Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

A Density in [0; 1]

  1. May 13, 2016 #1
    Hello, let be ##x \in \mathbb{R} - \mathbb{Q}##, do we have de density of ##\{nx - \lfloor{nx} / n \in \mathbb{N}\}## in ##[0; 1]## please?

    I think yes but it's just an intuition : if I take a and b in ##[0; 1]## with a < b, I have an irrationnal between them let call it c but I don't know hat to do with it.

    Could you help me please?

    Thank you in advance and have a nice afternoon:oldbiggrin:.
     
  2. jcsd
  3. May 13, 2016 #2

    fresh_42

    Staff: Mentor

    In principle you already have said what you need. How is a dense subset formally defined? It might be a bit of work to formally prove that there are always irrational numbers as close to a given rational one as wanted. But given this fact, it's basically the definition of density itself.
    E.g.: https://en.wikipedia.org/wiki/Dense_set
     
  4. May 13, 2016 #3
    No that's not what I mean : the irrationnal number are dense in ##\mathbb{R}## so in ##[0; 1]## too (1)
    And I wanted to use that to show that
    ##\{nx - \lfloor nx \rfloor / n \in \mathbb{N}\}## in [0; 1], where x is an irrationnal number, by using (1).
     
  5. May 13, 2016 #4

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

  6. May 13, 2016 #5

    fresh_42

    Staff: Mentor

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted