Why do we call complex numbers “numbers” but we don’t consider 2-vectors numbers?












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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 terminology philosophy







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  • 10




    $begingroup$
    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 $mathbb{Z}$ a group or a ring? It depends on which operations are of interest for the question you're studying.
    $endgroup$
    – Nate Eldredge
    yesterday








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    What's a "number" anyway?
    $endgroup$
    – Asaf Karagila♦
    yesterday






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    @QthePlatypus: No, that's exactly the point. The question of whether the set $mathbb{R}^2$ 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
    yesterday






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    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
    yesterday








  • 1




    $begingroup$
    Some considerations can be found here: math.stackexchange.com/q/865409
    $endgroup$
    – user
    yesterday


















27












$begingroup$


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 terminology philosophy







share|cite|improve this question











$endgroup$








  • 10




    $begingroup$
    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 $mathbb{Z}$ a group or a ring? It depends on which operations are of interest for the question you're studying.
    $endgroup$
    – Nate Eldredge
    yesterday








  • 12




    $begingroup$
    What's a "number" anyway?
    $endgroup$
    – Asaf Karagila♦
    yesterday






  • 12




    $begingroup$
    @QthePlatypus: No, that's exactly the point. The question of whether the set $mathbb{R}^2$ 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
    yesterday






  • 7




    $begingroup$
    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
    yesterday








  • 1




    $begingroup$
    Some considerations can be found here: math.stackexchange.com/q/865409
    $endgroup$
    – user
    yesterday
















27












27








27


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$begingroup$


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 terminology philosophy







share|cite|improve this question











$endgroup$




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 terminology philosophy




matrices complex-numbers terminology philosophy






share|cite|improve this question















share|cite|improve this question













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share|cite|improve this question








edited 14 hours ago









Basj

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3941528










asked yesterday









Q the PlatypusQ the Platypus

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  • 10




    $begingroup$
    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 $mathbb{Z}$ a group or a ring? It depends on which operations are of interest for the question you're studying.
    $endgroup$
    – Nate Eldredge
    yesterday








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    What's a "number" anyway?
    $endgroup$
    – Asaf Karagila♦
    yesterday






  • 12




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    @QthePlatypus: No, that's exactly the point. The question of whether the set $mathbb{R}^2$ 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
    yesterday






  • 7




    $begingroup$
    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
    yesterday








  • 1




    $begingroup$
    Some considerations can be found here: math.stackexchange.com/q/865409
    $endgroup$
    – user
    yesterday
















  • 10




    $begingroup$
    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 $mathbb{Z}$ a group or a ring? It depends on which operations are of interest for the question you're studying.
    $endgroup$
    – Nate Eldredge
    yesterday








  • 12




    $begingroup$
    What's a "number" anyway?
    $endgroup$
    – Asaf Karagila♦
    yesterday






  • 12




    $begingroup$
    @QthePlatypus: No, that's exactly the point. The question of whether the set $mathbb{R}^2$ 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
    yesterday






  • 7




    $begingroup$
    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
    yesterday








  • 1




    $begingroup$
    Some considerations can be found here: math.stackexchange.com/q/865409
    $endgroup$
    – user
    yesterday










10




10




$begingroup$
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 $mathbb{Z}$ a group or a ring? It depends on which operations are of interest for the question you're studying.
$endgroup$
– Nate Eldredge
yesterday






$begingroup$
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 $mathbb{Z}$ a group or a ring? It depends on which operations are of interest for the question you're studying.
$endgroup$
– Nate Eldredge
yesterday






12




12




$begingroup$
What's a "number" anyway?
$endgroup$
– Asaf Karagila♦
yesterday




$begingroup$
What's a "number" anyway?
$endgroup$
– Asaf Karagila♦
yesterday




12




12




$begingroup$
@QthePlatypus: No, that's exactly the point. The question of whether the set $mathbb{R}^2$ 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
yesterday




$begingroup$
@QthePlatypus: No, that's exactly the point. The question of whether the set $mathbb{R}^2$ 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
yesterday




7




7




$begingroup$
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
yesterday






$begingroup$
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
yesterday






1




1




$begingroup$
Some considerations can be found here: math.stackexchange.com/q/865409
$endgroup$
– user
yesterday






$begingroup$
Some considerations can be found here: math.stackexchange.com/q/865409
$endgroup$
– user
yesterday












6 Answers
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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. $mathbb{N}$, $mathbb{Z}$, $mathbb{Q}$, $mathbb{R}$ and $mathbb{C}$ 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?






