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Irrationality of Difference of two numbers

  1. Nov 16, 2007 #1
    Proof of Irrationality

    How can I prove that the square root of 8 minus the square root of 3 is an irrational number using the fact that the square root of 3 is an irrational number? I know I need to use a proof by contradiction, but I am stuck after that.
  2. jcsd
  3. Nov 16, 2007 #2
    1. The problem statement, all variables and given/known data
    [tex]\sqrt{8}[/tex]-[tex]\sqrt{3}[/tex] is an irrational number.
    Use the fact that [tex]\sqrt{3}[/tex] is an irrational number to prove the following theorem.

    2. Relevant equations
    A rational number can be written in the form [tex]\frac{p}{q}[/tex] where p and q are integers in lowest terms.

    3. The attempt at a solution
    I know that I need to use a proof by contradiction to solve this problem. Therefore, we would assume that [tex]\sqrt{8}[/tex]-[tex]\sqrt{3}[/tex] is an rational number and that [tex]\sqrt{3}[/tex] is an irrational number and try and reason to a contradiction. I am stuck and don't know how to get to the contradiction.
  4. Nov 16, 2007 #3
    well it's actually asking to prove that
    is irrational.
    assume it equals p/q where (p,q)=1
    then multiply by sqrt(8)-sqrt(3)
    youll get that sqrt(8)-sqrt(3)=5q/p
    and you also have sqrt(8)+sqrt(3)=p/q
    so you get that 4sqrt(2)=5q/p+p/q
    which yields: sqrt(2)=(5q/p+p/q)/4 which is a contradiction.
    you could have easily have done it with sqrt(3) instead but it doesnt matter.
    Last edited: Nov 16, 2007
  5. Nov 16, 2007 #4
    well, i had mistaken sqrt(8)-sqrt(3) with sqrt(8)+sqrt(3) but it doenst matter the same idea will work as well as in this case.
  6. Nov 16, 2007 #5

    matt grime

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    Homework Helper

    Suppose that it is, then this implies that the square root of 8 is what?
  7. Nov 16, 2007 #6


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    Just to add something to review my number theory.Another idea:

    I think a nice general result is that for x a pos. integer, sqr(x) is rational iff x is a perfect square (an integer, of course). Think x=a^2/b^2 , so a^2x=b^2 . Then think of what the factorization of x needs to satisfy in order for a^2x to be a perfect square.

    x=p_1^e_1...p_ne^n .

    Then , re your problem, think of what happens when you square your expression.
  8. Nov 17, 2007 #7


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    I believe this is the third time you have posted this same question in a separate thread!
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