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abdul_shabeer

  • 2 years ago

Does a point have no dimension?

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  1. mathslover
    • 2 years ago
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    yes a point is dimensionless

  2. zepp
    • 2 years ago
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    0D = Dot 1D = Line 2D = Plane 3D = Solid 4D = Space-time 5D = ...idk :P

  3. abdul_shabeer
    • 2 years ago
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    Then how does it make a line?

  4. zepp
    • 2 years ago
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    What do you mean?

  5. zepp
    • 2 years ago
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    . <-- This is not a point, it's just a representation because a point has no height, no length, no thickness.

  6. zepp
    • 2 years ago
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    One dimension: A line, numbers, they have only a length;

  7. zepp
    • 2 years ago
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    2D: Shape, Length and width 3D: Solids, Length, width and thickness

  8. abdul_shabeer
    • 2 years ago
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    What is the definition for a line?

  9. zepp
    • 2 years ago
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    4D Spacetime: 3 dimentional object travelling/moving through time and space.

  10. zepp
    • 2 years ago
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    Straight objects without depth and width.

  11. abdul_shabeer
    • 2 years ago
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    Is it made of many points?

  12. zepp
    • 2 years ago
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    Um, you can see it that way :)

  13. abdul_shabeer
    • 2 years ago
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    When a point has no dimension then how can a collection of points make a line?

  14. zepp
    • 2 years ago
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    A point has no length, depth, width, when you align then, you just created a length

  15. TuringTest
    • 2 years ago
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    take a point and send it in some direction through space the path that it traces out is a line the real number line is infinitely dense so now we are getting into some of the trickyness of infinity

  16. TuringTest
    • 2 years ago
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    |dw:1341410594698:dw|here is the real number line (still a line, consisting of infinitely many points) I can ask you to pick out the point 1

  17. TuringTest
    • 2 years ago
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    |dw:1341410679112:dw|and there it is I can also ask you to find numbers between numbers so if I ask you to find 1.5 it should be on the line

  18. abdul_shabeer
    • 2 years ago
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    My doubt is when a point has zero dimension, then how can it make a line that has one dimension?

  19. TuringTest
    • 2 years ago
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    |dw:1341410742343:dw|similarly I can keep asking you to find points between the two we have already found 1.75 must be on the line 1.85 1.8 1.8243574687654365768.... etc. must all be on the line therefore there are infintiely many points on a line

  20. TuringTest
    • 2 years ago
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    between any two points there are infinitely many numbers, do you agree?

  21. TuringTest
    • 2 years ago
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    like between 1 and 2 are there not an infinity of numbers?

  22. abdul_shabeer
    • 2 years ago
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    Though there are infinitely many points, each has zero length, zero breadth etc.

  23. TuringTest
    • 2 years ago
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    but you are trying to add an infinity of infinitely tiny lengths!\[\infty\cdot\frac1\infty=\text{undefined}\]you can't treat infinity like a number and perform addition on it; it is a concept, rather than a number.

  24. TuringTest
    • 2 years ago
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    it would require an infinity of points lined up in a row, and they still would have not length this may seem paradoxical, but it reflects the infinite density of number distribution on the real line, so it is not trivial

  25. abdul_shabeer
    • 2 years ago
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    Wouldn't 0+0+0+0........ be zero?

  26. TuringTest
    • 2 years ago
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    no, because \(0\cdot\infty\) is undefined

  27. TuringTest
    • 2 years ago
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    again, I know this does not make instinctive sense infinity is a tricky topic that has driven many mathematicians mad!

  28. abdul_shabeer
    • 2 years ago
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    When 0*infinity is undefined, then how does it produce a length?

  29. TuringTest
    • 2 years ago
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    because you don't have to build a line one point at a time; it's a mathematical construct that contains infinitely many points

  30. TuringTest
    • 2 years ago
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    if there were some finite number of points that you could put together to make a line, then the real number line would not be infinitely dense (i.e. some points would be missing from the line because there would have to be a limit on the number of times I can tell you to find the midpoint between two points)

  31. TuringTest
    • 2 years ago
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    |dw:1341411610187:dw|take any line segment; I want the midpoint (whatever it is)

  32. TuringTest
    • 2 years ago
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    |dw:1341411643038:dw|now I want the midpoint of the right half

  33. TuringTest
    • 2 years ago
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    |dw:1341411666632:dw|now again, the midpoint of the right half

  34. TuringTest
    • 2 years ago
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    |dw:1341411687412:dw|we could do this forever, right? that means there are infinitely many points on that line; I can always ask you to find the midpoint between any two points.

  35. TuringTest
    • 2 years ago
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    yet even though it is an infinity of points, they only add up to a finite line segment, so an infinity of things can add to something finite. Reference to Zeno's paradox.

  36. abdul_shabeer
    • 2 years ago
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    What does "though it is an infinity of points, they only add up to a finite line segment" mean?

  37. zepp
    • 2 years ago
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    TuringTest just showed that there's an amount of point between two point, but when this infinite adds up, it gives something finite, something you can just count on your fingers.

  38. zepp
    • 2 years ago
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    an infinite amount*

  39. abdul_shabeer
    • 2 years ago
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    How can infinite things add and give some finite value?

  40. nbouscal
    • 2 years ago
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    It is important to understand here that points and lines are simply geometric realizations of abstract concepts. As mentioned earlier, a . is not a point, nor is a drawing of a line actually a line. A point takes up no space, so it can't be shown, and similarly a line can't be shown because it has no height. Even a plane can't be shown really, because it has no depth. So, when you're asking these questions, it is best to avoid relying too heavily on your geometric interpretation of the concepts. Instead, it is better to look at them using the tools of analysis. TuringTest is referring to some relevant results of analysis, for example, the uncountability of the real line. Not only is the real line infinite in the number of points, it is uncountably infinite. This ends up implying that the number of points between 0 and 1 is actually the same as the number of points on the entire real line. So, to interpret this somewhat geometrically, you can zoom in as close as you would like to the real line and still be looking at the same number of points. Of course, this doesn't work like any real-world object. Another relevant way of looking at the real line is through the lens of topology. We can talk about the real line as a metric space, and we define the metric abstractly. We simply define how distance works. One sees by learning some introductory topology that the real line is actually just one way to look at the set of real numbers, and there are many others. So, this is another way to look at the problem.

  41. TuringTest
    • 2 years ago
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    in answer to how an infinity of things can add up to something finite do you know what an infinite series is @abdul_shabeer ?

  42. abdul_shabeer
    • 2 years ago
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    No @TuringTest

  43. zepp
    • 2 years ago
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    The sum of all terms of the geometric sequence that halves each time would be the perfect example :D

  44. TuringTest
    • 2 years ago
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    \[\sum_{n=1}^\infty(\frac12)^n=\frac12+\frac14+\frac18+...=1\]when you study them this may make a little more sense the one above can be said to represent Zeno's paradox

  45. nbouscal
    • 2 years ago
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    In general, we don't have an infinity of things simply "adding up" to finite value. We have an infinity of things approaching a limit that has finite value.

  46. asnaseer
    • 2 years ago
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    Hey guys - I'm at work at the moment but spotted this discussion and remembered a thread I saw that might help explain this to to you @abdul_shabeer : http://mathforum.org/library/drmath/view/55297.html Hope it helps - back to work now... :)

  47. UnkleRhaukus
    • 2 years ago
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    |dw:1341412717556:dw|