Thread: 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!

But is not unique. and both satisfy the equation .

Originally Posted by Prove It

But is not unique. and both satisfy the equation .

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

Tonio

Originally Posted by jstarks44444

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, since both numbers are positive and we're done.

Tonio

Thank you! This should do the trick.