Negative Square Roots

Date: 01/25/2003 at 15:55:44
From: Aimee
Subject: Square roots

Dear Dr. Math,

When you take the square root of 100 you get 10.  Why can't -10 be an
answer?  If you square -10 you get 100 also, so why can't 10 and -10
be the answers?

Sincerely,
Aimee

Date: 01/25/2003 at 15:59:18
From: Doctor Ian
Subject: Re: Square roots

Dear Aimee,

You've asked a good question.  The answer depends on the context in
which you're taking the square root.

If you're looking for all the solutions to an equation like

  x^2 = 100

then you want to find _every_ value of x that satisfies the equation.
There are clearly two such values: -10 and 10.

However, suppose the equation arises in a context in which you're
trying to find the length of the hypotenuse of a right triangle:

  h^2 = 6^2 + 8^2

      = 36 + 64

      = 100

In this context, it's clear that only the positive root is meaningful.
It's not so much that

    h = -10

isn't a solution to the equation, but we don't know what to _do_ with
a solution like that.  So we ignore it.

Now, suppose you turn in an exam paper with an answer like "The length
of the hypotenuse of triangle ABC is either 10 cm or -10 cm".  What
should your teacher do with something like that?

On the one hand, you've provided both solutions to the equation that
you ended up with.  But on the other hand, the equation was just a
tool that you were using to find the length of a particular line segment.
   
Other tools would have returned only positive values.  So while -10 is
a solution to the _equation_, it's not really an answer to the _question_.

Do you see the difference?

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/