Complex numbers

z = a + bi

where a, b are real numbers    (doubles)

re(z) = a    (reeal part of z)
im(a) = b    (imaginary part)

To visualize, associate a + bi with (a,b)

addition:
    a = 5 + 2i 
    b = 6 - 3i

    a + b = 5 + 2i + 6 - 3i = 11 - 5i

Combine like terms, and i terms are unlike non-i terms.

subtraction:

    a - b = 5 + 2i - (6 - 3i)
          = -1 + 5i


subtract and combine like terms. Mind the parens!!!!

multiply
    key fact:  i^2 = -1

    a*b = (5 + 2i)*(6 - 3i)
        = 30 - 15i + 12i - 6i^2
          30 - 3i + 6
          36 - 3i

FOIL then use the fact taht i^2 = -1.

If z = a + bi is a complex number, we define the 
conjugagte of z by 

   conjugate(z) = a - bi.

Let's go fission for a reciprocal

         1           (a - bi)               a - bi
     ----------- =   ----------------  =  --------------
     a  + bi         (a - bi)(a + bi)     a^2 + b^2

                          a               b
                 =   -----------  -  i ------------
                      a^2 + b^2        a^2 + b^2

When is the RHS defined?

    provided a + bi is not 0 + 0i.

absolute value of a real number is the distance of a real
number to the origin.

Same for complex numbers.

 |a + bi| = sqrt(a*a + b*b)

theorem:
Every complex polynomial can be factored into linear factors

a = 7 + 24i
What is |a|

z = 3 + 4i
w = 5 +12i

z + w = 8 + 16i
z - w = -2 - 8i
z*w   = 33 + 56i
           63   16i
z/w   =    -- - ---
           13   13

|z| = 5
|w| = 13