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abdul_shabeer Group Title

Does a point have no dimension?

  • 2 years ago
  • 2 years ago

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  1. mathslover Group Title
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    yes a point is dimensionless

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

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

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

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

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

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

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

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

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

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

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

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

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

    • 2 years ago
  15. TuringTest Group Title
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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

    • 2 years ago
  16. TuringTest Group Title
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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

    • 2 years ago
  17. TuringTest Group Title
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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

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

    • 2 years ago
  19. TuringTest Group Title
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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

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

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

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

    • 2 years ago
  23. TuringTest Group Title
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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.

    • 2 years ago
  24. TuringTest Group Title
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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

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

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

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

    • 2 years ago
  28. TuringTest Group Title
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    http://math.stackexchange.com/questions/28940/why-is-infinity-multiplied-by-zero-not-an-easy-zero-answer

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

    • 2 years ago
  30. TuringTest Group Title
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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

    • 2 years ago
  31. TuringTest Group Title
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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)

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

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

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

    • 2 years ago
  35. TuringTest Group Title
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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.

    • 2 years ago
  36. TuringTest Group Title
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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.

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

    • 2 years ago
  38. zepp Group Title
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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.

    • 2 years ago
  39. zepp Group Title
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    an infinite amount*

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

    • 2 years ago
  41. nbouscal Group Title
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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.

    • 2 years ago
  42. TuringTest Group Title
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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 ?

    • 2 years ago
  43. abdul_shabeer Group Title
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    No @TuringTest

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

    • 2 years ago
  45. TuringTest Group Title
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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

    • 2 years ago
  46. nbouscal Group Title
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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.

    • 2 years ago
  47. asnaseer Group Title
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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... :)

    • 2 years ago
  48. UnkleRhaukus Group Title
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    |dw:1341412717556:dw|

    • 2 years ago