Graphing Sums of Complex Numbers

Date: 12/06/2006 at 18:36:13
From: Michael
Subject: Graphing sums of complex numbers, why is it a parallelogram?

Why is it that when two complex numbers are graphed, then the sum of
those two complex numbers is graphed (all of this on the same graph),
and then lines are drawn to connect the parts of each graph farthest
from the origin, a parallelogram is formed?

I have a hunch that there is a proof for this, but if there is any 
other explanation that isn't a proof, I would be glad to have both 
that and the proof.

Date: 12/06/2006 at 23:01:24
From: Doctor Peterson
Subject: Re: Graphing sums of complex numbers, why is it a parallelogram?

Hi, Michael.

Addition of complex numbers is the same as addition of vectors, which
follow this parallelogram rule as well.  In both cases, it works
because addition is done by adding coordinates.

The direct way to think of addition is to represent each number by an
arrow, and to put these arrows together head to tail, one after
another.  To add (a+bi) + (c+di), you would do this:

   |               o
   |           /   |d
   |       /       |
   |    o----------+
   |   /|     c    |
   |  / |          |
   | /  |b         |b
   |/   |          |
   +----+----------+--
      a       c

Clearly the result is (a+c) + (b+d)i; we have added the x-coordinates
and added the y-coordinates.

Adding into the picture arrows parallel to these (essentially adding
the same numbers in the opposite order, ending up at the same point),
we get

   |               o
   |           /  /
   |       /     /
   |    o       /
   |   /       /
   |  /       o
   | /     /
   |/  /
   +------------------

There's your parallelogram!

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 12/07/2006 at 19:38:31
From: Michael
Subject: Thank you (Graphing sums of complex numbers, why is it a
parallelogram?)

Thank you so much for answering my question!  It is very clear now.

- Michael