What Are Imaginary Numbers?

Date: 7/24/96 at 16:52:29
From: Anonymous
Subject: What are Imaginary Numbers?

A fellow 6th grade teacher and I were lamenting the amount of math 
knowledge we have forgotten since our own high school and college 
educations.  Her son is currently having trouble with imaginary 
numbers.  I can't even remember what they are, their purpose, and how 
to use them.  Can you give me a brief refresher summary?  Thanks!

Date: 7/25/96 at 14:29:48
From: Doctor Pete
Subject: Re: What are Imaginary Numbers?

Hello,

Before I begin, let's define some notation.  When used in an equation, 
* is multiplication, so 4*7 = 28.  x^2 means "x squared," so ^ denotes 
exponentiation.  Sqrt[ ] is the square root of the expression in 
brackets.

Imaginary numbers were conceived in response to the question of whether
or not we could think about the square root of negative numbers, or
equivalently whether or not there existed a value that satisfied the
equation x^2 + 1 = 0.  If we decide that this equation _does_ have a
solution, then we can give that solution a name: let's call it i.  Then
it's not too hard to show that i^2 = -1, and that -i also satisfies the
equation x^2 + 1 = 0.  The reason mathematicians chose "i" as this new
number's name is that they still questioned its validity as a number,
and questioned its right to co-mingle with the real numbers.

After all, they had a lot to worry about - what physical significance 
does "i" have?  (Several, in particular, in the physics of electric 
circuits.) Does its introduction lead to logical inconsistencies? 
(No - as a matter of fact, it opens up vast fields of study from 
abstract algebra to complex analysis.)  But as time went on, people 
began to realize that "i" was a generally good idea - though the term 
"imaginary" stuck.

How does one use "i" in calculation?  Well, i follows most 
mathematical conventions, though care needs to be taken occasionally 
during multiplication and root extraction. For example, i+i = 2i, and 
(1+i)+(3-i) = 4.  i^2 = i*i = -1, from the definition; note 
Sqrt[-4]*Sqrt[-1] is not equal to sqrt[(-4)(-1)] = 2, but rather 
Sqrt[-4]*Sqrt[-1] = i*Sqrt[4]*i = -2.  This apparent contradiction 
arises simply because the rule where Sqrt[p]*Sqrt[q] = Sqrt[p*q] is 
only applicable when p and q are non-negative.

Thus we have two sets of numbers: "real" and "imaginary." They are 
mutually exclusive (except perhaps for 0, which could be considered as 
both real and imaginary, though nearly always it is thought of as 
belonging to the former). The union of these sets forms the *complex* 
numbers; these are numbers of the form a+b*i, where a and b are reals.  
As can be verified, addition and multiplication are well-defined (non-
ambiguous), there is an *additive identity* (zero), a *multiplicative 
identity* (one), there is always an *additive inverse* (i.e., for 
every complex number a+bi there exists a unique c+di such that (a+bi)+
(c+di) = 0), and a *multiplicative inverse* (i.e., for every a+bi 
there is a unique c+di such that (a+bi)(c+di) = 1).  Finally, addition 
and multiplication are associative, commutative, and distributive.

The most common purpose of the imaginary numbers is in the 
representation of roots of a polynomial equation in one variable.  For 
example, what are the roots of x^2 + 2*x + 5 ?  Using the quadratic 
formula, we find

         -2 + Sqrt[4 - 4*5]   -2 - Sqrt[4 - 4*5]
     x = ------------------ , ------------------
                 2                     2
or
     x = {-1+2*i, -1-2*i}.

An important theorem in algebra (which I stated for a particular case 
at the beginning of this letter) states that for a polynomial in one 
variable of degree n, there are exactly n roots, counting multiple 
roots.  So a cubic equation has 3 roots, a quartic 4, and so on.

There are other applications of complex numbers, some of them quite 
abstract - often they deal with the algebraic structure of these 
numbers, rather than the computational aspects.

Hope this helps....  :)


-Doctor Pete,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 7/25/96 at 19:17:28
From: Doctor Robert
Subject: Re: imaginary numbers

I think that the moniker "imaginary" is unfortunate, since in one 
sense, all numbers are imaginary in that they exist only in our minds.  
But, in mathematics, there is a distinction made between real and 
imaginary numbers. The real numbers are those that show up on the 
number line.  Imaginary numbers arise because mathematicians could not 
find a solution to the equation

   x^2 + 1 = 0 

in the set of real numbers. So, they decided to designate the square 
root of negative one by the small case letter i.  If i = the square 
root of -1, then the square root of -4 is 2i. Now, complex numbers 
are those which have both a real part and an imaginary part. An 
example would be the number 2+3i. The young man studying imaginary 
numbers is probably learning to do basic operations with complex 
numbers like addition, subtraction, multiplication, and division.  

I hope that this helps.

-Doctor Robert,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/