# Uniqueness of Square Root proof

• Nov 15th 2010, 06:37 PM
jstarks44444
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!
• Nov 15th 2010, 07:14 PM
Prove It
But $\displaystyle x$ is not unique. $\displaystyle x = \sqrt{r}$ and $\displaystyle x = -\sqrt{r}$ both satisfy the equation $\displaystyle x^2 = r$.
• Nov 15th 2010, 07:32 PM
tonio
Quote:

Originally Posted by Prove It
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
• Nov 15th 2010, 07:34 PM
tonio
Quote:

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

Tonio
• Nov 16th 2010, 01:19 PM
jstarks44444
Thank you! This should do the trick.