Polar Number Multiplication and Division

Date: 02/13/99 at 13:53:42
From: Matt Malensek
Subject: Polar Number Multiplication and Division

I understand that when two polar numbers are multiplied, the real parts 
multiply and the angles add. I also understand that when two polar 
numbers are divided, the real parts divide and the angles subtract. 
However, I do not understand how these rules work. Why is it that when 
I multiply or divide polar numbers, the angles add or subtract? I tried 
to convert the polar numbers into complex numbers and then FOIL (First 
Outer Inner Last) them out. Then I took the complex answer and 
converted it back to polar. This seemed to show me that two arctangents 
produce a third arctangent.

For multiplication of polar numbers:

  arctan(imag1/real1) + arctan(imag2/real2) = yet another arctangent

For division of polar numbers:

  arctan(imag1/real1) - arctan(imag2/real2) = yet another arctangent

This did not seem to prove anything. Is there a proof that will show 
me the validity of these two polar rules? Do I have the right idea 
with the arctangent approach? I think I am missing something.

Thank you for your interest and help.

Date: 02/16/99 at 20:03:21
From: Doctor Schwa
Subject: Re: Polar Number Multiplication and Division

On the contrary, this proved exactly what you wanted, since  

  arctan(imag1/real1) = angle1
  arctan(imag2/real2) = angle2

yet another, 

  arctan = angle3

then you have proved that, for multiplication,

  angle1 + angle2 = angle3

which is just as it should be.

Another way to do the proof is to write the polar numbers like this
r1 (cos theta1 + i sin theta1) * r2 (cos theta2 + i sin theta2)

Then after foiling, you can use some trigonometric identities to show
that the answer is 

  r1*r2 (cos (theta1 + theta2) + i sin (theta1 + theta2)),

which proves what you needed.

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