Imaginary and complex numbers
      Multiplication and division of complex numbers
      Polar or trigonometric notation of complex numbers
         Multiplication and division of complex numbers in the polar form
Multiplication and division of complex numbers
Multiplication:   z1 z2 = (a + bi) (c + di) = ac + bci + adi + bdi2 = (ac - bd) + (ad + bc)i  
Division:        
Examples:  Given are complex numbers,  z1 = -3 + 2i  and  z2 = 4 + 3i, find  z1   z2  and  z1 / z2.
Solutions:      z1   z2 = (-3 + 2i) (4 + 3i) = -3 4 + 2 4i + (-3) 3i +  2 3 i2 = -18 - i
and    
Example:  For what real number a the real part of the complex number equals 1.  
Solution:  
 
Example:  Evaluate the expression   where  z = 1 - i.  
Solution:  
Polar or trigonometric notation of complex numbers
A point (x, y) of the complex plane that represents the complex number z can also be specified by its distance r from the origin and the angle j between the line joining the point to the origin and the positive x-axis. 
Cartesian coordinates expressed by polar coordinates:
x = r cosj
y = r sinj
plugged into  z = x + yi  give
z = r (cos j + i sin j),
 where  
 
Thus, obtained is the polar or trigonometric form of a complex number where polar coordinates are r, called the absolute value or modulus, and j, that is called the argument, written j = arg (z).
By using Euler's formula  eij = cosj + i sinj,  a complex number can also be written as
  z = r eij which is called the exponential form.
To show the equivalence between the algebraic and the trigonometric form of a complex number,
 z = r eij = r (cosj + isinj)
express the sine and the cosine functions in terms of the tangent
and substitute into above expression
   
Example:  Given the complex number  z = 1 - 3i,  express  z = x + yi  in the trigonometric form.
Solution:  The modulus  
                 the argument  
   
    the trigonometric form is  
Multiplication and division of complex numbers in the polar form
If given  z1 = r1(cosj1 + isinj1)  and  z2 = r2(cosj2 + isinj2)  then
  z1   z2 = r1 r2  [cos(j1 + j2) + isin(j1 + j2)]  
 and   
College algebra contents
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