Calculator for finding the Area of a circle
 So things that look like "a + bi" are called complex numbers.
Related Chapters
 Could we graph complex numbers?
 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?
 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

cos q = a so a = r cos q

 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°

copyright 2008 Bruce Kirkpatrick