Graphing Sums of Complex NumbersDate: 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 |
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