex numbers and rotation matricesIf there is anything else introduced into equations like the complex numbers.
$begingroup$
We refer to complex numbers as numbers. However we refer to vectors as arrays of numbers. There doesn’t seem to be anything that makes one more numeric than the other. Is this just a quirk of history and naming or is there something more fundamental?
matrices complex-numbers philosophy
edited 6 hours ago
Bernard
122k740116
asked 7 hours ago
Q the PlatypusQ the Platypus
2,808933
$endgroup$
|
show 1 more comment
$begingroup$
We refer to complex numbers as numbers. However we refer to vectors as arrays of numbers. There doesn’t seem to be anything that makes one more numeric than the other. Is this just a quirk of history and naming or is there something more fundamental?
matrices complex-numbers philosophy
edited 6 hours ago
Bernard
122k740116
asked 7 hours ago
Q the PlatypusQ the Platypus
2,808933
$endgroup$
7
$begingroup$
Usually, the difference would be whether you're in a context where complex multiplication would be interesting. It's kind of like the question: is $mathbbZ$ a group or a ring? It depends on which operations are of interest for the question you're studying.
$endgroup$
– Nate Eldredge
7 hours ago
4
$begingroup$
What's a "number" anyway?
$endgroup$
– Asaf Karagila♦
6 hours ago
$begingroup$
I would say that $mathbbZ$ is not a ring or a group as you have to have an operator in order to be a magma. However that would be over pedantic so I get where you are coming from.
$endgroup$
– Q the Platypus
6 hours ago
5
$begingroup$
@QthePlatypus: No, that's exactly the point. The question of whether the set $mathbbR^2$ of pairs of real numbers should be considered as the set of complex numbers, or as a set of vectors, depends on which operations you wish to equip that set with.
$endgroup$
– Nate Eldredge
6 hours ago
2
$begingroup$
I think the correct question were: is there a rigorous definition of the object "number" (without further characteristics such as "integer", "real", "complex" etc.)?
$endgroup$
– user
6 hours ago
|
show 1 more comment
$begingroup$
We refer to complex numbers as numbers. However we refer to vectors as arrays of numbers. There doesn’t seem to be anything that makes one more numeric than the other. Is this just a quirk of history and naming or is there something more fundamental?
matrices complex-numbers philosophy
edited 6 hours ago
Bernard
122k740116
asked 7 hours ago
Q the PlatypusQ the Platypus
2,808933
$endgroup$
We refer to complex numbers as numbers. However we refer to vectors as arrays of numbers. There doesn’t seem to be anything that makes one more numeric than the other. Is this just a quirk of history and naming or is there something more fundamental?
matrices complex-numbers philosophy
matrices complex-numbers philosophy
edited 6 hours ago
Bernard
122k740116
asked 7 hours ago
Q the PlatypusQ the Platypus
2,808933
edited 6 hours ago
Bernard
122k740116
asked 7 hours ago
Q the PlatypusQ the Platypus
2,808933
edited 6 hours ago
Bernard
122k740116
edited 6 hours ago
Bernard
122k740116
edited 6 hours ago
Bernard
122k740116
122k740116
asked 7 hours ago
Q the PlatypusQ the Platypus
2,808933
asked 7 hours ago
Q the PlatypusQ the Platypus
2,808933
asked 7 hours ago
Q the PlatypusQ the Platypus
2,808933
2,808933
7
$begingroup$
Usually, the difference would be whether you're in a context where complex multiplication would be interesting. It's kind of like the question: is $mathbbZ$ a group or a ring? It depends on which operations are of interest for the question you're studying.
$endgroup$
– Nate Eldredge
7 hours ago
4
$begingroup$
What's a "number" anyway?
$endgroup$
– Asaf Karagila♦
6 hours ago
$begingroup$
I would say that $mathbbZ$ is not a ring or a group as you have to have an operator in order to be a magma. However that would be over pedantic so I get where you are coming from.
$endgroup$
– Q the Platypus
6 hours ago
5
$begingroup$
@QthePlatypus: No, that's exactly the point. The question of whether the set $mathbbR^2$ of pairs of real numbers should be considered as the set of complex numbers, or as a set of vectors, depends on which operations you wish to equip that set with.
$endgroup$
– Nate Eldredge
6 hours ago
2
$begingroup$
I think the correct question were: is there a rigorous definition of the object "number" (without further characteristics such as "integer", "real", "complex" etc.)?
