complex numbers addition subtraction

Mathematics
Tutor: None Selected Time limit: 1 Day

Explain how complex numbers combine algebraically and graphically (solely using the graph, meaning just graphing the result of the algebraic computation is not sufficient) under the following operations:

a.  Addition

b.  Division 

I understand the first part but the graph part of not be sufficient has stumped 

Apr 17th, 2015

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.

Apr 17th, 2015

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:

a.  Addition

b.  Division  just don't quite know how to put it into words

Apr 17th, 2015

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



Apr 17th, 2015

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Apr 17th, 2015
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