What is i?

Date: 9/24/95 at 14:31:58
From: Anonymous
Subject: SQR(-1) ??

Hi Dr. Math,

I want to know if it is correct to say:
   i = SQR(-1) ?

Or am I only allowed to say:
   i^2 = -1 ?

Because:
   -1 = SQR(-1) * SQR(-1) = SQR((-1)*(-1)) = SQR(+1) = 1
and -1 is not equal 1!!

I really don't know what is right, but I've read both in 
separate math  books.

Thanks for your help (hope so).

Bye TOETI

Date: 9/27/95 at 11:30:30
From: Doctor Ken
Subject: Re: SQR(-1) ??

Hello!

Actually, you're right.  You just proved that -1 = 1.

Just kidding.

Well, the problem is that i isn't defined to be the square root 
of -1.   You see, any number has two square roots.  For 
instance, 4 has the square roots 2 and -2.  For convenience, we 
almost always define the square root function to give us the 
positive square root of a real number (we pick 2 instead of 
-2).  Well, it's similar with imaginaries.   There are actually 
two square roots of -1, there's i and there's -i.  So we say "i 
is defined to be a square root of -1 and that makes -i the  
other one," not "i is _the_ square root of -1."

With that in mind, it's more correct to say that i^2 = -1, 
although people will usually know what you mean if you say i = 
Sqr{-1}, just like people will usually know what you mean if 
you say 2 = Sqr{4}, although the concept of "taking the 
positive square root" is a little weird when the answer is 
imaginary.

So the flawed step in that chain of equations you wrote is 
right at the end; it's because 1 and -1 are both square roots 
of 1.

It's similar to saying

 -2 = Sqr{4} since (-2)^2 = 4
   = 2.

This is pretty fun stuff to think about.  Hope you get 
something good out of it.

- Doctor Ken,  The Geometry Forum