Lattice Points in x-y plane












0












$begingroup$



  1. What are Lattice Points?


  2. Which points in x-y planes are Lattice Points?



    Is (m,n) a lattice point where m,n are any integers?








plane-curves







share|cite|improve this question









$endgroup$

















    0












    $begingroup$



    1. What are Lattice Points?


    2. Which points in x-y planes are Lattice Points?



      Is (m,n) a lattice point where m,n are any integers?








    plane-curves







    share|cite|improve this question









    $endgroup$















      0












      0








      0


      1



      $begingroup$



      1. What are Lattice Points?


      2. Which points in x-y planes are Lattice Points?



        Is (m,n) a lattice point where m,n are any integers?








      plane-curves







      share|cite|improve this question









      $endgroup$





      1. What are Lattice Points?


      2. Which points in x-y planes are Lattice Points?



        Is (m,n) a lattice point where m,n are any integers?








      plane-curves




      plane-curves






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jul 12 '14 at 21:54









      user3481652user3481652

      386




      386






















          2 Answers
          2






          active

          oldest

          votes


















          0












          $begingroup$

          No, that's not accurate. The points $(m,n)inBbb Z^2$ are a lattice, but they are not the only lattice in $Bbb R^2$, consider the sets:



          $${(a,bsqrt 2): a,binBbb Z},quad left{left(a+{bover 2}, b{sqrt{3}over 2}right): a,binBbb Zright}tag{$*$}$$



          These are also a lattices.



          Generally a lattice in $Bbb R^2$ is a $Bbb Z$ module of rank $2$ which contains a basis for $Bbb R^2$.



          As Cameron notes, this just means that you have integer combinations of two $Bbb R$-linearly independent vectors from $Bbb R^2$ (it's important that they be linearly independent over $Bbb R$ and not something like $Bbb Q$)






          share|cite|improve this answer











          $endgroup$









          • 2




            $begingroup$
            Or in perhaps friendlier terms: given two linearly independent vectors in $mathbb{R}^2$, a lattice is all integer linear combinations of these vectors.
            $endgroup$
            – Cameron Williams
            Jul 12 '14 at 22:09



















          0












          $begingroup$

          That is correct. The term "lattice points" usually refers to the points with integer coordinates.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            okay thank you for the help.But can i know why?
            $endgroup$
            – user3481652
            Jul 12 '14 at 21:58










          • $begingroup$
            does 6x+8y=25 pass through any lattice point?
            $endgroup$
            – user3481652
            Jul 12 '14 at 22:00










          • $begingroup$
            @user3481652 why? because that's the definition
            $endgroup$
            – leonbloy
            Jul 12 '14 at 22:18






          • 1




            $begingroup$
            About your $6x+8y=25$: Your book is probably talking about the integer lattice, the points with integer coordinates. If $x$ and $y$ are integers, then $6x+8y$ is even, so cannot be $25$. The line does not pass through any points $(x,y)$ of the integer lattice.
            $endgroup$
            – André Nicolas
            Jul 12 '14 at 23:11













          Your Answer





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          2 Answers
          2






          active

          oldest

          votes








          2 Answers
          2






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          0












          $begingroup$

          No, that's not accurate. The points $(m,n)inBbb Z^2$ are a lattice, but they are not the only lattice in $Bbb R^2$, consider the sets:



          $${(a,bsqrt 2): a,binBbb Z},quad left{left(a+{bover 2}, b{sqrt{3}over 2}right): a,binBbb Zright}tag{$*$}$$



          These are also a lattices.



          Generally a lattice in $Bbb R^2$ is a $Bbb Z$ module of rank $2$ which contains a basis for $Bbb R^2$.



          As Cameron notes, this just means that you have integer combinations of two $Bbb R$-linearly independent vectors from $Bbb R^2$ (it's important that they be linearly independent over $Bbb R$ and not something like $Bbb Q$)






          share|cite|improve this answer











          $endgroup$









          • 2




            $begingroup$
            Or in perhaps friendlier terms: given two linearly independent vectors in $mathbb{R}^2$, a lattice is all integer linear combinations of these vectors.
            $endgroup$
            – Cameron Williams
            Jul 12 '14 at 22:09
















          0












          $begingroup$

          No, that's not accurate. The points $(m,n)inBbb Z^2$ are a lattice, but they are not the only lattice in $Bbb R^2$, consider the sets:



          $${(a,bsqrt 2): a,binBbb Z},quad left{left(a+{bover 2}, b{sqrt{3}over 2}right): a,binBbb Zright}tag{$*$}$$



          These are also a lattices.



