Ah!!! I misunderstood the nature of our disagreement, but I think I have it now:

"You have an overarching square root and that could be either positive or negative."

Usually in math square root means positive square root. That's why you have to specify /- in equations

k

References:
This one talks about sqrt meaning only positive:
http://www.mathsrevision.net/gcse/pages.php?page=34
http://math.arizona.edu/~diffeq/2004-4/notes/notes01_Lectures 01-02.pdf

If you look here, you see the definition of length or norm meaning only positive:
http://www.maths.sussex.ac.uk/Staff/JWPH/TEACH/ALG04/root.pdf

This stuff is not derivable. They're just conventions. However, this doesn't mean your wrong. I've discovered a few other cases where conventions have changed over the years - for example, the definition of natural numbers.

Thanks, AS,
k