What does ${mathbb R^n}$ mean?












2












$begingroup$


I know that $mathbb R$ refers to all real numbers. But what does $mathbb R^2$ mean?



For that matter, what does $mathbb R^n$ mean when $n$ equals any natural number?






real-numbers







share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    en.wikipedia.org/wiki/Real_coordinate_space
    $endgroup$
    – T. Bongers
    Dec 2 '18 at 22:40










  • $begingroup$
    Its an $n$ dimensional vector over the reals
    $endgroup$
    – Seth
    Dec 2 '18 at 22:43










  • $begingroup$
    $mathbb R^n$ is the set of all ordered $n$-tuples of real numbers.
    $endgroup$
    – littleO
    Dec 2 '18 at 22:47










  • $begingroup$
    These questions may also help: math.stackexchange.com/questions/2423055/… math.stackexchange.com/questions/855437/what-does-mathbbrj-mean
    $endgroup$
    – Jean-Claude Arbaut
    Dec 2 '18 at 22:47












  • $begingroup$
    Thank you all for these answers
    $endgroup$
    – Xavier Stanton
    Dec 3 '18 at 2:07
















2












$begingroup$


I know that $mathbb R$ refers to all real numbers. But what does $mathbb R^2$ mean?



For that matter, what does $mathbb R^n$ mean when $n$ equals any natural number?






real-numbers







share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    en.wikipedia.org/wiki/Real_coordinate_space
    $endgroup$
    – T. Bongers
    Dec 2 '18 at 22:40










  • $begingroup$
    Its an $n$ dimensional vector over the reals
    $endgroup$
    – Seth
    Dec 2 '18 at 22:43










  • $begingroup$
    $mathbb R^n$ is the set of all ordered $n$-tuples of real numbers.
    $endgroup$
    – littleO
    Dec 2 '18 at 22:47










  • $begingroup$
    These questions may also help: math.stackexchange.com/questions/2423055/… math.stackexchange.com/questions/855437/what-does-mathbbrj-mean
    $endgroup$
    – Jean-Claude Arbaut
    Dec 2 '18 at 22:47












  • $begingroup$
    Thank you all for these answers
    $endgroup$
    – Xavier Stanton
    Dec 3 '18 at 2:07














2












2








2





$begingroup$


I know that $mathbb R$ refers to all real numbers. But what does $mathbb R^2$ mean?



For that matter, what does $mathbb R^n$ mean when $n$ equals any natural number?






real-numbers







share|cite|improve this question











$endgroup$




I know that $mathbb R$ refers to all real numbers. But what does $mathbb R^2$ mean?



For that matter, what does $mathbb R^n$ mean when $n$ equals any natural number?






real-numbers




real-numbers






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 3 '18 at 0:11









amWhy

1




1










asked Dec 2 '18 at 22:38









Xavier StantonXavier Stanton

311211




311211








  • 2




    $begingroup$
    en.wikipedia.org/wiki/Real_coordinate_space
    $endgroup$
    – T. Bongers
    Dec 2 '18 at 22:40










  • $begingroup$
    Its an $n$ dimensional vector over the reals
    $endgroup$
    – Seth
    Dec 2 '18 at 22:43










  • $begingroup$
    $mathbb R^n$ is the set of all ordered $n$-tuples of real numbers.
    $endgroup$
    – littleO
    Dec 2 '18 at 22:47










  • $begingroup$
    These questions may also help: math.stackexchange.com/questions/2423055/… math.stackexchange.com/questions/855437/what-does-mathbbrj-mean
    $endgroup$
    – Jean-Claude Arbaut
    Dec 2 '18 at 22:47












  • $begingroup$
    Thank you all for these answers
    $endgroup$
    – Xavier Stanton
    Dec 3 '18 at 2:07














  • 2




    $begingroup$
    en.wikipedia.org/wiki/Real_coordinate_space
    $endgroup$
    – T. Bongers
    Dec 2 '18 at 22:40










  • $begingroup$
    Its an $n$ dimensional vector over the reals
    $endgroup$
    – Seth
    Dec 2 '18 at 22:43










  • $begingroup$
    $mathbb R^n$ is the set of all ordered $n$-tuples of real numbers.
    $endgroup$
    – littleO
    Dec 2 '18 at 22:47