          Generally a lattice in $Bbb R^2$ is a $Bbb Z$ module of rank $2$ which contains a basis for $Bbb R^2$.



          As Cameron notes, this just means that you have integer combinations of two $Bbb R$-linearly independent vectors from $Bbb R^2$ (it's important that they be linearly independent over $Bbb R$ and not something like $Bbb Q$)






          share|cite|improve this answer











          $endgroup$









          • 2




            $begingroup$
            Or in perhaps friendlier terms: given two linearly independent vectors in $mathbb{R}^2$, a lattice is all integer linear combinations of these vectors.
            $endgroup$
            – Cameron Williams
            Jul 12 '14 at 22:09














          0












          0








          0





          $begingroup$

          No, that's not accurate. The points $(m,n)inBbb Z^2$ are a lattice, but they are not the only lattice in $Bbb R^2$, consider the sets:



          $${(a,bsqrt 2): a,binBbb Z},quad left{left(a+{bover 2}, b{sqrt{3}over 2}right): a,binBbb Zright}tag{$*$}$$



          These are also a lattices.



          Generally a lattice in $Bbb R^2$ is a $Bbb Z$ module of rank $2$ which contains a basis for $Bbb R^2$.



          As Cameron notes, this just means that you have integer combinations of two $Bbb R$-linearly independent vectors from $Bbb R^2$ (it's important that they be linearly independent over $Bbb R$ and not something like $Bbb Q$)






          share|cite|improve this answer











          $endgroup$



          No, that's not accurate. The points $(m,n)inBbb Z^2$ are a lattice, but they are not the only lattice in $Bbb R^2$, consider the sets:



          $${(a,bsqrt 2): a,binBbb Z},quad left{left(a+{bover 2}, b{sqrt{3}over 2}right): a,binBbb Zright}tag{$*$}$$



          These are also a lattices.



          Generally a lattice in $Bbb R^2$ is a $Bbb Z$ module of rank $2$ which contains a basis for $Bbb R^2$.



          As Cameron notes, this just means that you have integer combinations of two $Bbb R$-linearly independent vectors from $Bbb R^2$ (it's important that they be linearly independent over $Bbb R$ and not something like $Bbb Q$)







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Jul 12 '14 at 22:16

























          answered Jul 12 '14 at 22:06









          Adam HughesAdam Hughes

          32.3k83770




          32.3k83770








          • 2




            $begingroup$
            Or in perhaps friendlier terms: given two linearly independent vectors in $mathbb{R}^2$, a lattice is all integer linear combinations of these vectors.
            $endgroup$
            – Cameron Williams
            Jul 12 '14 at 22:09














          • 2




            $begingroup$
            Or in perhaps friendlier terms: given two linearly independent vectors in $mathbb{R}^2$, a lattice is all integer linear combinations of these vectors.
            $endgroup$
            – Cameron Williams
            Jul 12 '14 at 22:09








          2




          2




          $begingroup$
          Or in perhaps friendlier terms: given two linearly independent vectors in $mathbb{R}^2$, a lattice is all integer linear combinations of these vectors.
          $endgroup$
          – Cameron Williams
          Jul 12 '14 at 22:09




          $begingroup$
          Or in perhaps friendlier terms: given two linearly independent vectors in $mathbb{R}^2$, a lattice is all integer linear combinations of these vectors.
          $endgroup$
          – Cameron Williams
          Jul 12 '14 at 22:09











          0












          $begingroup$

          That is correct. The term "lattice points" usually refers to the points with integer coordinates.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            okay thank you for the help.But can i know why?
            $endgroup$
            – user3481652
            Jul 12 '14 at 21:58










          • $begingroup$
            does 6x+8y=25 pass through any lattice point?
            $endgroup$
            – user3481652
            Jul 12 '14 at 22:00










          • $begingroup$
            @user3481652 why? because that's the definition
            $endgroup$
            – leonbloy
            Jul 12 '14 at 22:18






          • 1




            $begingroup$
            About your $6x+8y=25$: Your book is probably talking about the integer lattice, the points with integer coordinates. If $x$ and $y$ are integers, then $6x+8y$ is even, so cannot be $25$. The line does not pass through any points $(x,y)$ of the integer lattice.
            $endgroup$
            – André Nicolas
            Jul 12 '14 at 23:11


















