Principal Square Root Positive

Date: 10/24/2002 at 14:40:31
From: Arthur
Subject: Principal Square Roots

I was just wondering why it's always taught that the principal square 
root of a number is positive. Is there any reason, (except that a 
function must get only one value of f(x) for every value of x) that 
the second, negative square root is rejected? If so, why not reject 
the positive root, accepting the negative root to be the principal 
one?

Arthur

Date: 10/24/2002 at 17:09:55
From: Doctor Peterson
Subject: Re: Principal Square Roots

Hi, Arthur.

Probably the main reason we choose the positive root is that positive 
numbers are more familiar and useful. If we use the Pythagorean 
theorem to find the length of a hypotenuse, we expect to get a 
positive number.

In addition, if the principal root were negative, we could not say

    sqrt(a) * sqrt(b) = sqrt(ab)

because, for example,

    sqrt(4) * sqrt(9) = -2 * -3 = 6

while

    sqrt(4*9) = sqrt(36) = -6

So it's a lot easier to take it the way we do.

If you have any further questions, feel free to write back.

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