$endgroup$
– user
6 hours ago
|
show 1 more comment
7
$begingroup$
Usually, the difference would be whether you're in a context where complex multiplication would be interesting. It's kind of like the question: is $mathbbZ$ a group or a ring? It depends on which operations are of interest for the question you're studying.
$endgroup$
– Nate Eldredge
7 hours ago
4
$begingroup$
What's a "number" anyway?
$endgroup$
– Asaf Karagila♦
6 hours ago
$begingroup$
I would say that $mathbbZ$ is not a ring or a group as you have to have an operator in order to be a magma. However that would be over pedantic so I get where you are coming from.
$endgroup$
– Q the Platypus
6 hours ago
5
$begingroup$
@QthePlatypus: No, that's exactly the point. The question of whether the set $mathbbR^2$ of pairs of real numbers should be considered as the set of complex numbers, or as a set of vectors, depends on which operations you wish to equip that set with.
$endgroup$
– Nate Eldredge
6 hours ago
2
$begingroup$
I think the correct question were: is there a rigorous definition of the object "number" (without further characteristics such as "integer", "real", "complex" etc.)?
$endgroup$
– user
6 hours ago
7
7
$begingroup$
Usually, the difference would be whether you're in a context where complex multiplication would be interesting. It's kind of like the question: is $mathbbZ$ a group or a ring? It depends on which operations are of interest for the question you're studying.
$endgroup$
– Nate Eldredge
7 hours ago
$begingroup$
Usually, the difference would be whether you're in a context where complex multiplication would be interesting. It's kind of like the question: is $mathbbZ$ a group or a ring? It depends on which operations are of interest for the question you're studying.
$endgroup$
– Nate Eldredge
7 hours ago
4
4
$begingroup$
What's a "number" anyway?
$endgroup$
– Asaf Karagila♦
6 hours ago
$begingroup$
What's a "number" anyway?
$endgroup$
– Asaf Karagila♦
6 hours ago
$begingroup$
I would say that $mathbbZ$ is not a ring or a group as you have to have an operator in order to be a magma. However that would be over pedantic so I get where you are coming from.
$endgroup$
– Q the Platypus
6 hours ago
$begingroup$
I would say that $mathbbZ$ is not a ring or a group as you have to have an operator in order to be a magma. However that would be over pedantic so I get where you are coming from.
$endgroup$
– Q the Platypus
6 hours ago
5
5
$begingroup$
@QthePlatypus: No, that's exactly the point. The question of whether the set $mathbbR^2$ of pairs of real numbers should be considered as the set of complex numbers, or as a set of vectors, depends on which operations you wish to equip that set with.
$endgroup$
– Nate Eldredge
6 hours ago
$begingroup$
@QthePlatypus: No, that's exactly the point. The question of whether the set $mathbbR^2$ of pairs of real numbers should be considered as the set of complex numbers, or as a set of vectors, depends on which operations you wish to equip that set with.
$endgroup$
– Nate Eldredge
6 hours ago
2
2
$begingroup$
I think the correct question were: is there a rigorous definition of the object "number" (without further characteristics such as "integer", "real", "complex" etc.)?
$endgroup$
– user
6 hours ago
$begingroup$
I think the correct question were: is there a rigorous definition of the object "number" (without further characteristics such as "integer", "real", "complex" etc.)?
$endgroup$
– user
6 hours ago
|
show 1 more comment
4 Answers
4
active
oldest
votes
$begingroup$
They're called "numbers" for historical reasons, since the motivation in the development of the complex numbers was solving polynomial equations. They were viewed as natural extensions of the real numbers. It's somehow quite natural and satisfying to say "every polynomial equation can be solved by some (complex) number". Is it more natural to regard $i$ as being a number which, when squared, is equal to $-1$, or is it more natural to regard $i$ as being some non-number thingamajig which when squared is equal to $-1$? Clearly the former.
"Higher" number systems, like quaternions, aren't really called numbers very often, for the simple fact that they are not as intimately connected with number theory and analysis in the same way that complex numbers are.
Beyond these social conventions, I can't see any other reason. The word "number" doesn't have a strict or absolute definition in pure mathematics. $mathbbN$, $mathbbZ$, $mathbbQ$, $mathbbR$ and $mathbbC$ are technically just sets with a certain algebraic structure.
