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This is not as simple as complex number multiplication, and it works on complex numbers in the representation a + bi. See below for a (shorter) solution to polar form complex numbers.
      a + bi
?  =  ------
      c + di
      
      a + bi   c - di
   =  ------ x ------
      c + di   c - di
   
      (a + bi)(c - di)
   =  ----------------
      (c + di)(c - di)
   
      (a + bi)(c - di)
   =  ----------------       
        c^2 - d^2i^2

      (a + bi)(c - di)
   =  ----------------      since i^2 = -1
          c^2 + d^2
(c - di is called the conjugate of c + di)

The point of all that was to get a real denominator, which we can easily divide into the numerator. As stupot says, "We multiply the numerator and denominator by the complex conjugate of the denominator in order to obtain a real denominator". The (a + bi)(c - di) term may be expanded by complex number multiplication, to yield the final answer:



a + bi      (ac + bd) + (bc - ad)i
------   =  ----------------------       
c + di             c^2 + d^2

A simpler notation can be developed as follows:

         _
 z      zw
 -  =  -----
 w     |w|^2

where
 _
 w = conjugate of w

|w| = absolute value of w = sqrt(real^2 + imaginary^2)

Polar form
ariels and jpfed have informed me of a simple solution for polar form complex numbers. In this representation, any complex number z = r*exp(θ*i), due to certain features of exp(θ*i). I will seperate z and w as follows: z = r1*exp(θ1*i) and w = r2*exp(θ2*i). Then,


z     r1*exp(θ1*i)
-  =  ------------  =  (r1/r2) * exp((θ1-θ2)i)
w     r2*exp(θ2*i)
So, in English, you "divide radii and subtract thetas".

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