  • $begingroup$
    These questions may also help: math.stackexchange.com/questions/2423055/… math.stackexchange.com/questions/855437/what-does-mathbbrj-mean
    $endgroup$
    – Jean-Claude Arbaut
    Dec 2 '18 at 22:47












  • $begingroup$
    Thank you all for these answers
    $endgroup$
    – Xavier Stanton
    Dec 3 '18 at 2:07








2




2




$begingroup$
en.wikipedia.org/wiki/Real_coordinate_space
$endgroup$
– T. Bongers
Dec 2 '18 at 22:40




$begingroup$
en.wikipedia.org/wiki/Real_coordinate_space
$endgroup$
– T. Bongers
Dec 2 '18 at 22:40












$begingroup$
Its an $n$ dimensional vector over the reals
$endgroup$
– Seth
Dec 2 '18 at 22:43




$begingroup$
Its an $n$ dimensional vector over the reals
$endgroup$
– Seth
Dec 2 '18 at 22:43












$begingroup$
$mathbb R^n$ is the set of all ordered $n$-tuples of real numbers.
$endgroup$
– littleO
Dec 2 '18 at 22:47




$begingroup$
$mathbb R^n$ is the set of all ordered $n$-tuples of real numbers.
$endgroup$
– littleO
Dec 2 '18 at 22:47












$begingroup$
These questions may also help: math.stackexchange.com/questions/2423055/… math.stackexchange.com/questions/855437/what-does-mathbbrj-mean
$endgroup$
– Jean-Claude Arbaut
Dec 2 '18 at 22:47






$begingroup$
These questions may also help: math.stackexchange.com/questions/2423055/… math.stackexchange.com/questions/855437/what-does-mathbbrj-mean
$endgroup$
– Jean-Claude Arbaut
Dec 2 '18 at 22:47














$begingroup$
Thank you all for these answers
$endgroup$
– Xavier Stanton
Dec 3 '18 at 2:07




$begingroup$
Thank you all for these answers
$endgroup$
– Xavier Stanton
Dec 3 '18 at 2:07










4 Answers
4






active

oldest

votes


















2












$begingroup$

In general a set $X$ will have a Cartesian product with itself sometimes called $X^2$. We can again take the Cartesian product of $X^2$ and $X$ to get $X^3$
and so on.



Another way to interpret this notation is by considering two sets $X$ and $Y$ then define $Y^X$ as the set of all function from $X$ to $Y$. Then we can interpret $X^2$ with $2$ meaning the set of $2$ elements rather than as the number $2$ itself. This can be done with $X^n$ being the set of function from a set of $n$ elements to $X$ since they are all unique up to isomorphism.






share|cite|improve this answer









$endgroup$





















    1












    $begingroup$

    $mathbb R^n$ refers to the space of all $n$-dimensional vectors $(x_1, x_2, dots, x_n)$ where each coordinate $x_i$ is a real number i.e. $x_i in mathbb R$.






    share|cite|improve this answer











    $endgroup$





















      0












      $begingroup$

      Without giving you a word definition, let's see if you understand this way:



      $xinmathbb{R}$



      $(x_1, x_2) in mathbb{R^2}$



      $(x_1, x_2, x_3) in mathbb{R^3}$



      Therefore,



      $(x_1, x_2, x_3, dots , x_n) in mathbb{R^n}$



      where each $x_i$ is a member of the real numbers $mathbb{R}$.



      Thus, When we say something is a member of $mathbb{R}^n$, its saying that its a $n$ dimensional vector with elements of type $mathbb{R}$






      share|cite|improve this answer









      $endgroup$





















        0












        $begingroup$

        $mathbb{R}^n$ is the set of all n-tuples with real elements. They are NOT a vector space by themselves, just a set. For a vector space, we would need an extra scalar field and 2 operations: addition between the vectors (elements of $mathbb{R}^n$) and multiplication between the scalars and vectors. But usually we just denote the vector space of $mathbb{R}^n$ over the $mathbb{R}$, with the usual product and sum as $mathbb{R}^n$ for simplicity reasons, but they are not the same.






        share|cite|improve this answer









        $endgroup$













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          4 Answers
          4






          active

          oldest

          votes








          4 Answers
          4






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          2












          $begingroup$

          In general a set $X$ will have a Cartesian product with itself sometimes called $X^2$. We can again take the Cartesian product of $X^2$ and $X$ to get $X^3$
          and so on.