          0












          $begingroup$

          That is correct. The term "lattice points" usually refers to the points with integer coordinates.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            okay thank you for the help.But can i know why?
            $endgroup$
            – user3481652
            Jul 12 '14 at 21:58










          • $begingroup$
            does 6x+8y=25 pass through any lattice point?
            $endgroup$
            – user3481652
            Jul 12 '14 at 22:00










          • $begingroup$
            @user3481652 why? because that's the definition
            $endgroup$
            – leonbloy
            Jul 12 '14 at 22:18






          • 1




            $begingroup$
            About your $6x+8y=25$: Your book is probably talking about the integer lattice, the points with integer coordinates. If $x$ and $y$ are integers, then $6x+8y$ is even, so cannot be $25$. The line does not pass through any points $(x,y)$ of the integer lattice.
            $endgroup$
            – André Nicolas
            Jul 12 '14 at 23:11
















          0












          0








          0





          $begingroup$

          That is correct. The term "lattice points" usually refers to the points with integer coordinates.






          share|cite|improve this answer











          $endgroup$



          That is correct. The term "lattice points" usually refers to the points with integer coordinates.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Jul 13 '14 at 10:35

























          answered Jul 12 '14 at 21:57









          DavidButlerUofADavidButlerUofA

          2,672821




          2,672821












          • $begingroup$
            okay thank you for the help.But can i know why?
            $endgroup$
            – user3481652
            Jul 12 '14 at 21:58










          • $begingroup$
            does 6x+8y=25 pass through any lattice point?
            $endgroup$
            – user3481652
            Jul 12 '14 at 22:00










          • $begingroup$
            @user3481652 why? because that's the definition
            $endgroup$
            – leonbloy
            Jul 12 '14 at 22:18






          • 1




            $begingroup$
            About your $6x+8y=25$: Your book is probably talking about the integer lattice, the points with integer coordinates. If $x$ and $y$ are integers, then $6x+8y$ is even, so cannot be $25$. The line does not pass through any points $(x,y)$ of the integer lattice.
            $endgroup$
            – André Nicolas
            Jul 12 '14 at 23:11




















          • $begingroup$
            okay thank you for the help.But can i know why?
            $endgroup$
            – user3481652
            Jul 12 '14 at 21:58










          • $begingroup$
            does 6x+8y=25 pass through any lattice point?
            $endgroup$
            – user3481652
            Jul 12 '14 at 22:00










          • $begingroup$
            @user3481652 why? because that's the definition
            $endgroup$
            – leonbloy
            Jul 12 '14 at 22:18






          • 1




            $begingroup$
            About your $6x+8y=25$: Your book is probably talking about the integer lattice, the points with integer coordinates. If $x$ and $y$ are integers, then $6x+8y$ is even, so cannot be $25$. The line does not pass through any points $(x,y)$ of the integer lattice.
            $endgroup$
            – André Nicolas
            Jul 12 '14 at 23:11


















          $begingroup$
          okay thank you for the help.But can i know why?
          $endgroup$
          – user3481652
          Jul 12 '14 at 21:58




          $begingroup$
          okay thank you for the help.But can i know why?
          $endgroup$
          – user3481652
          Jul 12 '14 at 21:58












          $begingroup$
          does 6x+8y=25 pass through any lattice point?
          $endgroup$
          – user3481652
          Jul 12 '14 at 22:00




          $begingroup$
          does 6x+8y=25 pass through any lattice point?
          $endgroup$
          – user3481652
          Jul 12 '14 at 22:00












          $begingroup$
          @user3481652 why? because that's the definition
          $endgroup$
          – leonbloy
          Jul 12 '14 at 22:18




          $begingroup$
          @user3481652 why? because that's the definition
          $endgroup$
          – leonbloy
          Jul 12 '14 at 22:18




          1




          1




          $begingroup$
          About your $6x+8y=25$: Your book is probably talking about the integer lattice, the points with integer coordinates. If $x$ and $y$ are integers, then $6x+8y$ is even, so cannot be $25$. The line does not pass through any points $(x,y)$ of the integer lattice.
          $endgroup$
          – André Nicolas
          Jul 12 '14 at 23:11






          $begingroup$
          About your $6x+8y=25$: Your book is probably talking about the integer lattice, the points with integer coordinates. If $x$ and $y$ are integers, then $6x+8y$ is even, so cannot be $25$. The line does not pass through any points $(x,y)$ of the integer lattice.
          $endgroup$
          – André Nicolas
          Jul 12 '14 at 23:11




















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