Principal or Positive Square Roots

Date: 08/31/2006 at 20:57:54
From: Heidi
Subject: Why the square roots of numbers are positive?

Why are the square roots of numbers assumed to be positive even though
negative numbers also work, such as when my teacher said that the
square root of 16 can only be classified in the positive numbers' domain?

I find it very illogical to discount the negative answer simply
because it doesn't comply with the theorem sqrt(a)*sqrt(b) = sqrt(ab).

I read one of your answers but it didn't answer my specific question.
 I am confused as to why both numbers can't be used for the square root.

Date: 08/31/2006 at 21:45:12
From: Doctor Rick
Subject: Re: Why the square roots of numbers are positive?

Hi, Heidi.

Sometimes we *do* use both the positive and negative square roots; 
when we do so, we make it explicit by using the "plus-or-minus" sign 
(plus above minus, which I'll have to write as +-), as in the 
quadratic formula

  x = (-b +- sqrt(b^2 - 4ac))/(2a)

Other times it is clearly the positive square root that we want: for 
instance when we use the Pythagorean theorem to find the hypotenuse 
of a right triangle,

  c = sqrt(a^2 + b^2)

Lengths are always positive, so it's the positive square root that 
we need.

We want an expression to represent a single number (when values have 
been assigned to any variables in the expression).  Therefore we want 
an operation on numbers to return a single number.  For instance, if 
2+3 had two different values at the same time, it would get very 
confusing--especially in expressions with more than one addition: 
2+3+4 could have FOUR different values!  I don't know about you, but 
I'm very glad math isn't that complicated!

Therefore the operation "square root" should have a single value: 
the square root of 9 can't be *both* 3 and -3, or we'd have the same 
confusions I just mentioned.  As I said above, when we do want to 
talk about both square roots, we have a way to do so: the plus-or-
minus sign, which is a shorthand for two different expressions: one 
with the plus sign and one with the minus sign.  (In the same way, I 
can write 2 +- 3 +- 4.  This stands for four different expressions: 
2+3+4, 2+3-4, 2-3+4, and 2-3-4, which equal 9, 1, 3, and -5 
respectively.)

Elsewhere we have written about the "square root function".  Is one 
of these the one you read?

  Square Root Function
    http://mathforum.org/library/drmath/view/52645.html 

  Square Root of 100
    http://mathforum.org/library/drmath/view/52650.html 

I'm giving a slightly different perspective in hopes that it will 
get at your specific question.  Did it?

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

Date: 09/02/2006 at 10:48:16
From: Heidi
Subject: Thank you (Why the square roots of numbers are positive?)

Thank you for answering my question, I understand a lot better now
that you have explained it to me.  Thanks again!

Heidi