Lattice Points in x-y plane
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
- What are Lattice Points?
Which points in x-y planes are Lattice Points?
Is (m,n) a lattice point where m,n are any integers?
plane-curves
asked Jul 12 '14 at 21:54
user3481652
376
add a comment |Â
up vote
0
down vote
favorite
- What are Lattice Points?
Which points in x-y planes are Lattice Points?
Is (m,n) a lattice point where m,n are any integers?
plane-curves
asked Jul 12 '14 at 21:54
user3481652
376
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
- What are Lattice Points?
Which points in x-y planes are Lattice Points?
Is (m,n) a lattice point where m,n are any integers?
plane-curves
asked Jul 12 '14 at 21:54
user3481652
376
- What are Lattice Points?
Which points in x-y planes are Lattice Points?
Is (m,n) a lattice point where m,n are any integers?
plane-curves
asked Jul 12 '14 at 21:54
user3481652
376
asked Jul 12 '14 at 21:54
user3481652
376
asked Jul 12 '14 at 21:54
user3481652
376
asked Jul 12 '14 at 21:54
user3481652
376
376
add a comment |Â
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
0
down vote
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:
$$(a,bsqrt 2): a,binBbb Z,quad leftleft(a+bover 2, bsqrt3over 2right): a,binBbb Zrighttag$*$$$
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 $Bbb Q$)
answered Jul 12 '14 at 22:06
Adam Hughes
31.6k83670
2
Or in perhaps friendlier terms: given two linearly independent vectors in $mathbbR^2$, a lattice is all integer linear combinations of these vectors.
– Cameron Williams
Jul 12 '14 at 22:09
add a comment |Â
up vote
0
down vote
That is correct. The term "lattice points" usually refers to the points with integer coordinates.
answered Jul 12 '14 at 21:57
DavidButlerUofA
2,614721
okay thank you for the help.But can i know why?
– user3481652
Jul 12 '14 at 21:58
does 6x+8y=25 pass through any lattice point?
– user3481652
Jul 12 '14 at 22:00
@user3481652 why? because that's the definition
– leonbloy
Jul 12 '14 at 22:18
1
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.
– André Nicolas
Jul 12 '14 at 23:11
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
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:
$$(a,bsqrt 2): a,binBbb Z,quad leftleft(a+bover 2, bsqrt3over 2right): a,binBbb Zrighttag$*$$$
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 $Bbb Q$)
answered Jul 12 '14 at 22:06
Adam Hughes
31.6k83670
2
Or in perhaps friendlier terms: given two linearly independent vectors in $mathbbR^2$, a lattice is all integer linear combinations of these vectors.
– Cameron Williams
Jul 12 '14 at 22:09
add a comment |Â
up vote
0
down vote
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:
$$(a,bsqrt 2): a,binBbb Z,quad leftleft(a+bover 2, bsqrt3over 2right): a,binBbb Zrighttag$*$$$
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 $Bbb Q$)
answered Jul 12 '14 at 22:06
Adam Hughes
31.6k83670
2
Or in perhaps friendlier terms: given two linearly independent vectors in $mathbbR^2$, a lattice is all integer linear combinations of these vectors.
– Cameron Williams
Jul 12 '14 at 22:09
add a comment |Â
up vote
0
down vote
up vote
0
down vote
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:
$$(a,bsqrt 2): a,binBbb Z,quad leftleft(a+bover 2, bsqrt3over 2right): a,binBbb Zrighttag$*$$$
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 $Bbb Q$)
answered Jul 12 '14 at 22:06
Adam Hughes
31.6k83670
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:
$$(a,bsqrt 2): a,binBbb Z,quad leftleft(a+bover 2, bsqrt3over 2right): a,binBbb Zrighttag$*$$$
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 $Bbb Q$)
answered Jul 12 '14 at 22:06
Adam Hughes
31.6k83670
edited Jul 12 '14 at 22:16
answered Jul 12 '14 at 22:06
Adam Hughes
31.6k83670
answered Jul 12 '14 at 22:06
Adam Hughes
31.6k83670
answered Jul 12 '14 at 22:06
Adam Hughes
31.6k83670
31.6k83670
2
Or in perhaps friendlier terms: given two linearly independent vectors in $mathbbR^2$, a lattice is all integer linear combinations of these vectors.