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    Wikipedia calls the quaternions a number system.
    $endgroup$
    – JJJ
    20 hours ago






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    You can also tell what opinion people had of a number system based on its name, generally (it is not a coincidence that the word rational means both a number system and a way of thinking). At least the old ones.
    $endgroup$
    – PyRulez
    18 hours ago










  • $begingroup$
    Agreed; the use of the word number is definitely a matter of convention. As for opinions; the use of the word 'real' numbers is regarded as an incredibly triggering piece of propaganda by physical finitists...
    $endgroup$
    – Eelco Hoogendoorn
    12 hours ago






  • 2




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    @PyRulez I would have expected that "rational" comes from "ratio", not from "ration".
    $endgroup$
    – Thern
    8 hours ago










  • $begingroup$
    @Thern The English words "ratio", "rational" and "reason" all come from the same Latin root "ratio", meaning "reckoning" or "reasoning"
    $endgroup$
    – David Richerby
    4 hours ago



















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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.






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    So Quaternions which have Non commutativity multiplication would also be considered as non numbers?
    $endgroup$
    – Q the Platypus
    yesterday






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    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
    yesterday






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    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
    22 hours ago






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    @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
    22 hours ago








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    @Todd Sewell - I think this may relate to positive and negative zero. Positive and negative zero are equal, yet if you divide something by one or the other you may get positive or negative infinity.
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    – Hammerite
    11 hours ago



















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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: $mathbb{N}$. Afterwards, people started to "count backwards" to get $mathbb{Z}$ (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 $mathbb{Q}$ made its way to the world. A certain idea of geometric continuity gives us $mathbb{R}$. Finally we want all equations to have a root, that's how $mathbb{C}$ comes into play.



I guess what is considered "numbers" is rather a social question. The use of $mathbb{C}$
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 $mathbb{C}$ as a $mathbb{R}$-vector space of dimension $2$, not more natural than to think of $mathbb{R}$ as an infinite-dimensional $mathbb{Q}$-vector space, nor of $mathbb{Q}$ as a non-infinitely-generated $mathbb{Z}$-module.






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  • 3




    $begingroup$
    People do think of complex numbers as points on the number plane.
    $endgroup$
    – Q the Platypus
    yesterday






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    Thank you. I'm sorry, that final part is just a personal opinion. edited.
    $endgroup$
    – Leaning
    yesterday





















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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.






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    Both, complex numbers and vector spaces are mathematical spaces, i.e., sets equipped it with some structure, e.g., operations (such as addition or multiplication) and rules obeyed by them (such as commutativity or associativity).
    Without that structure, these sets are pretty useless:
    You can hardly make any interesting statements about them or connect them to reality.



    For example, without any structure, the only difference between ℝ and ℝ² is how we tend to name the objects within: We can find bijective maps between them.
    Of course, once we impose the usual structure onto these sets, we will note that these maps do not preserve any of it (they are not even continuous) and thus are not very relevant.



    Now, for ℝ² (the space of 2-vectors) and ℂ, some structure such as addition is the same, but some isn’t, e.g., there is no equivalent to complex multiplication in ℝ².
    This distinction in structure is the only relevant difference between ℝ² and ℂ.
    If you define a multiplication of 2-vectors that just mirrors complex multiplication, you would just be inventing complex numbers under a different name.



    So much for why it is justified to apply different labels to complex numbers and 2-vectors.
    As for why we call complex numbers numbers in the first place, there’s certainly a bit of history involved (of which I know little) and number is not a mathematically defined concept anyway.
    That being said, calling complex numbers numbers is not completely deviod of consistency, given that complex numbers feature most of the structural properties of other sets that we call numbers (for even more historical reasons), e.g., the vast majority of things you can do with real numbers, you can also do with complex numbers.
    This does not hold for 2-vectors.






