Definition of Negative Square Roots

Date: 03/08/2004 at 14:26:46
From: Linda
Subject: Square Roots

I know that the square root of 49 = 7 since 7 x 7 = 49.  But the
negative square root of 49 is -7.  Is this because (-7) x (-7) also
equals 49 or because the square root of 49 is 7 and the negative stays
because it is not involved with the operation?

The teacher wrote -SQRT(49) = -7 because (-7) x (-7) = 49.

Date: 03/08/2004 at 16:28:48
From: Doctor Peterson
Subject: Re: Square Roots

Hi, Linda.

Both statements in English are valid; but what your teacher wrote in 
symbols is not quite right.  When we say "the negative square root of
49," we may mean either of two things. 

The first option is "of the two square roots of 49, the one that is
negative," which says that the negative number -7 is a square root of
49, and therefore is the negative square root of 49:

  (-7)*(-7) = 49 


The second option is "the negative of the (principal) square root of
49," which is what we are saying when we write
     __
  -\/49  or  -sqrt(49)  which is -(7) or -7

Since the radical symbol refers specifically to the principal square 
root, which is the non-negative one, the expression above means "take
the principal square root, which is 7, and negate it to get -7."  This
is not a statement about what you get when you square -7.

So when your teacher wrote (presumably using symbols)

  -sqrt(49) = -7 because (-7)*(-7) = 49

the first equation technically means only that -7 is the negative of 
the square root of 49, and not that -7 itself, squared, is 49.  But 
the two facts are so closely related that it's hard to really say the 
teacher is wrong!  The expression "-sqrt(X)," while it properly means 
"the negative OF THE square root," does always refer to the same 
number as "the negative square root," so it becomes natural to think 
of it as the latter.  In that sense, the expression becomes sort of a 
mathematical idiom, as if "-sqrt" meant "the negative root."

But it can be dangerous to teach it that way.  Sometimes students see 
the quadratic formula and think that the "+/-" before the radical just 
means to take either sign for the root, and think that is multiplied 
by -b.  In fact, the "+/-" means to add or subtract the (principal) 
root from -b.  If we make it clear that the radical represents the 
positive root, and anything outside of that is just an operation on 
that number, such misunderstandings should not arise.

Here is probably the best reason not to say it as your teacher did: 
If I use "cbrt" to mean the cube root, then it is true that

  -cbrt(343) = -7

but it is NOT true that

  (-7)^3 = 343

Here it is absolutely wrong to think of "-cbrt" as meaning "the 
negative cube root"; there is only one cube root, and it is positive!

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

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