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Rationals dense in Q

  1. Jan 6, 2013 #1
    1. The problem statement, all variables and given/known data
    Prove that the dyadic rationals are dense in Q.
    That is the rationals of the form [itex] \frac{m}{2^n} [/itex]
    m is an integer and n is a natural
    3. The attempt at a solution
    Lets say we have two arbitrary rationals x and y. where x<y
    Now I will pick a rational smaller than x such that it is of the form
    [itex] \frac{s}{2^k} [/itex] and i will call this P ,
    now I will pick a rational larger than y that is of the same form
    and i will call it O .
    Now I will add P and O together and then divide by 2, find the midpoint
    Now this new rational has a denominator that is a power of 2 because
    everything we did had a denominator of 2. Now I will keep doing this,
    I will keep finding mid points between these sets of rationals
    that I created and I might have to pick the left or right one and then
    keep finding the midpoints. Eventually i will get in between x and y.
    I realize this is informal but Is my general idea in the right direction.
     
  2. jcsd
  3. Jan 6, 2013 #2

    Dick

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    Science Advisor
    Homework Helper

    You can probably prove it in a less elaborate way. If x<y then y-x is positive and there must be an n such that 1/2^n is less than y-x, yes?
     
  4. Jan 7, 2013 #3
    yes I could do it that way. Thats the cool thing about pure math is that it is very creative.
     
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