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I know that line is a one dimension but i cannot understand then why

  1. Sep 4, 2011 #1
    i know that line is a one dimension but i cannot understand then why do sometime need two spatial variable to specify the st line for example x+y=c is a straight line but we require two coordinates to define it then how it is one dimensional?
     
  2. jcsd
  3. Sep 4, 2011 #2
    Re: cannot understand dimension

    Because the line is embedded in a two dimensional space, namely the plane. You need one relation between the two coordinates to get a one-parameter manifold. This is expressed by the relation you wrote. Because it is a linear relationship and we are in a Euclidean space, it represents a straight line.
     
  4. Sep 4, 2011 #3
    Re: dimensions

    A line is a one dimensional object because you COULD take it and lie it down right onto the x-axis, for example; and, if you need to specify any position along the line, all you need is a single number.

    The reason why you sometimes need 2 variables is because you are now placing such line onto a 2-dimensional space and so you need two numbers to specify every position along the line in such space...but this is due to the space, not to the line.
     
  5. Sep 4, 2011 #4

    Hurkyl

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    Re: dimensions

    Exercise: for the line in the plane defined by the equation x+y=2, find a scheme for labeling its points with a single numbers. You should be able to do both of the following:
    • Given the (x,y) coordinates of a point on the line, you should be able to compute the number that labels it
    • Given a number t, you should be able to compute the (x,y) coordinates of the point that has that label
     
  6. Sep 4, 2011 #5

    HallsofIvy

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    Re: cannot understand dimension

    You are assuming, without saying it and perhaps without realizing it yourself, that the graph of x+ y= c is drawn in a plane. A plane (or any surface) is two dimensional- an arbitrary point is determined by two numbers, x and y. But with the equation x+ y= c, given either x or y, you could determine the other. That is, it only requires one number to determine a point on that line so it is one dimensional.

    And, you must specify what space you are "embedding" your figure in. If you were to draw the graph of x+ y= c in a a three dimensional coordinate system, it would be a plane, not a line. An arbitrary point in three dimensions, requires three numbers, (x, y, z), to specify it. With the requirement that x+ y= c, we could, given either x or y, calculate the other. But z would still be undetermined. With that equation, we still require two numbers to specify a point so the set is a two dimensional subset of three dimensional space.

    You will probably soon, if you have not already, see the concept of "parametric equations". We could write equations for "x+ y= c", assuming it is in two dimensions as x= t, y= c- t. Choosing any value for t gives a point on the line. Two equation, because the line is "embedded" in two dimensions but depending on one parameter. It is the one parameter that makes this one dimensional.

    But if this were in three dimensions, we would have something like x= t, y= c- t, z= s. Three equations, because it is not embedded in three dimensions but now depending on two parameters. If we had three equation x= f(t), y= g(t), z= h(t), all depending on a single parameter, that would give a one dimensional curve in three dimensional space. Since you posted this in the "Physics" section rather than "Mathematics", you can think of "t" as the time and the three equations as telling the path of some moving object. And a path is one dimensional, of course.
     
    Last edited: Sep 4, 2011
  7. Sep 4, 2011 #6

    HallsofIvy

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    Re: dimensions

    This is the same as a thread asad1111 posted in "General Physics" so I am merging the two threads.

    asad1111, you can get into trouble double posting!
     
  8. Sep 4, 2011 #7
    Re: dimensions

    lol, nice one. troll the troll.
     
  9. Sep 8, 2011 #8
    Re: dimensions

    Saying that something is n dimensional if you can specify points in the object by using n coordinates is something that you have to take with a pinch of salt- it does give you a good idea of what n dimensional objects are like (in particular, what their local neighbourhoods look like) but it isn't the best of definitions.

    For example, I can fill a square with a 1 dimensional line:
    http://en.wikipedia.org/wiki/Peano_curve
    This means that I only need 1 coordinate to specify a point in the square, even though it should really be two dimensional.

    So use this definition as a guide, but not as a definition unto itself. Remember, dimension is just a mathematical concept, and needs a definition (and has many definitions depending on the context). Annoyingly, popular media tries convey it as some sort of mystical thing that has a real definition outside of mathematics.

    However, there are definitions of it, several in fact. For example, in topology there is the notion of the Lebesgue covering dimension, the topological dimension and more. If you are working with manifolds, you say it is n-dimensional if it locally looks like a piece of n dimensional Euclidean space.

    I'm sure you have probably also heard of "fractal dimensions" which can have dimensions which are fractions (or irrational numbers, of course!). This immediately confuses people because they try to apply the above definition to what it means to be, say 1.43 dimensional... you need 1 and 0.43 coordinates to specify a point? Sounds very odd, but not as odd when you realise that the quantity is just a mathematical tool which is trying to describe some characteristic of the object.


    So, in your example, most definitions of dimension will tell you your line is one dimensional. It's covering dimension, topological dimension and manifold dimension will all be one (it locally looks like a piece of 1 dimensional space!). And, as others have pointed out, although your description requires 2 coordinates to specify the point, you can alter your coordinate system so that you now only need one input, so that all your points can be labeled with only 1 coordinate (e.g. you could label them with "length along the line past the intersection with the y axis", or many more).
     
  10. Sep 8, 2011 #9
    Re: dimensions

    thanx jamma it is exactly kind of answer i was looking for it clears alot
     
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