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How to prove √X is irrational number

  1. Mar 5, 2010 #1
    when X is even number,it's easy to prove
    but how about the condition which X is odd number?
    I have no idea of this
     
  2. jcsd
  3. Mar 5, 2010 #2

    Hurkyl

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    [itex]\sqrt{4}[/itex] is irrational?
     
  4. Mar 5, 2010 #3
    no,i may add despite 1,4,9,16,25...etc
     
  5. Mar 5, 2010 #4

    Char. Limit

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    So in other words...

    [tex]\sqrt{x}[/tex] is irrational iff x=/=n^2 for n belonging to the integer set.
     
  6. Mar 5, 2010 #5
    yes, but how to prove?:confused:
     
  7. Mar 6, 2010 #6
    Fundamental theorem of arithmetic. Assume p^2/q^2=x with gcd(p,q)=1, and see what has to divide what.
     
  8. Mar 6, 2010 #7
    what does gcd mean?
     
  9. Mar 6, 2010 #8
    Greatest common divisor. If gcd(p,q)=1, it means the fraction p/q is in lowest terms.

    Look at the proof for sqrt(2), and adapt it. Remember that "even" just means "is divisible by 2", so that if you're checking a number other than 2, you won't be thinking about "even" anymore.
     
  10. Mar 6, 2010 #9
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