I partially disagree with the other answers which claim that complex numbers are numbers simply by virtue of the fact that you can add and multiply them. Well, if that's the rationale, is every ring also a set of numbers?
answered 6 hours ago
MathematicsStudent1122MathematicsStudent1122
8,81622468
$endgroup$
$begingroup$
Wikipedia calls the quaternions a number system.
$endgroup$
– JJJ
1 hour ago
add a comment |
$begingroup$
The two fundamental operations for numbers are "addition" and "multiplication" which obey very nice "laws" of arithmetic. Taking powers is also important. You can do all of those things with complex numbers. You can add two vectors but the "dot" product of two vectors is not a vector and the "cross" product of two vectors does not satisfy the "nice laws". Neither the dot product nor the cross product of vectors can be used to define powers.
answered 6 hours ago
user247327user247327
11.3k1515
$endgroup$
$begingroup$
So Quaternions which have Non commutativity multiplication would also be considered as non numbers?
$endgroup$
– Q the Platypus
6 hours ago
3
$begingroup$
That's a good point- though I have never liked quaternions! Multiplication of quaternions is not commutative but it is associative which allows powers. That's what I was really thinking of.
$endgroup$
– user247327
6 hours ago
1
$begingroup$
You can also add and multiply matrices, but I don't think I've ever heard anyone call a matrix a "number".
$endgroup$
– Hong Ooi
3 hours ago
2
$begingroup$
@user247327: It's worth pointing out that IEEE 754 floating point numbers are non-associative, closed under division, non-reflexive (i.e. they violate $forall a:a = a$), non-substitutable (i.e. numbers which are "equal" cannot always be substituted for one another in expressions or equations), and also the most widely used approximation of the reals in practical computing applications.
$endgroup$
– Kevin
3 hours ago
add a comment |
$begingroup$
Numbers appeared first when we started to count things. $1$ tree, $2$ trees, $3$ trees, and so forth. That make up the set of non-zero natural numbers: $mathbbN$. Afterwards, people started to "count backwards" to get $mathbbZ$ (I'm just kidding, you can do some research on how negative numbers appeared historically). With the urge to divide things without remainders, the set of rational numbers $mathbbQ$ made its way to the world. A certain idea of geometric continuity gives us $mathbbR$. Finally we want all equations to have a root, that's how $mathbbC$ comes into play.
I guess what is considered "numbers" is rather a social question. The use of $mathbbC$
in physics and its $2$-D representation must have given a good intuition for a large set of people to accept it's intuitive enough to be considered "numbers".
I think it's not that natural to think of $mathbbC$ as a $mathbbR$-vector space of dimension $2$, not more natural than to think of $mathbbR$ as an infinite-dimensional $mathbbQ$-vector space, nor of $mathbbQ$ as a non-infinitely-generated $mathbbZ$-module.
answered 6 hours ago
LeaningLeaning
1,221718
$endgroup$
2
$begingroup$
People do think of complex numbers as points on the number plane.
$endgroup$
– Q the Platypus
6 hours ago
1
$begingroup$
Thank you. I'm sorry, that final part is just a personal opinion. edited.
$endgroup$
– Leaning
6 hours ago
add a comment |
$begingroup$
I think naming is a rather soft topic, so there may not be a hard answer. However, I think it is worth noting that complex numbers are one of the three associative real division algebras (real numbers, complex numbers, and quaternions). These are all linked by the idea that division is meaningful in those three systems.
In general, a 2 dimensional real vector cannot admit a concept of division which matches what we expect division to do. However, if said real vector defines addition and multiplication the way complex numbers do, division is a natural result of those definitions.
answered 1 hour ago
Cort AmmonCort Ammon
2,421716
$endgroup$
add a comment |
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4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
They're called "numbers" for historical reasons, since the motivation in the development of the complex numbers was solving polynomial equations. They were viewed as natural extensions of the real numbers. It's somehow quite natural and satisfying to say "every polynomial equation can be solved by some (complex) number". Is it more natural to regard $i$ as being a number which, when squared, is equal to $-1$, or is it more natural to regard $i$ as being some non-number thingamajig which when squared is equal to $-1$? Clearly the former.
"Higher" number systems, like quaternions, aren't really called numbers very often, for the simple fact that they are not as intimately connected with number theory and analysis in the same way that complex numbers are.
Beyond these social conventions, I can't see any other reason. The word "number" doesn't have a strict or absolute definition in pure mathematics. $mathbbN$, $mathbbZ$, $mathbbQ$, $mathbbR$ and $mathbbC$ are technically just sets with a certain algebraic structure.