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      $begingroup$

      [Possibly not a direct answer, but]



      Notice ordinal sum and product are not commutative, that full distributivity of multiplication over addition and of exponentiation over product fails, and they're still regarded as numbers. OTOH, the corresponding operations for vector spaces are well-behaved (probably because they do become cardinal-like once you identify the isomorphic ones) and yet we don't tend to think of or to refer to them as numbers.






      share|cite|improve this answer









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        6 Answers
        6






        active

        oldest

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        6 Answers
        6






        active

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        active

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        41












        $begingroup$

        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. $mathbb{N}$, $mathbb{Z}$, $mathbb{Q}$, $mathbb{R}$ and $mathbb{C}$ 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?






        share|cite|improve this answer











        $endgroup$









        • 6




          $begingroup$
          Wikipedia calls the quaternions a number system.
          $endgroup$
          – JJJ
          20 hours ago






        • 5




          $begingroup$
          You can also tell what opinion people had of a number system based on its name, generally (it is not a coincidence that the word rational means both a number system and a way of thinking). At least the old ones.
          $endgroup$
          – PyRulez
          18 hours ago










        • $begingroup$
          Agreed; the use of the word number is definitely a matter of convention. As for opinions; the use of the word 'real' numbers is regarded as an incredibly triggering piece of propaganda by physical finitists...
          $endgroup$
          – Eelco Hoogendoorn
          12 hours ago






        • 2




          $begingroup$
          @PyRulez I would have expected that "rational" comes from "ratio", not from "ration".
          $endgroup$
          – Thern
          8 hours ago










        • $begingroup$
          @Thern The English words "ratio", "rational" and "reason" all come from the same Latin root "ratio", meaning "reckoning" or "reasoning"
          $endgroup$
          – David Richerby
          4 hours ago
















        41












        $begingroup$

        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. $mathbb{N}$, $mathbb{Z}$, $mathbb{Q}$, $mathbb{R}$ and $mathbb{C}$ 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?






        share|cite|improve this answer











        $endgroup$









        • 6




          $begingroup$
          Wikipedia calls the quaternions a number system.
          $endgroup$
          – JJJ
          20 hours ago






        • 5




          $begingroup$
          You can also tell what opinion people had of a number system based on its name, generally (it is not a coincidence that the word rational means both a number system and a way of thinking). At least the old ones.
          $endgroup$
          – PyRulez
          18 hours ago










        • $begingroup$
          Agreed; the use of the word number is definitely a matter of convention. As for opinions; the use of the word 'real' numbers is regarded as an incredibly triggering piece of propaganda by physical finitists...
          $endgroup$
          – Eelco Hoogendoorn
          12 hours ago






        • 2




          $begingroup$
          @PyRulez I would have expected that "rational" comes from "ratio", not from "ration".
          $endgroup$
          – Thern
          8 hours ago










        • $begingroup$
          @Thern The English words "ratio", "rational" and "reason" all come from the same Latin root "ratio", meaning "reckoning" or "reasoning"
          $endgroup$
          – David Richerby
          4 hours ago














        41












        41








        41





        $begingroup$

        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. $mathbb{N}$, $mathbb{Z}$, $mathbb{Q}$, $mathbb{R}$ and $mathbb{C}$ 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?






        share|cite|improve this answer











        $endgroup$



        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. $mathbb{N}$, $mathbb{Z}$, $mathbb{Q}$, $mathbb{R}$ and $mathbb{C}$ 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?







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited 21 hours ago

























        answered yesterday









        MathematicsStudent1122MathematicsStudent1122

        8,92622668




        8,92622668








        • 6




          $begingroup$
          Wikipedia calls the quaternions a number system.
          $endgroup$
          – JJJ
          20 hours ago






        • 5




          $begingroup$
          You can also tell what opinion people had of a number system based on its name, generally (it is not a coincidence that the word rational means both a number system and a way of thinking). At least the old ones.
          $endgroup$
          – PyRulez
          18 hours ago










