Multiplying Radicals of Negative Numbers

Date: 07/12/2000 at 17:07:26
From: Amy 
Subject: i(imaginary)

Hi,

This question appeared in a math problem in school:

     i * sqrt(-98) - sqrt(98)

This is how it is meant to be solved (according to our book):

     i * sqrt(-98) - sqrt(98)
     = i * i * sqrt(98) - sqrt(98)
     = -1 * sqrt(98) - sqrt(98)
     = -2 sqrt(98)   < the answer in the book

However...I solved it another way and got 0 as an answer (not listed 
as an answer in the book):

     i = sqrt(-1)  so...
     i * sqrt(-98) - sqrt(98)
     = sqrt(-1)* sqrt(-98) - sqrt(98)
     = sqrt(98) - sqrt(98)
     = 0

Can you multiply square roots of negative numbers? You can multiply 
radicals when they're positive.  It looks as if it should work, but 
the answer in the book is the first one (more simplified but you get 
the idea).

Thanks for the help.
Amy

Date: 07/12/2000 at 22:44:20
From: Doctor Peterson
Subject: Re: i(imaginary)

Hi, Amy.

What you did is not quite legal, but there's an interesting twist: 
what they did isn't quite legal either. Before we look at why, I'll 
simplify the problem to make it obviously wrong. Watch carefully 
(nothing up my sleeve):

     i = sqrt(-1)
     i*i = sqrt(-1)*sqrt(-1)
     -1 = sqrt(-1*-1)
     -1 = sqrt(1)
     -1 = 1

Think maybe there's something wrong here?

The only questionable step is where I did just what you did, assuming 
that

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

This is valid when a and b are positive, because the square root of a 
is defined as the positive number whose square is a; since the left 
side is positive, and its square is ab, it is, in fact, the square 
root of ab.

But when we move into the complex numbers, we can no longer so easily 
define the square root as a function. There is no such thing as a 
"positive complex number." Rather, when we say i is sqrt(-1), we 
really mean i is one of two numbers whose square is -1, and there's 
no way to distinguish them. As a result, we can no longer be sure of 
the sign; we can only say that sqrt(a)*sqrt(b) is _a_ square root of 
ab, but not which one. All we can really say, in my "proof" that 
-1 = 1, is that -1 is equal to + or - 1, which of course is true.

Now look back at your problem. You were asked to find

     i * sqrt(-98) - sqrt(98)

I've just told you that we can't uniquely define the square root for 
all complex numbers; sqrt(-98) can equal either +-i sqrt(98). So in 
fact, your answer is one of the two correct answers; and the book's 
answer is not complete. Of course, you can choose to define the 
square root of a negative number to be a positive multiple of i; but 
if your book hasn't done that, it doesn't have a right to ask this 
question.

You may enjoy this page, which you can get to from our FAQ on "false 
proofs"; it has a nice explanation of the fallacy I've described:

  1=2: A Proof using Complex Numbers - University of Toronto
  http://www.math.toronto.edu/mathnet/falseProofs/second1eq2.html   

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