I partially disagree with the other answers which claim that complex numbers are numbers simply by virtue of the fact that you can add and multiply them. Well, if that's the rationale, is every ring also a set of numbers?
answered 6 hours ago
MathematicsStudent1122MathematicsStudent1122
8,81622468
$endgroup$
$begingroup$
Wikipedia calls the quaternions a number system.
$endgroup$
– JJJ
1 hour ago
add a comment |
$begingroup$
They're called "numbers" for historical reasons, since the motivation in the development of the complex numbers was solving polynomial equations. They were viewed as natural extensions of the real numbers. It's somehow quite natural and satisfying to say "every polynomial equation can be solved by some (complex) number". Is it more natural to regard $i$ as being a number which, when squared, is equal to $-1$, or is it more natural to regard $i$ as being some non-number thingamajig which when squared is equal to $-1$? Clearly the former.
"Higher" number systems, like quaternions, aren't really called numbers very often, for the simple fact that they are not as intimately connected with number theory and analysis in the same way that complex numbers are.
Beyond these social conventions, I can't see any other reason. The word "number" doesn't have a strict or absolute definition in pure mathematics. $mathbbN$, $mathbbZ$, $mathbbQ$, $mathbbR$ and $mathbbC$ are technically just sets with a certain algebraic structure.
I partially disagree with the other answers which claim that complex numbers are numbers simply by virtue of the fact that you can add and multiply them. Well, if that's the rationale, is every ring also a set of numbers?
answered 6 hours ago
MathematicsStudent1122MathematicsStudent1122
8,81622468
$endgroup$
$begingroup$
Wikipedia calls the quaternions a number system.
$endgroup$
– JJJ
1 hour ago
add a comment |
$begingroup$
They're called "numbers" for historical reasons, since the motivation in the development of the complex numbers was solving polynomial equations. They were viewed as natural extensions of the real numbers. It's somehow quite natural and satisfying to say "every polynomial equation can be solved by some (complex) number". Is it more natural to regard $i$ as being a number which, when squared, is equal to $-1$, or is it more natural to regard $i$ as being some non-number thingamajig which when squared is equal to $-1$? Clearly the former.
"Higher" number systems, like quaternions, aren't really called numbers very often, for the simple fact that they are not as intimately connected with number theory and analysis in the same way that complex numbers are.
Beyond these social conventions, I can't see any other reason. The word "number" doesn't have a strict or absolute definition in pure mathematics. $mathbbN$, $mathbbZ$, $mathbbQ$, $mathbbR$ and $mathbbC$ are technically just sets with a certain algebraic structure.
I partially disagree with the other answers which claim that complex numbers are numbers simply by virtue of the fact that you can add and multiply them. Well, if that's the rationale, is every ring also a set of numbers?
answered 6 hours ago
MathematicsStudent1122MathematicsStudent1122
8,81622468
$endgroup$
They're called "numbers" for historical reasons, since the motivation in the development of the complex numbers was solving polynomial equations. They were viewed as natural extensions of the real numbers. It's somehow quite natural and satisfying to say "every polynomial equation can be solved by some (complex) number". Is it more natural to regard $i$ as being a number which, when squared, is equal to $-1$, or is it more natural to regard $i$ as being some non-number thingamajig which when squared is equal to $-1$? Clearly the former.
"Higher" number systems, like quaternions, aren't really called numbers very often, for the simple fact that they are not as intimately connected with number theory and analysis in the same way that complex numbers are.
Beyond these social conventions, I can't see any other reason. The word "number" doesn't have a strict or absolute definition in pure mathematics. $mathbbN$, $mathbbZ$, $mathbbQ$, $mathbbR$ and $mathbbC$ are technically just sets with a certain algebraic structure.
I partially disagree with the other answers which claim that complex numbers are numbers simply by virtue of the fact that you can add and multiply them. Well, if that's the rationale, is every ring also a set of numbers?
answered 6 hours ago
MathematicsStudent1122MathematicsStudent1122
8,81622468
edited 3 hours ago
answered 6 hours ago
MathematicsStudent1122MathematicsStudent1122
8,81622468
answered 6 hours ago
MathematicsStudent1122MathematicsStudent1122
8,81622468
answered 6 hours ago
MathematicsStudent1122MathematicsStudent1122
8,81622468
8,81622468
$begingroup$
Wikipedia calls the quaternions a number system.