          Another way to interpret this notation is by considering two sets $X$ and $Y$ then define $Y^X$ as the set of all function from $X$ to $Y$. Then we can interpret $X^2$ with $2$ meaning the set of $2$ elements rather than as the number $2$ itself. This can be done with $X^n$ being the set of function from a set of $n$ elements to $X$ since they are all unique up to isomorphism.






          share|cite|improve this answer









          $endgroup$


















            2












            $begingroup$

            In general a set $X$ will have a Cartesian product with itself sometimes called $X^2$. We can again take the Cartesian product of $X^2$ and $X$ to get $X^3$
            and so on.



            Another way to interpret this notation is by considering two sets $X$ and $Y$ then define $Y^X$ as the set of all function from $X$ to $Y$. Then we can interpret $X^2$ with $2$ meaning the set of $2$ elements rather than as the number $2$ itself. This can be done with $X^n$ being the set of function from a set of $n$ elements to $X$ since they are all unique up to isomorphism.






            share|cite|improve this answer









            $endgroup$
















              2












              2








              2





              $begingroup$

              In general a set $X$ will have a Cartesian product with itself sometimes called $X^2$. We can again take the Cartesian product of $X^2$ and $X$ to get $X^3$
              and so on.



              Another way to interpret this notation is by considering two sets $X$ and $Y$ then define $Y^X$ as the set of all function from $X$ to $Y$. Then we can interpret $X^2$ with $2$ meaning the set of $2$ elements rather than as the number $2$ itself. This can be done with $X^n$ being the set of function from a set of $n$ elements to $X$ since they are all unique up to isomorphism.






              share|cite|improve this answer









              $endgroup$



              In general a set $X$ will have a Cartesian product with itself sometimes called $X^2$. We can again take the Cartesian product of $X^2$ and $X$ to get $X^3$
              and so on.



              Another way to interpret this notation is by considering two sets $X$ and $Y$ then define $Y^X$ as the set of all function from $X$ to $Y$. Then we can interpret $X^2$ with $2$ meaning the set of $2$ elements rather than as the number $2$ itself. This can be done with $X^n$ being the set of function from a set of $n$ elements to $X$ since they are all unique up to isomorphism.







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered Dec 2 '18 at 22:52









              CyclotomicFieldCyclotomicField

              2,3481314




              2,3481314























                  1












                  $begingroup$

                  $mathbb R^n$ refers to the space of all $n$-dimensional vectors $(x_1, x_2, dots, x_n)$ where each coordinate $x_i$ is a real number i.e. $x_i in mathbb R$.






                  share|cite|improve this answer











                  $endgroup$


















                    1












                    $begingroup$

                    $mathbb R^n$ refers to the space of all $n$-dimensional vectors $(x_1, x_2, dots, x_n)$ where each coordinate $x_i$ is a real number i.e. $x_i in mathbb R$.






                    share|cite|improve this answer











                    $endgroup$
















                      1












                      1








                      1





                      $begingroup$

                      $mathbb R^n$ refers to the space of all $n$-dimensional vectors $(x_1, x_2, dots, x_n)$ where each coordinate $x_i$ is a real number i.e. $x_i in mathbb R$.






                      share|cite|improve this answer











                      $endgroup$



                      $mathbb R^n$ refers to the space of all $n$-dimensional vectors $(x_1, x_2, dots, x_n)$ where each coordinate $x_i$ is a real number i.e. $x_i in mathbb R$.







                      share|cite|improve this answer














                      share|cite|improve this answer



                      share|cite|improve this answer








                      edited Dec 2 '18 at 22:49

























                      answered Dec 2 '18 at 22:45









                      RebellosRebellos

                      14.5k31246




                      14.5k31246























                          0












                          $begingroup$

                          Without giving you a word definition, let's see if you understand this way:



                          $xinmathbb{R}$



                          $(x_1, x_2) in mathbb{R^2}$



                          $(x_1, x_2, x_3) in mathbb{R^3}$



                          Therefore,



                          $(x_1, x_2, x_3, dots , x_n) in mathbb{R^n}$



                          where each $x_i$ is a member of the real numbers $mathbb{R}$.



