Graphing Complex and Imaginary Numbers

Date: 10/23/97 at 09:13:39
From: Liberal high School
Subject: Algebra II

How do you graph imaginary numbers?

Date: 11/23/97 at 18:10:42
From: Doctor Mark
Subject: Re: Algebra II

Helloooooo, Liberal!

As you probably know, imaginary numbers are just a particular case of
complex numbers.  A complex number is written as a + ib, where i is 
the imaginary unit (the square root of - 1), and a and b are real 
numbers. An imaginary number is one where the real part (the "a") is 
zero, so an imaginary number is written as just ib, where b is a real 
number. So I'm assuming that you really want to know how to graph 
complex numbers, not just imaginary ones. (Once you know how to graph 
complex numbers, you automatically know how to graph imaginary 
numbers: how's that for a good deal?)  This is done using the *complex 
plane*.  Here's how it goes:

Since any complex number z = a + ib has two real numbers associated 
with it, we can associate with each complex number z = a + ib an 
ordered pair of real numbers (a,b). Why is it an ordered pair? Because 
the order is important: the ordered pair (2,3), for instance, 
corresponds to the complex number 2 + 3i, and the ordered pair (3,2) 
corresponds to the complex number 3 + 2i, which is different from 
2 + 3i.

Does that remind you of something?  I hope it reminds you of the 
regular old xy plane, where you have two coordinate axes (the x and 
y axes), oriented perpendicular to one another, and a point in the 
plane is associated with an ordered pair of numbers (a,b), where the 
number a gives the x coordinate of the point, and the number y gives 
the y coordinate of the point.

Now notice that if we have a complex number z = a + ib, we can 
associate with it the ordered pair of numbers (a,b), and that is 
associated with the point whose x coordinate is a, and whose y 
coordinate is b. This allows us to associate with every complex number 
a point in the xy plane. Similarly, every point in the xy plane is 
associated with an ordered pair of numbers (a,b), and we can associate 
with this ordered pair the complex number z = a + ib.

Thus, every complex number is associated with a point in the xy plane, 
and every point in the xy plane is associated with a complex number.  
This allows us to represent every complex number as a point in the 
xy plane, and vice-versa.  When we do this, we call the xy plane the 
*complex plane*.

For instance, the complex number 2 - 3i (= 2 + (-3)i) can be 
represented as the point (2,-3). The complex number 4i = 0 + 4i can be 
represented as the point (0,4). The *real* number 2 = 2 + 0i (since it 
is still true that 0 times any number (real *or* complex) is 0, the 
number 0i is just equal to 0) can be represented by the point (2,0).  
Do you see that real numbers (numbers of the form a + 0i) correspond 
to points on the "x" axis, and that purely imaginary numbers (numbers 
of the form 0 + bi, where b is real) correspond to points on the "y" 
axis?  

As a result, when we talk about the complex plane, we rename the x and 
y axes: the x axis is now called the "real axis" (since that is where 
all the points correspond to real numbers), and the y axis is now 
called the "imaginary axis" (since that is where all the points
correspond to purely imaginary numbers).

The complex plane is an interesting place, and you can find out more 
about it in any precalculus book. For instance, if you note that any 
point in the complex plane can be represented by the distance of the 
point from the origin (the number 0 + 0i) and an angle that the line 
between the origin and the point makes with the x (Real) axis, you can 
use complex numbers to do trigonometry, and to find easy ways of 
understanding various trigonometric identities.  The complex plane can 
also be used to do some calculations in calculus (though you wouldn't 
get this until about 5 courses after AP calculus!).  It can also be 
used to understand how to find the solutions to cubic and quartic 
equations (which you never learn about in high school - or even in 
college - because the formulas which are the analogues of the 
quadratic formula for solving a quadratic equation are pretty 
complicated), and how to find all seven of the seventh roots of, 
say, 6.

Hope this has been of help.  Write back if there's anything I said 
that was confusing.

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