$endgroup$
– JJJ
1 hour ago
add a comment |
$begingroup$
Wikipedia calls the quaternions a number system.
$endgroup$
– JJJ
1 hour ago
$begingroup$
Wikipedia calls the quaternions a number system.
$endgroup$
– JJJ
1 hour ago
$begingroup$
Wikipedia calls the quaternions a number system.
$endgroup$
– JJJ
1 hour ago
add a comment |
$begingroup$
The two fundamental operations for numbers are "addition" and "multiplication" which obey very nice "laws" of arithmetic. Taking powers is also important. You can do all of those things with complex numbers. You can add two vectors but the "dot" product of two vectors is not a vector and the "cross" product of two vectors does not satisfy the "nice laws". Neither the dot product nor the cross product of vectors can be used to define powers.
answered 6 hours ago
user247327user247327
11.3k1515
$endgroup$
$begingroup$
So Quaternions which have Non commutativity multiplication would also be considered as non numbers?
$endgroup$
– Q the Platypus
6 hours ago
3
$begingroup$
That's a good point- though I have never liked quaternions! Multiplication of quaternions is not commutative but it is associative which allows powers. That's what I was really thinking of.
$endgroup$
– user247327
6 hours ago
1
$begingroup$
You can also add and multiply matrices, but I don't think I've ever heard anyone call a matrix a "number".
$endgroup$
– Hong Ooi
3 hours ago
2
$begingroup$
@user247327: It's worth pointing out that IEEE 754 floating point numbers are non-associative, closed under division, non-reflexive (i.e. they violate $forall a:a = a$), non-substitutable (i.e. numbers which are "equal" cannot always be substituted for one another in expressions or equations), and also the most widely used approximation of the reals in practical computing applications.
$endgroup$
– Kevin
3 hours ago
add a comment |
$begingroup$
The two fundamental operations for numbers are "addition" and "multiplication" which obey very nice "laws" of arithmetic. Taking powers is also important. You can do all of those things with complex numbers. You can add two vectors but the "dot" product of two vectors is not a vector and the "cross" product of two vectors does not satisfy the "nice laws". Neither the dot product nor the cross product of vectors can be used to define powers.
answered 6 hours ago
user247327user247327
11.3k1515
$endgroup$
$begingroup$
So Quaternions which have Non commutativity multiplication would also be considered as non numbers?
$endgroup$
– Q the Platypus
6 hours ago
3
$begingroup$
That's a good point- though I have never liked quaternions! Multiplication of quaternions is not commutative but it is associative which allows powers. That's what I was really thinking of.
$endgroup$
– user247327
6 hours ago
1
$begingroup$
You can also add and multiply matrices, but I don't think I've ever heard anyone call a matrix a "number".
$endgroup$
– Hong Ooi
3 hours ago
2
$begingroup$
@user247327: It's worth pointing out that IEEE 754 floating point numbers are non-associative, closed under division, non-reflexive (i.e. they violate $forall a:a = a$), non-substitutable (i.e. numbers which are "equal" cannot always be substituted for one another in expressions or equations), and also the most widely used approximation of the reals in practical computing applications.
$endgroup$
– Kevin
3 hours ago
add a comment |
$begingroup$
The two fundamental operations for numbers are "addition" and "multiplication" which obey very nice "laws" of arithmetic. Taking powers is also important. You can do all of those things with complex numbers. You can add two vectors but the "dot" product of two vectors is not a vector and the "cross" product of two vectors does not satisfy the "nice laws". Neither the dot product nor the cross product of vectors can be used to define powers.
answered 6 hours ago
user247327user247327
11.3k1515
$endgroup$
The two fundamental operations for numbers are "addition" and "multiplication" which obey very nice "laws" of arithmetic. Taking powers is also important. You can do all of those things with complex numbers. You can add two vectors but the "dot" product of two vectors is not a vector and the "cross" product of two vectors does not satisfy the "nice laws". Neither the dot product nor the cross product of vectors can be used to define powers.
answered 6 hours ago
user247327user247327
11.3k1515
answered 6 hours ago
user247327user247327
11.3k1515
answered 6 hours ago
user247327user247327
11.3k1515
answered 6 hours ago
user247327user247327
11.3k1515
11.3k1515
$begingroup$
So Quaternions which have Non commutativity multiplication would also be considered as non numbers?
