1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Small understanding

  1. Mar 18, 2007 #1
    Hi,

    This is not a homework question since I am out of college for a long time.

    I was trying to understand the following that [0,1] /\ Q is not closed in R.
    My understanding is that u must take a sequence (since this is a metric space) of the form m/n s.t m < n and create a sequence.

    So I was trying to construct sequences like 1/2, 2/3, 3/4 but they seemed to be all ending within [0,1] /\ Q. I am not sure but do I have to take a sum or sth but I am not sure how to prove it.
     
  2. jcsd
  3. Mar 18, 2007 #2

    mathman

    User Avatar
    Science Advisor
    Gold Member

    (I am assuming that Q means rational nos. and R means real nos.). Take any irrational number between 0 and 1 and let the rational number sequence be the the sequence where the nth term is the truncation of the decimal expansion of the irrational after n decimal places.
     
  4. Mar 18, 2007 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Just giving an example of what mathman said: [itex]\sqrt{2}{2}[/itex] is irrational and is 0.70710678118654752440084436210485...
    Each number in the sequence 0.7, 0.70, 0.707, 0.7071, 0.70710, 0.707106,... is a rational number because it is a terminating decimal; but the sequence as a whole converges to the irrational number [itex]\frac{\sqrt{2}}{{2}}[/itex]
     
    Last edited: Mar 20, 2007
  5. Mar 18, 2007 #4

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    You're working from the wrong direction. Rather than trying to come up with a sequence of rationals and hope that its limit is irrational... you should pick the irrational, and then try and find a sequence of rationals that converges to it.
     
  6. Mar 19, 2007 #5
    Thanks a lot for clearing the lacunae in my understanding. I think the basic idea is that u need to show that if u wiggle the rational number a bit u will fall into the set of irrational numbers, which is clearly proved by the sqrt(22) example.

    Thank u a lot for the same.
     
  7. Mar 19, 2007 #6
    Actually that was sqrt(2)/2 (how could the square root of 22 be less than 1?). You should read what Hurkyl said. The simplest irrational you can think of in the interval is sqrt(2)/2, and constructing a sequence that converges to it is easy, as shown by HallsofIvy.
     
  8. Mar 19, 2007 #7

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    'Cause I'm nothing if not simple!
     
  9. Mar 19, 2007 #8
    Do you understand why this argument suffices?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Small understanding
  1. Understanding skewness (Replies: 5)

Loading...