        • $begingroup$
          Agreed; the use of the word number is definitely a matter of convention. As for opinions; the use of the word 'real' numbers is regarded as an incredibly triggering piece of propaganda by physical finitists...
          $endgroup$
          – Eelco Hoogendoorn
          12 hours ago






        • 2




          $begingroup$
          @PyRulez I would have expected that "rational" comes from "ratio", not from "ration".
          $endgroup$
          – Thern
          8 hours ago










        • $begingroup$
          @Thern The English words "ratio", "rational" and "reason" all come from the same Latin root "ratio", meaning "reckoning" or "reasoning"
          $endgroup$
          – David Richerby
          4 hours ago














        • 6




          $begingroup$
          Wikipedia calls the quaternions a number system.
          $endgroup$
          – JJJ
          20 hours ago






        • 5




          $begingroup$
          You can also tell what opinion people had of a number system based on its name, generally (it is not a coincidence that the word rational means both a number system and a way of thinking). At least the old ones.
          $endgroup$
          – PyRulez
          18 hours ago










        • $begingroup$
          Agreed; the use of the word number is definitely a matter of convention. As for opinions; the use of the word 'real' numbers is regarded as an incredibly triggering piece of propaganda by physical finitists...
          $endgroup$
          – Eelco Hoogendoorn
          12 hours ago






        • 2




          $begingroup$
          @PyRulez I would have expected that "rational" comes from "ratio", not from "ration".
          $endgroup$
          – Thern
          8 hours ago










        • $begingroup$
          @Thern The English words "ratio", "rational" and "reason" all come from the same Latin root "ratio", meaning "reckoning" or "reasoning"
          $endgroup$
          – David Richerby
          4 hours ago








        6




        6




        $begingroup$
        Wikipedia calls the quaternions a number system.
        $endgroup$
        – JJJ
        20 hours ago




        $begingroup$
        Wikipedia calls the quaternions a number system.
        $endgroup$
        – JJJ
        20 hours ago




        5




        5




        $begingroup$
        You can also tell what opinion people had of a number system based on its name, generally (it is not a coincidence that the word rational means both a number system and a way of thinking). At least the old ones.
        $endgroup$
        – PyRulez
        18 hours ago




        $begingroup$
        You can also tell what opinion people had of a number system based on its name, generally (it is not a coincidence that the word rational means both a number system and a way of thinking). At least the old ones.
        $endgroup$
        – PyRulez
        18 hours ago












        $begingroup$
        Agreed; the use of the word number is definitely a matter of convention. As for opinions; the use of the word 'real' numbers is regarded as an incredibly triggering piece of propaganda by physical finitists...
        $endgroup$
        – Eelco Hoogendoorn
        12 hours ago




        $begingroup$
        Agreed; the use of the word number is definitely a matter of convention. As for opinions; the use of the word 'real' numbers is regarded as an incredibly triggering piece of propaganda by physical finitists...
        $endgroup$
        – Eelco Hoogendoorn
        12 hours ago




        2




        2




        $begingroup$
        @PyRulez I would have expected that "rational" comes from "ratio", not from "ration".
        $endgroup$
        – Thern
        8 hours ago




        $begingroup$
        @PyRulez I would have expected that "rational" comes from "ratio", not from "ration".
        $endgroup$
        – Thern
        8 hours ago












        $begingroup$
        @Thern The English words "ratio", "rational" and "reason" all come from the same Latin root "ratio", meaning "reckoning" or "reasoning"
        $endgroup$
        – David Richerby
        4 hours ago




        $begingroup$
        @Thern The English words "ratio", "rational" and "reason" all come from the same Latin root "ratio", meaning "reckoning" or "reasoning"
        $endgroup$
        – David Richerby
        4 hours ago











        19












        $begingroup$

        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.






        share|cite|improve this answer









        $endgroup$













        • $begingroup$
          So Quaternions which have Non commutativity multiplication would also be considered as non numbers?
          $endgroup$
          – Q the Platypus
          yesterday






        • 5




          $begingroup$
          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
          yesterday






        • 4




          $begingroup$
          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
          22 hours ago






        • 7




          $begingroup$
          @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
          22 hours ago