$endgroup$
– Q the Platypus
6 hours ago
3
$begingroup$
That's a good point- though I have never liked quaternions! Multiplication of quaternions is not commutative but it is associative which allows powers. That's what I was really thinking of.
$endgroup$
– user247327
6 hours ago
1
$begingroup$
You can also add and multiply matrices, but I don't think I've ever heard anyone call a matrix a "number".
$endgroup$
– Hong Ooi
3 hours ago
2
$begingroup$
@user247327: It's worth pointing out that IEEE 754 floating point numbers are non-associative, closed under division, non-reflexive (i.e. they violate $forall a:a = a$), non-substitutable (i.e. numbers which are "equal" cannot always be substituted for one another in expressions or equations), and also the most widely used approximation of the reals in practical computing applications.
$endgroup$
– Kevin
3 hours ago
add a comment |
$begingroup$
So Quaternions which have Non commutativity multiplication would also be considered as non numbers?
$endgroup$
– Q the Platypus
6 hours ago
3
$begingroup$
That's a good point- though I have never liked quaternions! Multiplication of quaternions is not commutative but it is associative which allows powers. That's what I was really thinking of.
$endgroup$
– user247327
6 hours ago
1
$begingroup$
You can also add and multiply matrices, but I don't think I've ever heard anyone call a matrix a "number".
$endgroup$
– Hong Ooi
3 hours ago
2
$begingroup$
@user247327: It's worth pointing out that IEEE 754 floating point numbers are non-associative, closed under division, non-reflexive (i.e. they violate $forall a:a = a$), non-substitutable (i.e. numbers which are "equal" cannot always be substituted for one another in expressions or equations), and also the most widely used approximation of the reals in practical computing applications.
$endgroup$
– Kevin
3 hours ago
$begingroup$
So Quaternions which have Non commutativity multiplication would also be considered as non numbers?
$endgroup$
– Q the Platypus
6 hours ago
$begingroup$
So Quaternions which have Non commutativity multiplication would also be considered as non numbers?
$endgroup$
– Q the Platypus
6 hours ago
3
3
$begingroup$
That's a good point- though I have never liked quaternions! Multiplication of quaternions is not commutative but it is associative which allows powers. That's what I was really thinking of.
$endgroup$
– user247327
6 hours ago
$begingroup$
That's a good point- though I have never liked quaternions! Multiplication of quaternions is not commutative but it is associative which allows powers. That's what I was really thinking of.
$endgroup$
– user247327
6 hours ago
1
1
$begingroup$
You can also add and multiply matrices, but I don't think I've ever heard anyone call a matrix a "number".
$endgroup$
– Hong Ooi
3 hours ago
$begingroup$
You can also add and multiply matrices, but I don't think I've ever heard anyone call a matrix a "number".
$endgroup$
– Hong Ooi
3 hours ago
2
2
$begingroup$
@user247327: It's worth pointing out that IEEE 754 floating point numbers are non-associative, closed under division, non-reflexive (i.e. they violate $forall a:a = a$), non-substitutable (i.e. numbers which are "equal" cannot always be substituted for one another in expressions or equations), and also the most widely used approximation of the reals in practical computing applications.
$endgroup$
– Kevin
3 hours ago
$begingroup$
@user247327: It's worth pointing out that IEEE 754 floating point numbers are non-associative, closed under division, non-reflexive (i.e. they violate $forall a:a = a$), non-substitutable (i.e. numbers which are "equal" cannot always be substituted for one another in expressions or equations), and also the most widely used approximation of the reals in practical computing applications.
$endgroup$
– Kevin
3 hours ago
add a comment |
$begingroup$
Numbers appeared first when we started to count things. $1$ tree, $2$ trees, $3$ trees, and so forth. That make up the set of non-zero natural numbers: $mathbbN$. Afterwards, people started to "count backwards" to get $mathbbZ$ (I'm just kidding, you can do some research on how negative numbers appeared historically). With the urge to divide things without remainders, the set of rational numbers $mathbbQ$ made its way to the world. A certain idea of geometric continuity gives us $mathbbR$. Finally we want all equations to have a root, that's how $mathbbC$ comes into play.
I guess what is considered "numbers" is rather a social question. The use of $mathbbC$
in physics and its $2$-D representation must have given a good intuition for a large set of people to accept it's intuitive enough to be considered "numbers".
