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A proof regarding Rational numbers

  1. Jun 25, 2013 #1
    I have found some trouble in trying to prove this question.please help mw with that.

    Q1) If (a+b)/2 is a rational number can we say that a and b are also rational numbers.? Justify your answer.


    I have tried the sum in the following way.

    Assume (a+b)/2=p/q (As it is rational)
    Lets assume a and b are also rational. Then a=m/n , b=x/y where m,n,x,y ε Z and n,y not equal to 0.

    ∴ p/q = (my+nx)/2ny = (a+b)/2

    ∴ (a+b)/2 = m/2n + x/2y
    = 1/2(m/n+x/y)

    for a and b to be rational they has to be equal to m/n and x/y..That is not the case always so we can't say if (a+b)/2 is rational a and b are also rational.

    I doubt that this proof is wrong.Please correct that if there's any wrong.
     
  2. jcsd
  3. Jun 25, 2013 #2

    lurflurf

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    We do not want to start with assume a and b are also rational as that is what we are trying to show.
    The equations do note prove anything.
    The question does not make it clear what a and b are, I assume they are from some common number system like real, complex, or algebraic numbers.

    The most straight forward thing to do is think of an example of a and b so that (a+b)/2 is rational and a and b are not.
     
  4. Jun 25, 2013 #3
    I thought about that too but could'nt figure out 2 examples for a and be..But just figured out we can use a=√2 and b=-√2 so that a+b=0 and (a+b)/2=0 which is rational. Any way thanx a lot.
     
  5. Jun 25, 2013 #4

    tiny-tim

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    hi thudda! :wink:
    yes that's fine …

    the answer to the question is "no", and you've justified it by showing a counter-example! :smile:
     
  6. Jun 26, 2013 #5
    it would seem that if a is irrational and b= (a rational number) - a , then this would be a perfect example of your equation coming out rational with a+b as the numerator. for example a = pi and b = (6-pi). add them together and you get 6 even though both numbers are irrational.
     
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