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Math Help - Uniqueness of Square Root proof

  1. #1
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    Uniqueness of Square Root proof

    Given any r which is an element of the positive rational numbers, the number root(r) is unique in the sense that, if x is a positive real number such that x^2 = r, then x = root(r).

    I know that in order to prove uniqueness, one needs to show that if x^2 = r = y^2 where x and y are elements from the positive rationals, then x = y. But I am not sure how to go about doing this. Any help would be appreciated!
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  2. #2
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    But \displaystyle x is not unique. \displaystyle x = \sqrt{r} and \displaystyle x = -\sqrt{r} both satisfy the equation \displaystyle x^2 = r.
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    Quote Originally Posted by Prove It View Post
    But \displaystyle x is not unique. \displaystyle x = \sqrt{r} and \displaystyle x = -\sqrt{r} both satisfy the equation \displaystyle x^2 = r.

    But he's talking only about positive numbers....

    Tonio
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    Quote Originally Posted by jstarks44444 View Post
    Given any r which is an element of the positive rational numbers, the number root(r) is unique in the sense that, if x is a positive real number such that x^2 = r, then x = root(r).

    I know that in order to prove uniqueness, one needs to show that if x^2 = r = y^2 where x and y are elements from the positive rationals, then x = y. But I am not sure how to go about doing this. Any help would be appreciated!

    I think you should be working with positive real numbers and not only rationals, since otherwise the square

    root of most of them aren't even rational...

    Anyway, x^2=y^2\Longleftrightarrow (x-y)(x+y)=0\Longrightarrow x=y since both numbers are positive and we're done.

    Tonio
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    Thank you! This should do the trick.
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