Multiplying Square Roots of Negative Numbers

Date: 11/25/2003 at 04:28:09
From: Justin
Subject: sqrt(-2) * sqrt(-2) =2 or -2 ?

It seems to me that there are two ways to do a problem where you are 
multiplying square roots of negative numbers.  Here's an example:

Method 1: 
sqrt(-2) * sqrt(-2) = sqrt((-2)*(-2)) = sqrt(4) = 2

Method 2: where i = sqrt(-1)
sqrt(-2) * sqrt(-2) = sqrt(2)i * sqrt(2)i = 2(i^2) = -2

Which solution is correct?  What's wrong with the other approach?  Did 
I miss any mathematics concept?

Date: 11/25/2003 at 09:36:02
From: Doctor Ian
Subject: Re: sqrt(-2) * sqrt(-2) =2 or -2 ?

Hi Justin,

What you've missed is that the square root returns two values.  To
make it easier to see what's going on, let's use a perfect square.

Because we can square either (2i) or (-2i) to get -4, we have to
consider several cases:

  sqrt(-4) * sqrt(-4) = (2i) * (2i)      ->   -4

                           - OR -

                        (-2i) * (-2i)    ->   -4

                           - OR -

                         (2i) * (-2i)    ->    4

                           - OR -

                        (-2i) * (2i)     ->    4

Each of your derivations considers one of these cases separately, thus
losing the connection between them.

Here's one way to think of it:  Each person has a parent() function,
which returns two possible values.  Suppose the parents of Pat are
Chris and Jean.  Then your original question was sort of like this:

  Method 1:
  parent(Pat) + parent(Pat) = Chris + Chris

  Method 2:
  parent(Pat) + parent(Pat) = Jean + Jean

Should we then conclude that Chris is the same as Jean?  No.  Should
we conclude that there is a problem with the definition of the parent
function?  No.  Should we conclude that we have to be very careful
with multi-valued functions?  Bingo!

Does this help? 

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

Date: 11/25/2003 at 08:57:03
From: Doctor Peterson
Subject: Re: sqrt(-2) * sqrt(-2) =2 or -2 ?

Hi, Justin.

The problem is that the question is bad!  When we talk about complex 
numbers, there are two square roots; we can't define "the" square 
root, the principal root, which in the real numbers would be the 
positive one.  So the symbol "sqrt(-2)" doesn't have the meaning you 
expect; it has two values, not just one.  And the rule that sqrt(ab) = 
sqrt(a)sqrt(b) is not true any more.

Take a look at these pages for more explanations and comments:

  Multiplying Radicals of Negative Numbers
    http://mathforum.org/library/drmath/view/53873.html 

  Simplifying Complex Numbers
    http://mathforum.org/library/drmath/view/62973.html 

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

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

Date: 11/25/2003 at 09:52:38
From: Justin
Subject: Thank you (sqrt(-2) * sqrt(-2) =2 or -2 ?)

Thanks for your quick response!  It helps a lot.