                          Thus, When we say something is a member of $mathbb{R}^n$, its saying that its a $n$ dimensional vector with elements of type $mathbb{R}$






                          share|cite|improve this answer









                          $endgroup$


















                            0












                            $begingroup$

                            Without giving you a word definition, let's see if you understand this way:



                            $xinmathbb{R}$



                            $(x_1, x_2) in mathbb{R^2}$



                            $(x_1, x_2, x_3) in mathbb{R^3}$



                            Therefore,



                            $(x_1, x_2, x_3, dots , x_n) in mathbb{R^n}$



                            where each $x_i$ is a member of the real numbers $mathbb{R}$.



                            Thus, When we say something is a member of $mathbb{R}^n$, its saying that its a $n$ dimensional vector with elements of type $mathbb{R}$






                            share|cite|improve this answer









                            $endgroup$
















                              0












                              0








                              0





                              $begingroup$

                              Without giving you a word definition, let's see if you understand this way:



                              $xinmathbb{R}$



                              $(x_1, x_2) in mathbb{R^2}$



                              $(x_1, x_2, x_3) in mathbb{R^3}$



                              Therefore,



                              $(x_1, x_2, x_3, dots , x_n) in mathbb{R^n}$



                              where each $x_i$ is a member of the real numbers $mathbb{R}$.



                              Thus, When we say something is a member of $mathbb{R}^n$, its saying that its a $n$ dimensional vector with elements of type $mathbb{R}$






                              share|cite|improve this answer









                              $endgroup$



                              Without giving you a word definition, let's see if you understand this way:



                              $xinmathbb{R}$



                              $(x_1, x_2) in mathbb{R^2}$



                              $(x_1, x_2, x_3) in mathbb{R^3}$



                              Therefore,



                              $(x_1, x_2, x_3, dots , x_n) in mathbb{R^n}$



                              where each $x_i$ is a member of the real numbers $mathbb{R}$.



                              Thus, When we say something is a member of $mathbb{R}^n$, its saying that its a $n$ dimensional vector with elements of type $mathbb{R}$







                              share|cite|improve this answer












                              share|cite|improve this answer



                              share|cite|improve this answer










                              answered Dec 2 '18 at 22:52









                              K Split XK Split X

                              4,20611131




                              4,20611131























                                  0












                                  $begingroup$

                                  $mathbb{R}^n$ is the set of all n-tuples with real elements. They are NOT a vector space by themselves, just a set. For a vector space, we would need an extra scalar field and 2 operations: addition between the vectors (elements of $mathbb{R}^n$) and multiplication between the scalars and vectors. But usually we just denote the vector space of $mathbb{R}^n$ over the $mathbb{R}$, with the usual product and sum as $mathbb{R}^n$ for simplicity reasons, but they are not the same.






                                  share|cite|improve this answer









                                  $endgroup$


















                                    0












                                    $begingroup$

                                    $mathbb{R}^n$ is the set of all n-tuples with real elements. They are NOT a vector space by themselves, just a set. For a vector space, we would need an extra scalar field and 2 operations: addition between the vectors (elements of $mathbb{R}^n$) and multiplication between the scalars and vectors. But usually we just denote the vector space of $mathbb{R}^n$ over the $mathbb{R}$, with the usual product and sum as $mathbb{R}^n$ for simplicity reasons, but they are not the same.






                                    share|cite|improve this answer









                                    $endgroup$
















                                      0












                                      0








                                      0





                                      $begingroup$

                                      $mathbb{R}^n$ is the set of all n-tuples with real elements. They are NOT a vector space by themselves, just a set. For a vector space, we would need an extra scalar field and 2 operations: addition between the vectors (elements of $mathbb{R}^n$) and multiplication between the scalars and vectors. But usually we just denote the vector space of $mathbb{R}^n$ over the $mathbb{R}$, with the usual product and sum as $mathbb{R}^n$ for simplicity reasons, but they are not the same.






                                      share|cite|improve this answer









                                      $endgroup$



                                      $mathbb{R}^n$ is the set of all n-tuples with real elements. They are NOT a vector space by themselves, just a set. For a vector space, we would need an extra scalar field and 2 operations: addition between the vectors (elements of $mathbb{R}^n$) and multiplication between the scalars and vectors. But usually we just denote the vector space of $mathbb{R}^n$ over the $mathbb{R}$, with the usual product and sum as $mathbb{R}^n$ for simplicity reasons, but they are not the same.







                                      share|cite|improve this answer












                                      share|cite|improve this answer



                                      share|cite|improve this answer










                                      answered Dec 2 '18 at 22:55









                                      BotondBotond

                                      5,6822732




                                      5,6822732






























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