What is i?

Date: 12/15/95 at 15:45:36
From: David Waxman
Subject: What is i

Dear Doctor Math,

My name is Ezra.  I am in second grade and I heard about i being an
imaginary number.  Can you please tell me what i really means?

P.S. What is the square root of i?

Ezra

Date: 12/18/95 at 11:57:41
From: Doctor Ken
Subject: Re: What is i

Hello!

Wow, we're really impressed that you're looking at imaginary numbers in 
second grade!  They're really cool; I hope you keep wondering about 
them.

Here's how they work.  You know how a normal square root works, right?  
Like the square root of 9 is 3, and the square root of 2 is about 
1.41421, and so on.  Well, you can take the square root of any positive 
number or zero, but what happens when you try to take the square root of 
a negative number?  Well, what SHOULD happen is that a lot of alarms and 
buzzers go off in your head, and you say "hey, you can't take the square 
root of a negative number!  That doesn't make sense!"  Well, in a sense 
you're right.  Because when you square any positive or negative number, 
you get a positive number, so going backwards you can't take the square 
root of a negative.

But some mathematicians decided they wanted to take the square roots of 
negative numbers anyway.  They were kind of renegade mathematicians at 
the time.  What they did is they said "hey, let's make up a new number 
and call it i.  It will have the property that when you square i, you 
get -1."  

Now, with this new number, you can take the square root of negative 
numbers.  For instance, the square root of -4 is 2 times i (which most 
mathematicians just write as 2i).  The square root of -2 is 1.41421i.  
And so on.  

With regard to your other question about the square root of i, that's a 
little more complicated.  It's one thing to know what imaginaries are, 
but it's another thing to know how to add and multiply them and take 
square roots.  However, I can tell you the answer: the square root of i 
is this:

The square root of 2         The square root of 2
--------------------    +    -------------------- times i
        2                              2


Enjoy!

-Doctor Ken,  The Geometry Forum

Date: 12/19/95 at 21:49:12
From: Anonymous
Subject: Re: What is i

Dear Doctor Math,

Thank you for the VERY good letter.  I thought the letter would be a
serious letter.  But when I read your letter I was really, I mean REALLY 
happy.  When my dad [first] kicked my sister of the phone, I
wondered why he needed the phone.  I asked him, then he told me that he
needed to communicate with his school to check your e.mail.

from Ezra Waxman

P.S.  I am going to learn more about what you said.

Sheila Waxman