– Cameron Williams
Jul 12 '14 at 22:09
add a comment |Â
2
Or in perhaps friendlier terms: given two linearly independent vectors in $mathbbR^2$, a lattice is all integer linear combinations of these vectors.
– Cameron Williams
Jul 12 '14 at 22:09
2
2
Or in perhaps friendlier terms: given two linearly independent vectors in $mathbbR^2$, a lattice is all integer linear combinations of these vectors.
– Cameron Williams
Jul 12 '14 at 22:09
Or in perhaps friendlier terms: given two linearly independent vectors in $mathbbR^2$, a lattice is all integer linear combinations of these vectors.
– Cameron Williams
Jul 12 '14 at 22:09
add a comment |Â
up vote
0
down vote
That is correct. The term "lattice points" usually refers to the points with integer coordinates.
answered Jul 12 '14 at 21:57
DavidButlerUofA
2,614721
okay thank you for the help.But can i know why?
– user3481652
Jul 12 '14 at 21:58
does 6x+8y=25 pass through any lattice point?
– user3481652
Jul 12 '14 at 22:00
@user3481652 why? because that's the definition
– leonbloy
Jul 12 '14 at 22:18
1
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.
– André Nicolas
Jul 12 '14 at 23:11
add a comment |Â
up vote
0
down vote
That is correct. The term "lattice points" usually refers to the points with integer coordinates.
answered Jul 12 '14 at 21:57
DavidButlerUofA
2,614721
okay thank you for the help.But can i know why?
– user3481652
Jul 12 '14 at 21:58
does 6x+8y=25 pass through any lattice point?
– user3481652
Jul 12 '14 at 22:00
@user3481652 why? because that's the definition
– leonbloy
Jul 12 '14 at 22:18
1
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.
– André Nicolas
Jul 12 '14 at 23:11
add a comment |Â
up vote
0
down vote
up vote
0
down vote
That is correct. The term "lattice points" usually refers to the points with integer coordinates.
answered Jul 12 '14 at 21:57
DavidButlerUofA
2,614721
That is correct. The term "lattice points" usually refers to the points with integer coordinates.
answered Jul 12 '14 at 21:57
DavidButlerUofA
2,614721
edited Jul 13 '14 at 10:35
answered Jul 12 '14 at 21:57
DavidButlerUofA
2,614721
answered Jul 12 '14 at 21:57
DavidButlerUofA
2,614721
answered Jul 12 '14 at 21:57
DavidButlerUofA
2,614721
2,614721
okay thank you for the help.But can i know why?
– user3481652
Jul 12 '14 at 21:58
does 6x+8y=25 pass through any lattice point?
– user3481652
Jul 12 '14 at 22:00
@user3481652 why? because that's the definition
– leonbloy
Jul 12 '14 at 22:18
1
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.
– André Nicolas
Jul 12 '14 at 23:11
add a comment |Â
okay thank you for the help.But can i know why?
– user3481652
Jul 12 '14 at 21:58
does 6x+8y=25 pass through any lattice point?
– user3481652
Jul 12 '14 at 22:00
@user3481652 why? because that's the definition
– leonbloy
Jul 12 '14 at 22:18
1
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.
– André Nicolas
Jul 12 '14 at 23:11
okay thank you for the help.But can i know why?
– user3481652
Jul 12 '14 at 21:58
okay thank you for the help.But can i know why?
– user3481652
Jul 12 '14 at 21:58
does 6x+8y=25 pass through any lattice point?
– user3481652
Jul 12 '14 at 22:00
does 6x+8y=25 pass through any lattice point?
– user3481652
Jul 12 '14 at 22:00
@user3481652 why? because that's the definition
– leonbloy
Jul 12 '14 at 22:18
@user3481652 why? because that's the definition
– leonbloy
Jul 12 '14 at 22:18
1
1
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.
– André Nicolas
Jul 12 '14 at 23:11
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.
– André Nicolas
Jul 12 '14 at 23:11
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f865505%2flattice-points-in-x-y-plane%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
