Square Root of 100

Date: 04/15/99 at 12:18:12
From: Cody Cry
Subject: Why can't -10 be a solution to 100 square rooted?

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?

Thanks,
Cody Cry

Date: 04/15/99 at 14:36:53
From: Doctor Rick
Subject: Re: Why can't -10 be a solution to 100 square rooted?

Hi, Cody! You've asked a good question.

What's going on here, I think, is that we have two separate concepts 
that can easily be confused. 

One concept is the process of finding the root of a square; for 
instance, the number x such that x^2 = 100. You know that this has two 
solutions: 10 and -10 are both roots of this equation.

The other concept is the square root FUNCTION. A function takes in one
number and returns another number. The number that it returns must be 
UNIQUE since a function is by definition single-valued. You can't put 
in 100 and get out both 10 and -10.

The square root function is therefore DEFINED so that sqrt(y) returns 
the NON-NEGATIVE root of x^2 = y. Then we say that the two roots of 
x^2 = 100 are +-sqrt(100) - that is, plus or minus the [non-negative] 
square root of 100.

Admittedly this is a rather arbitrary definition. But if functions 
were not single-valued, we'd have a mess. And when you use the square 
root function in real situations, it will make sense. For instance, 
the distance between the points (0, 0) and (x, y) is sqrt(x^2 + y^2), 
and it makes sense that the distance is a non-negative number.

The answer I just gave is already in our Dr. Math Archives. Here is
something else from our Archives that might add a bit to my reasoning.

  http://mathforum.org/dr.math/problems/ken.8.28.96.html   

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