Square Root of a Negative Number

Date: 01/25/97 at 19:04:00
From: Dustin Pedlar
Subject: Square root of negative numbers

Is it possible to find the square root of a negative number and if so,
to what number system do these square roots belong?  Could you please 
explain how this number system works?

Date: 01/25/97 at 20:47:34
From: Doctor Wallace
Subject: Re: Square root of negative numbers

Hi Dustin!

As you no doubt know, a square root is a number that when multiplied 
by itself is equal to a given number.  For example, 4 is the square 
root of 16, since 4 x 4 = 16.  Note, however, that -4 x -4 = 16, too.  
We call 4 the positive square root of 16, and -4 the negative square 
root of 16.

Now, you want to know if we can find the square root of a negative 
number.  Let's take -16.  We need to find a number, call it x, such 
that:
        x times x (x^2) = -16

Now, we know that any number times itself must be positive, not 
negative.  Therefore, there is no such number x in the set of real 
numbers.

A number x is defined, however, in the set of complex numbers.  The 
complex numbers are a superset of the real numbers.  That is, the 
complex numbers form a bigger set.  The reals are a subset of the 
complex.

A complex number has the form a + bi, where a and b are real numbers 
and the i is a special number.  The "a" is called the real part; the 
"bi" is called the imaginary part.  If we let a equal 0, then we have 
an imaginary number.  The set of imaginary numbers is also a subset of 
the complex numbers.  If we let b equal 0, then we have a regular real 
number.  This is why the reals are a subset of the complex: the reals 
are just complex numbers that all have b=0, that is, no imaginary 
part.

Now, the number i is defined to be equal to the square root of -1.  
This means that i^2 (i squared) is equal to -1.  So now we can find 
the square root of -16.

Since -16 = (-1) 16, we can write:

     sqr(-16) = sqr(-1) times sqr(16)      (property of square roots)
     sqr(-16) = i times 4

This is usually written as 4i.  We can check by squaring 4i.  We get 
4 x 4 = 16 times i x i = sqr(-1) times sqr(-1) = -1, giving 16 times 
-1 or -16.

There is much more to the complex and imaginary sets of numbers than I 
can go into here.  There are entire branches of mathematics (like 
complex analysis) which deal with these numbers.  For more 
information, consult the math section of your local library or check 
out this page in our FAQ:

   http://mathforum.org/dr.math/faq/faq.imag.num.html                        

-Doctor Wallace,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/