        • 3




          $begingroup$
          @Todd Sewell - I think this may relate to positive and negative zero. Positive and negative zero are equal, yet if you divide something by one or the other you may get positive or negative infinity.
          $endgroup$
          – Hammerite
          11 hours ago
















        19












        $begingroup$

        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.






        share|cite|improve this answer









        $endgroup$













        • $begingroup$
          So Quaternions which have Non commutativity multiplication would also be considered as non numbers?
          $endgroup$
          – Q the Platypus
          yesterday






        • 5




          $begingroup$
          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
          yesterday






        • 4




          $begingroup$
          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
          22 hours ago






        • 7




          $begingroup$
          @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
          22 hours ago








        • 3




          $begingroup$
          @Todd Sewell - I think this may relate to positive and negative zero. Positive and negative zero are equal, yet if you divide something by one or the other you may get positive or negative infinity.
          $endgroup$
          – Hammerite
          11 hours ago














        19












        19








        19





        $begingroup$

        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.






        share|cite|improve this answer









        $endgroup$



        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.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered yesterday









        user247327user247327

        11.4k1516




        11.4k1516












        • $begingroup$
          So Quaternions which have Non commutativity multiplication would also be considered as non numbers?
          $endgroup$
          – Q the Platypus
          yesterday






        • 5




          $begingroup$
          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
          yesterday






        • 4




          $begingroup$
          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
          22 hours ago






        • 7




          $begingroup$
          @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
          22 hours ago








        • 3




          $begingroup$
          @Todd Sewell - I think this may relate to positive and negative zero. Positive and negative zero are equal, yet if you divide something by one or the other you may get positive or negative infinity.
          $endgroup$
          – Hammerite
          11 hours ago


















        • $begingroup$
          So Quaternions which have Non commutativity multiplication would also be considered as non numbers?
          $endgroup$
          – Q the Platypus
          yesterday






        • 5




          $begingroup$
          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
          yesterday






        • 4




          $begingroup$
          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
          22 hours ago






        • 7




          $begingroup$
          @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
          22 hours ago








        • 3




          $begingroup$
          @Todd Sewell - I think this may relate to positive and negative zero. Positive and negative zero are equal, yet if you divide something by one or the other you may get positive or negative infinity.
          $endgroup$
          – Hammerite
          11 hours ago
















        $begingroup$
        So Quaternions which have Non commutativity multiplication would also be considered as non numbers?
        $endgroup$
        – Q the Platypus
        yesterday




        $begingroup$
        So Quaternions which have Non commutativity multiplication would also be considered as non numbers?
        $endgroup$
        – Q the Platypus
        yesterday




        5




        5




        $begingroup$
        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
        yesterday




        $begingroup$
        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
        yesterday




        4




        4




        $begingroup$
        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
        22 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
        22 hours ago




        7




        7




        $begingroup$
        @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
        22 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
        22 hours ago






        3




        3




        $begingroup$
        @Todd Sewell - I think this may relate to positive and negative zero. Positive and negative zero are equal, yet if you divide something by one or the other you may get positive or negative infinity.
        $endgroup$
        – Hammerite
        11 hours ago




        $begingroup$
        @Todd Sewell - I think this may relate to positive and negative zero. Positive and negative zero are equal, yet if you divide something by one or the other you may get positive or negative infinity.
        $endgroup$
        – Hammerite
        11 hours ago











        4












        $begingroup$

        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: $mathbb{N}$. Afterwards, people started to "count backwards" to get $mathbb{Z}$ (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 $mathbb{Q}$ made its way to the world. A certain idea of geometric continuity gives us $mathbb{R}$. Finally we want all equations to have a root, that's how $mathbb{C}$ comes into play.



        I guess what is considered "numbers" is rather a social question. The use of $mathbb{C}$
        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 $mathbb{C}$ as a $mathbb{R}$-vector space of dimension $2$, not more natural than to think of $mathbb{R}$ as an infinite-dimensional $mathbb{Q}$-vector space, nor of $mathbb{Q}$ as a non-infinitely-generated $mathbb{Z}$-module.






        share|cite|improve this answer











        $endgroup$









        • 3




          $begingroup$
          People do think of complex numbers as points on the number plane.
          $endgroup$
          – Q the Platypus
          yesterday






        • 1




          $begingroup$
          Thank you. I'm sorry, that final part is just a personal opinion. edited.
          $endgroup$
          – Leaning
          yesterday


















        4












        $begingroup$

        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: $mathbb{N}$. Afterwards, people started to "count backwards" to get $mathbb{Z}$ (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 $mathbb{Q}$ made its way to the world. A certain idea of geometric continuity gives us $mathbb{R}$. Finally we want all equations to have a root, that's how $mathbb{C}$ comes into play.



