The Imaginary Number

Date: 5 Jan 1995 10:47:27 -0500
From: Yoshifumi Takebuchi
Subject: What is the imaginary number?

I have a very simple, but confusing-me question.

I have learned that the imaginary number "i" is a number 
that doesn't really exist, defined as square root of -1.  However, 
I wondered if the imaginary number stands for OTHER NON-EXISTING 
NUMBERS. I mean, for example, if you are asked to find a number that is 
less than 1 and greater than 3, there is no real answer.  Therefore, I thought 
the answer for this could be the imaginary number, if the "imaginary 
number" means "imaginated number." 

What do you think?

Yoshi Takebuchi 

Date: 10 Jan 1995 23:09:58 -0500
From: Dr. Ken
Subject: Re: What is the imaginary number?

Hello there!

Sadly, the symbol "i" only means the square root of negative 1.  Or perhaps
we should be thankful. 

See, one of the aims of mathematics is to say what is possible and what is
not possible.  Having a number that's less than 1 and greater than three
would imply that 1 is greater than 3.  And that's just plain old false.  So
unless we _really_ start turning math on its ear, there is no number,
however imaginary you mean, that will satisfy these conditions.

It's unfortunate that "imaginary" is the term we use for numbers involving
i, the square root of negative 1.  There's really nothing more "imaginary"
about i than there is about the square root of 2, or some other "real"
numbers (because really, all those numbers just exist in the heads of
mathematicians anyway), but we call them imaginary because most
mathematicians thought they had all the "real" numbers already, before
imaginaries were invented.

However, this is all that the word imaginary usually means.  People don't
really use it to mean "some non-existent number that has a certain property."

-Ken "Dr." Math