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History and Discovery of Imaginary Numbers

Date: 02/28/2006 at 18:41:30
From: Lisa
Subject: Who thinks up stuff such as "imaginary numbers"

How do mathematicians decide on creating imaginary numbers such as the 
square root of -1 or (i)?  It just seems weird to have a number that
is the square root of a negative one.

Is it not easier to say that we deal with only "real" numbers instead
of these "imaginary" ones?  It would make life, and math class, a lot
easier!



Date: 02/28/2006 at 22:50:07
From: Doctor Peterson
Subject: Re: Who thinks up stuff such as 

Hi, Lisa.

There are two things that drive mathematicians to invent something
new, and then to accept it as worth keeping: curiosity and usefulness.

Curiosity is what makes us wonder, "What if there were a square root
of -1?", and then go ahead and see what happens.  We're interested in
new possibilities, because much of mathematics today is really a big
game of "what if".

Usefulness is what makes us say, "I'm stuck trying to solve this, so
I'm willing to try anything!"  It's also what later leads us to say,
"Weird as it seems, this 'i' business actually works, so we'd better
keep studying it and see if we can make sense of it."

Both are involved in the history of imaginary numbers, but especially
the second, to begin with.  I don't know that anyone tried out the 
idea from mere curiosity at first; that attitude is a more recent
development.  What started it all is that, in the 1500's, people were
trying to work out ways to solve more complicated kinds of equations,
having completely tamed quadratic equations.  They wanted a formula 
for cubic equations.  In the process, they found that for some 
equations (which they knew had solutions), they ran up against a brick 
wall: they had to take the square root of a negative number.  Some of 
them decided to go ahead and pretend that there was such a number; 
they just wrote sqrt(-16) as 4 sqrt(-1) and kept going as if sqrt(-1) 
were a number.  To their surprise, they found that the weird numbers
canceled out and left them with the correct solution!  Pretending the
number existed turned out to work.

It took a couple hundred years before mathematicians got comfortable
with thinking of the so-called "imaginary numbers" as really existing;
they thought of them as a cheap trick to get to the right answer.
(Oddly enough, during the same period, and even beyond, negative
numbers were likewise treated with suspicion, as if using them were
cheating.)  As they tried out more and more things with complex
numbers, they found that they were useful in many more ways: that
complex numbers could be solutions to polynomial equations, and that
including them actually simplified that subject, rather than making it
more complicated; that they unified the subjects of trigonometry and
exponents, so that the rules of one turned into mere applications of
the rules of the other; and so on.  They are now routinely used to
describe very real things like alternating electric currents.

Also, mathematicians became more comfortable, along the way, with the
idea that they could make any definition and see what would happen;
they found that complex numbers were not only useful, but also
interesting in themselves.  Having done this with complex numbers, 
they found themselves inventing many other new kinds of objects that
behaved sort of like numbers, and had interesting properties worth
studying.  Some of these, eventually, turned out also to be useful in
solving real problems.

So as it turns out, complex numbers actually make things easier rather
than harder, in the long run; and they make math beautiful in some
suprising ways.  Yes, it takes some getting used to, and some effort 
to learn how to work with them, but once you have them, you can do 
things you couldn't have done without them, or do familiar things much 
more easily.  Mathematicians were almost forced into inventing them in 
the first place, but now consider them one of their most valuable
creations (or discoveries, if you think of them as a world we didn't
know existed, but found ourselves in by accident).

Here is a page with some of the names and dates I've left out of this
summary:

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

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


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