Square root of a negative number

Date: Wed, 23 Nov 1994 12:58:34 -0800
From: (Paul Pragin)

Is it theoretically possible to calculate the square root of a negative 
number?

Date: Mon, 28 Nov 1994 16:51:23 -0500 (EST)
From: Dr. Sydney
Subject: Re: your mail

Hi there!  Thanks for writing to Dr. Math.  Taking the square root of a
negative number is a problem that mathematicians struggled with for a 
very long time.  It was not until the 1600's and 1700's that the great mathe-
maticians really got a firm grasp on what the square root of a negative
number is.  Why were there so many problems? From the basic axioms that
describe the real numbers we see that a positive real times a positive real
yields a positive real and a negative real times a negative real yields a
negative real.  So, in the real numbers it is impossible to square a number
and get a negative number.

So, a new field had to be introduced: the complex field.  Do you know what
complex numbers are?  The square root of -1 was given the name i (i is for
imaginary).  All numbers that are square roots of negative numbers can be
written as a real number times i.  Do you see how this works?  How would you
write the square root of -5 in terms of i?  

I hope this helps answer your question.  If you have any more
questions, please do write back.

--Sydney, a math doctor