



So things that look like "a + bi" are called complex numbers.




Could we graph
complex numbers? 





Sure! 





What we do is
call the horizontal axis (where X usually lives) the real axis, 


and call the
vertical axis (where Y usually lives) the imaginary axis. 





So the whole
thing looks like this: 








Let's graph 3
 4i: 








Wonderful, big deal, so what. 


What can we DO with this? 





We could find the distance from the origin (0,0), to our imaginary number. 





Remember the distance formula? 


It treats the
distances along the axis to the points as the sides of a right
triangle 


and the distance to our point as the hypotenuse. 


Then uses Pythagoras to find the distance. 





So in this case: 








This distance gets a special name. 


It is called the "modulus" of the complex number. 





Why is it called that? 


WHO KNOWS? 





If you really don't like that name, there is another name we can use. 





It is also called the absolute value of the complex number. 





So we can say: 


IF Z = 3 
4i then Z = 5 





While we've got Z here, 


remember when we were talking about complex conjugates in the last
chapter? 


We said that stuff like: 





3  4i
and 3 + 4i 





were complex
conjugates of each other. 





We have
notation that says "complex conjugate of." 


It's just a
line over the letter. 





So if Z = 3 
4i then Z = 3 +
4i 





Continuing to
dig down into the barrel of tricks we can do with complex numbers, 


we come up with our next trick. 





We can convert them to polar notation! 





Here's how it works. 


First, let our old pals a and b
stand for any number we might have, 


and graph them: 








Draw a line from the origin
to the complex number point. 


Label the
angle from the positive real axis to the line q
(greek letter
theta). 











We can make a right triangle
out of this. 


The line from
the origin to the point (a + bi) is the hypotenuse of the triangle, 


but we will call it
"r" for now.












So breaking
out our old sine and cosine definitions, we can say: 





sin
q 
= 
b 
so 
b 
= 
r
sin q 

r 







cos
q 
= 
a 
so 
a 
= 
r
cos q 

r 






Substituting
these into a + bi we get: 





a + bi = r
cos q
+ r sin q
i 





Factor out the "r"
and reposition the i a bit: 





a + bi = r (cos q
+ i sin q) 





This notation is actually the polar form. 


The cos + i
sin part is often shortened to CiS, and completed as CiSq 


This seems
like it could be confusing to me, 


but math types love it so you will
likely see it. 


But now you
know it's just one of those math buzz word short hand things 


to make
other people think that math types are o so smart. 





Next question. What is q
? 





Well, we can get at it using
the tangent function... 





tan
q
= 
a 
so 
q
= 
tan^{1} 
a 


b 
b 






Here's an example so you can see how all this works. 





Change 2 + 4i to polar
notation. 














a
= r cos q 
b
= r sin q 

q
= tan^{1} 
4 
=
63.4° 

2 



So: 





copyright 2008 Bruce Kirkpatrick

