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More numbers

  1. Dec 8, 2003 #1

    MathematicalPhysicist

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    between 0-1 there are infinite number of rational numbers now between 1-2 there are also infinite number of rational numbers, how can we proove that the number of rational numbers between 0-1 equals to those between 1-2?

    does the difference of the domains which equals to each other (1) has any significance?
     
  2. jcsd
  3. Dec 8, 2003 #2
    To show that the number of rationals on [0,1] is the same as the number of rationals on [1,2] you just need to find a bijection from [0,1] to [1,2].

    Clearly [itex]f(x)=x+1[/itex] is a suitable bijection.
     
  4. Dec 9, 2003 #3

    MathematicalPhysicist

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    what is x represnts in this context? (the number of rational numbers?).
     
  5. Dec 9, 2003 #4
    x represents any number in [0, 1]
     
  6. Dec 9, 2003 #5

    MathematicalPhysicist

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    let me see if i understand, x is in [0,1] then f(x) is in [1,2] therfore f:x->f(x) therfore the number of rationals in [0,1] equals to [1,2].
     
  7. Dec 9, 2003 #6

    Hurkyl

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    Also, using the function f(x)=2x, one can prove the number of rationals in [0, 1] is the same as the number of rationals in [0, 2]
     
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