        I guess what is considered "numbers" is rather a social question. The use of $mathbb{C}$
        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 $mathbb{C}$ as a $mathbb{R}$-vector space of dimension $2$, not more natural than to think of $mathbb{R}$ as an infinite-dimensional $mathbb{Q}$-vector space, nor of $mathbb{Q}$ as a non-infinitely-generated $mathbb{Z}$-module.






        share|cite|improve this answer











        $endgroup$









        • 3




          $begingroup$
          People do think of complex numbers as points on the number plane.
          $endgroup$
          – Q the Platypus
          yesterday






        • 1




          $begingroup$
          Thank you. I'm sorry, that final part is just a personal opinion. edited.
          $endgroup$
          – Leaning
          yesterday
















        4












        4








        4





        $begingroup$

        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: $mathbb{N}$. Afterwards, people started to "count backwards" to get $mathbb{Z}$ (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 $mathbb{Q}$ made its way to the world. A certain idea of geometric continuity gives us $mathbb{R}$. Finally we want all equations to have a root, that's how $mathbb{C}$ comes into play.



        I guess what is considered "numbers" is rather a social question. The use of $mathbb{C}$
        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 $mathbb{C}$ as a $mathbb{R}$-vector space of dimension $2$, not more natural than to think of $mathbb{R}$ as an infinite-dimensional $mathbb{Q}$-vector space, nor of $mathbb{Q}$ as a non-infinitely-generated $mathbb{Z}$-module.






        share|cite|improve this answer











        $endgroup$



        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: $mathbb{N}$. Afterwards, people started to "count backwards" to get $mathbb{Z}$ (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 $mathbb{Q}$ made its way to the world. A certain idea of geometric continuity gives us $mathbb{R}$. Finally we want all equations to have a root, that's how $mathbb{C}$ comes into play.



        I guess what is considered "numbers" is rather a social question. The use of $mathbb{C}$
        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 $mathbb{C}$ as a $mathbb{R}$-vector space of dimension $2$, not more natural than to think of $mathbb{R}$ as an infinite-dimensional $mathbb{Q}$-vector space, nor of $mathbb{Q}$ as a non-infinitely-generated $mathbb{Z}$-module.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited yesterday

























        answered yesterday









        LeaningLeaning

        1,296718




        1,296718








        • 3




          $begingroup$
          People do think of complex numbers as points on the number plane.
          $endgroup$
          – Q the Platypus
          yesterday






        • 1




          $begingroup$
          Thank you. I'm sorry, that final part is just a personal opinion. edited.
          $endgroup$
          – Leaning
          yesterday
















        • 3




          $begingroup$
          People do think of complex numbers as points on the number plane.
          $endgroup$
          – Q the Platypus
          yesterday






        • 1




          $begingroup$
          Thank you. I'm sorry, that final part is just a personal opinion. edited.
          $endgroup$
          – Leaning
          yesterday










        3




        3




        $begingroup$
        People do think of complex numbers as points on the number plane.
        $endgroup$
        – Q the Platypus
        yesterday




        $begingroup$
        People do think of complex numbers as points on the number plane.
        $endgroup$
        – Q the Platypus
        yesterday




        1




        1




        $begingroup$
        Thank you. I'm sorry, that final part is just a personal opinion. edited.
        $endgroup$
        – Leaning
        yesterday






        $begingroup$
        Thank you. I'm sorry, that final part is just a personal opinion. edited.
        $endgroup$
        – Leaning
        yesterday













        3












        $begingroup$

        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.






        share|cite|improve this answer









        $endgroup$


















          3












          $begingroup$

          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.






          share|cite|improve this answer









          $endgroup$
















            3












            3








            3





            $begingroup$

            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.






            share|cite|improve this answer









            $endgroup$



            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.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered 20 hours ago









            Cort AmmonCort Ammon

            2,451716




            2,451716























                1












                $begingroup$

                Both, complex numbers and vector spaces are mathematical spaces, i.e., sets equipped it with some structure, e.g., operations (such as addition or multiplication) and rules obeyed by them (such as commutativity or associativity).
                Without that structure, these sets are pretty useless:
                You can hardly make any interesting statements about them or connect them to reality.