I think it's not that natural to think of $mathbbC$ as a $mathbbR$-vector space of dimension $2$, not more natural than to think of $mathbbR$ as an infinite-dimensional $mathbbQ$-vector space, nor of $mathbbQ$ as a non-infinitely-generated $mathbbZ$-module.
answered 6 hours ago
LeaningLeaning
1,221718
$endgroup$
2
$begingroup$
People do think of complex numbers as points on the number plane.
$endgroup$
– Q the Platypus
6 hours ago
1
$begingroup$
Thank you. I'm sorry, that final part is just a personal opinion. edited.
$endgroup$
– Leaning
6 hours ago
add a comment |
$begingroup$
Numbers appeared first when we started to count things. $1$ tree, $2$ trees, $3$ trees, and so forth. That make up the set of non-zero natural numbers: $mathbbN$. Afterwards, people started to "count backwards" to get $mathbbZ$ (I'm just kidding, you can do some research on how negative numbers appeared historically). With the urge to divide things without remainders, the set of rational numbers $mathbbQ$ made its way to the world. A certain idea of geometric continuity gives us $mathbbR$. Finally we want all equations to have a root, that's how $mathbbC$ comes into play.
I guess what is considered "numbers" is rather a social question. The use of $mathbbC$
in physics and its $2$-D representation must have given a good intuition for a large set of people to accept it's intuitive enough to be considered "numbers".
I think it's not that natural to think of $mathbbC$ as a $mathbbR$-vector space of dimension $2$, not more natural than to think of $mathbbR$ as an infinite-dimensional $mathbbQ$-vector space, nor of $mathbbQ$ as a non-infinitely-generated $mathbbZ$-module.
answered 6 hours ago
LeaningLeaning
1,221718
$endgroup$
2
$begingroup$
People do think of complex numbers as points on the number plane.
$endgroup$
– Q the Platypus
6 hours ago
1
$begingroup$
Thank you. I'm sorry, that final part is just a personal opinion. edited.
$endgroup$
– Leaning
6 hours ago
add a comment |
$begingroup$
Numbers appeared first when we started to count things. $1$ tree, $2$ trees, $3$ trees, and so forth. That make up the set of non-zero natural numbers: $mathbbN$. Afterwards, people started to "count backwards" to get $mathbbZ$ (I'm just kidding, you can do some research on how negative numbers appeared historically). With the urge to divide things without remainders, the set of rational numbers $mathbbQ$ made its way to the world. A certain idea of geometric continuity gives us $mathbbR$. Finally we want all equations to have a root, that's how $mathbbC$ comes into play.
I guess what is considered "numbers" is rather a social question. The use of $mathbbC$
in physics and its $2$-D representation must have given a good intuition for a large set of people to accept it's intuitive enough to be considered "numbers".
I think it's not that natural to think of $mathbbC$ as a $mathbbR$-vector space of dimension $2$, not more natural than to think of $mathbbR$ as an infinite-dimensional $mathbbQ$-vector space, nor of $mathbbQ$ as a non-infinitely-generated $mathbbZ$-module.
answered 6 hours ago
LeaningLeaning
1,221718
$endgroup$
Numbers appeared first when we started to count things. $1$ tree, $2$ trees, $3$ trees, and so forth. That make up the set of non-zero natural numbers: $mathbbN$. Afterwards, people started to "count backwards" to get $mathbbZ$ (I'm just kidding, you can do some research on how negative numbers appeared historically). With the urge to divide things without remainders, the set of rational numbers $mathbbQ$ made its way to the world. A certain idea of geometric continuity gives us $mathbbR$. Finally we want all equations to have a root, that's how $mathbbC$ comes into play.
I guess what is considered "numbers" is rather a social question. The use of $mathbbC$
in physics and its $2$-D representation must have given a good intuition for a large set of people to accept it's intuitive enough to be considered "numbers".
I think it's not that natural to think of $mathbbC$ as a $mathbbR$-vector space of dimension $2$, not more natural than to think of $mathbbR$ as an infinite-dimensional $mathbbQ$-vector space, nor of $mathbbQ$ as a non-infinitely-generated $mathbbZ$-module.
answered 6 hours ago
LeaningLeaning
1,221718
edited 6 hours ago
answered 6 hours ago
LeaningLeaning
1,221718
answered 6 hours ago
LeaningLeaning
1,221718
answered 6 hours ago
LeaningLeaning
1,221718
1,221718
2
$begingroup$
People do think of complex numbers as points on the number plane.
$endgroup$
– Q the Platypus
6 hours ago
1
$begingroup$
Thank you. I'm sorry, that final part is just a personal opinion. edited.
