History of Complex Numbers

Date: 12/12/2005 at 22:01:49
From: Audrey
Subject: Why, and when were imaginary numbers created and by whom?

I'm doing a project on imaginary numbers and I was stuck with the 
why.  If you could explain without getting to in depth about 
hypercomplex numbers etc., (I'm a sophomore) that would be great!

Date: 12/12/2005 at 23:15:07
From: Doctor Peterson
Subject: Re: Why, and when were imaginary numbers created and by whom?

Hi, Audrey.

Here is a good place to start; I found this by searching our site for
the words "imaginary number history":

  History of Imaginary Numbers
    http://mathforum.org/library/drmath/view/52584.html 

The first link given in the answer has some more details.  It mentions
Cardan and Bombelli (in the 1500's), who were the first to work with
complex numbers.  

What happened was that, in solving quadratic equations, it had
previously been found that some equations could only be solved if you
could take the square root of a negative number, which had been
recognized to be impossible.  

But then when these people tried solving cubic equations, they found
that for some equations, if they went ahead and did their work as if
you COULD take the square root of a negative number, those "imaginary"
numbers would eventually cancel one another out and you would be left
with a real solution that WORKED.  In other words, they made a trip
through a world of numbers that presumably didn't exist, and came out
at a proper destination in the real world, just as if those numbers
they had used really meant something. In fact, they couldn't get to
those solutions WITHOUT using imaginary numbers!  That gave them the
beginning of a sense that imaginary numbers were worth thinking about.  

So, over a long period of time, the rules for working with them, and
appropriate notations, were developed, even though they were thought
of as a sort of fiction that didn't feel quite valid.  Eventually more
was discovered about them that made them seem more and more "real",
until they were finally accepted as being no less "real" than the real
numbers.

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/