                For example, without any structure, the only difference between ℝ and ℝ² is how we tend to name the objects within: We can find bijective maps between them.
                Of course, once we impose the usual structure onto these sets, we will note that these maps do not preserve any of it (they are not even continuous) and thus are not very relevant.



                Now, for ℝ² (the space of 2-vectors) and ℂ, some structure such as addition is the same, but some isn’t, e.g., there is no equivalent to complex multiplication in ℝ².
                This distinction in structure is the only relevant difference between ℝ² and ℂ.
                If you define a multiplication of 2-vectors that just mirrors complex multiplication, you would just be inventing complex numbers under a different name.



                So much for why it is justified to apply different labels to complex numbers and 2-vectors.
                As for why we call complex numbers numbers in the first place, there’s certainly a bit of history involved (of which I know little) and number is not a mathematically defined concept anyway.
                That being said, calling complex numbers numbers is not completely deviod of consistency, given that complex numbers feature most of the structural properties of other sets that we call numbers (for even more historical reasons), e.g., the vast majority of things you can do with real numbers, you can also do with complex numbers.
                This does not hold for 2-vectors.






                share|cite|improve this answer









                $endgroup$


















                  1












                  $begingroup$

                  Both, complex numbers and vector spaces are mathematical spaces, i.e., sets equipped it with some structure, e.g., operations (such as addition or multiplication) and rules obeyed by them (such as commutativity or associativity).
                  Without that structure, these sets are pretty useless:
                  You can hardly make any interesting statements about them or connect them to reality.



                  For example, without any structure, the only difference between ℝ and ℝ² is how we tend to name the objects within: We can find bijective maps between them.
                  Of course, once we impose the usual structure onto these sets, we will note that these maps do not preserve any of it (they are not even continuous) and thus are not very relevant.



                  Now, for ℝ² (the space of 2-vectors) and ℂ, some structure such as addition is the same, but some isn’t, e.g., there is no equivalent to complex multiplication in ℝ².
                  This distinction in structure is the only relevant difference between ℝ² and ℂ.
                  If you define a multiplication of 2-vectors that just mirrors complex multiplication, you would just be inventing complex numbers under a different name.



                  So much for why it is justified to apply different labels to complex numbers and 2-vectors.
                  As for why we call complex numbers numbers in the first place, there’s certainly a bit of history involved (of which I know little) and number is not a mathematically defined concept anyway.
                  That being said, calling complex numbers numbers is not completely deviod of consistency, given that complex numbers feature most of the structural properties of other sets that we call numbers (for even more historical reasons), e.g., the vast majority of things you can do with real numbers, you can also do with complex numbers.
                  This does not hold for 2-vectors.






                  share|cite|improve this answer









                  $endgroup$
















                    1












                    1








                    1





                    $begingroup$

                    Both, complex numbers and vector spaces are mathematical spaces, i.e., sets equipped it with some structure, e.g., operations (such as addition or multiplication) and rules obeyed by them (such as commutativity or associativity).
                    Without that structure, these sets are pretty useless:
                    You can hardly make any interesting statements about them or connect them to reality.



                    For example, without any structure, the only difference between ℝ and ℝ² is how we tend to name the objects within: We can find bijective maps between them.
                    Of course, once we impose the usual structure onto these sets, we will note that these maps do not preserve any of it (they are not even continuous) and thus are not very relevant.



                    Now, for ℝ² (the space of 2-vectors) and ℂ, some structure such as addition is the same, but some isn’t, e.g., there is no equivalent to complex multiplication in ℝ².
                    This distinction in structure is the only relevant difference between ℝ² and ℂ.
                    If you define a multiplication of 2-vectors that just mirrors complex multiplication, you would just be inventing complex numbers under a different name.



