The graph part is just plotting the complex number as a vector that starts at the origin and ends at the (x,y) coordinate where x = the real part, and y = the imaginary part.
For example: 3 + 2i is shown on a graph as a line drawn from the origin to the point (3,2).
Remember, the x-axis represents the REAL numbers, and the y-axis represents the IMAGINARY numbers.
When you add two complex numbers, it's the same thing. You add the REAL parts and then the IMAGINARY parts. Then plot it on the graph.
I understand that part my struggle is with this part (solely using the graph, meaning just graphing the result of the algebraic computation is not sufficient) under the following operations:
b. Division just don't quite know how to put it into words
Ok, so let's add 3 + 2i and 2 - 3i for example
You plot those two complex numbers on the graph. Then create a parallelogram with dotted lines. The resultant vector is the diagonal of the parallelogram from the origin to the vertex. It will be 5 - i, which is just the addition of the real and imaginary components of the two complex numbers. Draw it out and you'll see what I mean.
The division part is a little more complicated. You have to divide the magnitudes of each complex number and subtract their angles.
Using the same example:
magnitude of 3+2i = sqrt(3^2 + 2^2) = sqrt(13)
magnitude of 2-3i = sqrt(2^2 + (-3)^2) = sqrt(13)
The resultant magnitude is sqrt(13) / sqrt(13) = 1
The angle of 3+2i is found using:
tanx = 2/3
x = tan^-1(2/3) = 33.7 degrees
The angle of 2-3i is
x = tan^-1(-3/2) = -56.3 degrees
So, the final vector is 1 with angle of (33.7 -(-56.3) = 90 degrees
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