Line segment of length pi (Just a thought I've had)

  1. If you were to imagine a line segment of length pi, I would guess it would have to be finite. But since pi is an irrational number, it has infinitely many decimals so can't you just keep sort of zooming in on the end of the segment so that it sort of keeps on getting longer indefinitely?

    Pi is just an example, but I'm sure any irrational number would bring up the same idea. Any thoughts on this?
  2. jcsd
  3. NOT being able to precisely measure the length of the segment in this case, does not imply that the segment is/gets indefenitely long(er).
  4. HallsofIvy

    HallsofIvy 40,297
    Staff Emeritus
    Science Advisor

    But what do you mean by "precisely measure"? If you are talking about using some kind of measuring device then, of course, it cannot be "precisely measured". No interval can in that sense. If you mean "mathematically", in the same sense that we talk about an interval "of length 1", then it can be "precisely measured". Measure out an interval of length 1 and construct a circle about one end of that interval having the interval as radius. The circle will be of length [itex]]pi[/itex]. An interval of length "pi" is no different from an interval of any other length.
  5. Well, 3.141592653 is certainly longer than 3.1415, but 3.1415 isn't pi, so no, it is NOT getting longer.
  6. No, because at sufficiently small scales, atomic dimensions and quantum uncertainty would prevent the possibility of having something mesuring exactly [tex]\pi[/tex]. The same applies for any irrational number.
  7. Huh, are people aware that physical lines and geometric lines are two different things?

    Lines and line segments are abstract ideas. They're not what you draw on a piece of paper, nor are they anything that we see. To say that a physical line segment has irrational length is completely meaningless, as it is to say that a physical segment has some precise length.
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