                    So much for why it is justified to apply different labels to complex numbers and 2-vectors.
                    As for why we call complex numbers numbers in the first place, there’s certainly a bit of history involved (of which I know little) and number is not a mathematically defined concept anyway.
                    That being said, calling complex numbers numbers is not completely deviod of consistency, given that complex numbers feature most of the structural properties of other sets that we call numbers (for even more historical reasons), e.g., the vast majority of things you can do with real numbers, you can also do with complex numbers.
                    This does not hold for 2-vectors.






                    share|cite|improve this answer









                    $endgroup$



                    Both, complex numbers and vector spaces are mathematical spaces, i.e., sets equipped it with some structure, e.g., operations (such as addition or multiplication) and rules obeyed by them (such as commutativity or associativity).
                    Without that structure, these sets are pretty useless:
                    You can hardly make any interesting statements about them or connect them to reality.



                    For example, without any structure, the only difference between ℝ and ℝ² is how we tend to name the objects within: We can find bijective maps between them.
                    Of course, once we impose the usual structure onto these sets, we will note that these maps do not preserve any of it (they are not even continuous) and thus are not very relevant.



                    Now, for ℝ² (the space of 2-vectors) and ℂ, some structure such as addition is the same, but some isn’t, e.g., there is no equivalent to complex multiplication in ℝ².
                    This distinction in structure is the only relevant difference between ℝ² and ℂ.
                    If you define a multiplication of 2-vectors that just mirrors complex multiplication, you would just be inventing complex numbers under a different name.



                    So much for why it is justified to apply different labels to complex numbers and 2-vectors.
                    As for why we call complex numbers numbers in the first place, there’s certainly a bit of history involved (of which I know little) and number is not a mathematically defined concept anyway.
                    That being said, calling complex numbers numbers is not completely deviod of consistency, given that complex numbers feature most of the structural properties of other sets that we call numbers (for even more historical reasons), e.g., the vast majority of things you can do with real numbers, you can also do with complex numbers.
                    This does not hold for 2-vectors.







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                    answered 17 hours ago









                    WrzlprmftWrzlprmft

                    3,13111334




                    3,13111334























                        0












                        $begingroup$

                        [Possibly not a direct answer, but]



                        Notice ordinal sum and product are not commutative, that full distributivity of multiplication over addition and of exponentiation over product fails, and they're still regarded as numbers. OTOH, the corresponding operations for vector spaces are well-behaved (probably because they do become cardinal-like once you identify the isomorphic ones) and yet we don't tend to think of or to refer to them as numbers.






                        share|cite|improve this answer









                        $endgroup$


















                          0












                          $begingroup$

                          [Possibly not a direct answer, but]



                          Notice ordinal sum and product are not commutative, that full distributivity of multiplication over addition and of exponentiation over product fails, and they're still regarded as numbers. OTOH, the corresponding operations for vector spaces are well-behaved (probably because they do become cardinal-like once you identify the isomorphic ones) and yet we don't tend to think of or to refer to them as numbers.






                          share|cite|improve this answer









                          $endgroup$
















                            0












                            0








                            0





                            $begingroup$

                            [Possibly not a direct answer, but]



                            Notice ordinal sum and product are not commutative, that full distributivity of multiplication over addition and of exponentiation over product fails, and they're still regarded as numbers. OTOH, the corresponding operations for vector spaces are well-behaved (probably because they do become cardinal-like once you identify the isomorphic ones) and yet we don't tend to think of or to refer to them as numbers.






                            share|cite|improve this answer









                            $endgroup$



                            [Possibly not a direct answer, but]



                            Notice ordinal sum and product are not commutative, that full distributivity of multiplication over addition and of exponentiation over product fails, and they're still regarded as numbers. OTOH, the corresponding operations for vector spaces are well-behaved (probably because they do become cardinal-like once you identify the isomorphic ones) and yet we don't tend to think of or to refer to them as numbers.







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                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered 9 hours ago









                            alkchfalkchf

                            47337




                            47337






























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