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Homework Help: Proving equivalence

  1. Jan 28, 2006 #1
    Well I have to prove the following statements are equivalent:
    a. y is a rational number
    b. y/3 is a rational number
    c. 2y+5 is a rational number
    So a -> b -> c -> a

    I'm not quite sure how you are suppose to prove something is rational however.
    I started like this:

    * y = q/r Where q & r are integers, r is not 0, no common factors other than 1 [Therefore rational]

    * y/3
    q/r/3 ... (q/r)(1/3)= q/3r ... Which is rational because everything is integer math?

    * 2y+5 ... 2(q/r) + (5/1) ... same reason

    This doesn't seem to work like the ones I've done before with proving something is even (2n) or odd (2n+1). Is there some step I'm not getting?
  2. jcsd
  3. Jan 28, 2006 #2


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    I don't see why they can't have common factors (other than 1), 8/6 is a rational number too. I think it's sufficient that it can be written as p/q where q is non-zero and p and q are integers. It should be too hard then, it actually seems trivial.

    If y = p/q (with p and q as above), then y/3 is p/(3q) and 3q is still an integer since integer*integer is another integer, hence y/3 is rational too. Etcetera, I assume.
  4. Jan 28, 2006 #3


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    For 2y+5 you should condense it into a single fraction with nonzero integer numerator and denominator.
  5. Jan 28, 2006 #4


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    Why can't a fraction with a numerator of 0 be a rational number. As long as the denominator is not 0, we have y = 0 / r = 0 (a rational number).
    Yes, I agree with the nonzero integer denominator part, but not numerator!
    Am I missing something? :confused:
  6. Jan 28, 2006 #5


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    Oh yeah, right. I meant denominator. I kind of added the nonzero part at the end without thinking.
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