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What is i?

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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
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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
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
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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

Sheila Waxman

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