$endgroup$
– Leaning
6 hours ago
add a comment |
2
$begingroup$
People do think of complex numbers as points on the number plane.
$endgroup$
– Q the Platypus
6 hours ago
1
$begingroup$
Thank you. I'm sorry, that final part is just a personal opinion. edited.
$endgroup$
– Leaning
6 hours ago
2
2
$begingroup$
People do think of complex numbers as points on the number plane.
$endgroup$
– Q the Platypus
6 hours ago
$begingroup$
People do think of complex numbers as points on the number plane.
$endgroup$
– Q the Platypus
6 hours ago
1
1
$begingroup$
Thank you. I'm sorry, that final part is just a personal opinion. edited.
$endgroup$
– Leaning
6 hours ago
$begingroup$
Thank you. I'm sorry, that final part is just a personal opinion. edited.
$endgroup$
– Leaning
6 hours ago
add a comment |
$begingroup$
I think naming is a rather soft topic, so there may not be a hard answer. However, I think it is worth noting that complex numbers are one of the three associative real division algebras (real numbers, complex numbers, and quaternions). These are all linked by the idea that division is meaningful in those three systems.
In general, a 2 dimensional real vector cannot admit a concept of division which matches what we expect division to do. However, if said real vector defines addition and multiplication the way complex numbers do, division is a natural result of those definitions.
answered 1 hour ago
Cort AmmonCort Ammon
2,421716
$endgroup$
add a comment |
$begingroup$
I think naming is a rather soft topic, so there may not be a hard answer. However, I think it is worth noting that complex numbers are one of the three associative real division algebras (real numbers, complex numbers, and quaternions). These are all linked by the idea that division is meaningful in those three systems.
In general, a 2 dimensional real vector cannot admit a concept of division which matches what we expect division to do. However, if said real vector defines addition and multiplication the way complex numbers do, division is a natural result of those definitions.
answered 1 hour ago
Cort AmmonCort Ammon
2,421716
$endgroup$
add a comment |
$begingroup$
I think naming is a rather soft topic, so there may not be a hard answer. However, I think it is worth noting that complex numbers are one of the three associative real division algebras (real numbers, complex numbers, and quaternions). These are all linked by the idea that division is meaningful in those three systems.
In general, a 2 dimensional real vector cannot admit a concept of division which matches what we expect division to do. However, if said real vector defines addition and multiplication the way complex numbers do, division is a natural result of those definitions.
answered 1 hour ago
Cort AmmonCort Ammon
2,421716
$endgroup$
I think naming is a rather soft topic, so there may not be a hard answer. However, I think it is worth noting that complex numbers are one of the three associative real division algebras (real numbers, complex numbers, and quaternions). These are all linked by the idea that division is meaningful in those three systems.
In general, a 2 dimensional real vector cannot admit a concept of division which matches what we expect division to do. However, if said real vector defines addition and multiplication the way complex numbers do, division is a natural result of those definitions.
answered 1 hour ago
Cort AmmonCort Ammon
2,421716
answered 1 hour ago
Cort AmmonCort Ammon
2,421716
answered 1 hour ago
Cort AmmonCort Ammon
2,421716
answered 1 hour ago
Cort AmmonCort Ammon
2,421716
2,421716
add a comment |
add a comment |
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7
$begingroup$
Usually, the difference would be whether you're in a context where complex multiplication would be interesting. It's kind of like the question: is $mathbbZ$ a group or a ring? It depends on which operations are of interest for the question you're studying.
$endgroup$
– Nate Eldredge
7 hours ago
4
$begingroup$
What's a "number" anyway?
$endgroup$
– Asaf Karagila♦
6 hours ago
$begingroup$
I would say that $mathbbZ$ is not a ring or a group as you have to have an operator in order to be a magma. However that would be over pedantic so I get where you are coming from.
$endgroup$
– Q the Platypus
6 hours ago
5
$begingroup$
@QthePlatypus: No, that's exactly the point. The question of whether the set $mathbbR^2$ of pairs of real numbers should be considered as the set of complex numbers, or as a set of vectors, depends on which operations you wish to equip that set with.
$endgroup$
– Nate Eldredge
6 hours ago
2
$begingroup$
I think the correct question were: is there a rigorous definition of the object "number" (without further characteristics such as "integer", "real", "complex" etc.)?
$endgroup$
– user
6 hours ago