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November 15th, 2010, 06:32 PM   #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 uniquein 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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November 15th, 2010, 06:52 PM   #2
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Re: Uniqueness of Square Root proof

Try
x^2=y^2
x^2-y^2=0
(x+y)(x-y)=0
so x = y or x =-y since Q is a field.
But since they are both positives, x =-y is excluded.
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November 16th, 2010, 01:19 PM   #3
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Re: Uniqueness of Square Root proof

Thank you! This should do the trick.
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