Imaginary and Complex Numbers
      Addition and subtraction of 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
         Exponentiation and root extraction of complex numbers examples
Addition and subtraction of complex numbers
To add or subtract two complex numbers z1 = a + bi and z2 = c + di, we add or subtract the real parts and the imaginary parts.
Addition:       z1 + z2 = (a + bi) + (c + di) = (a + c) + (b + d)i  
Subtraction:       z1 - z2 = (a + bi) - (c + di) = (a - c) + (b - d)i  
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 + 3)i = 1 + 5i
  and             z1 - z2 = (4 + 3i) - (1 + 5i) = (4 - 1) + (3 - 5)i = 3 - 2i
Given addition and subtraction are shown in the complex plane in the figures below.
z1 + z2 = (-3 + 2i) + (4 + 3i) = 1 + 5i   z1 - z2 = (4 + 3i) - (1 + 5i) = 3 - 2i
   
Multiplication and division of complex numbers
Multiplication:   z1z2 = (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 (cosj + isinj),
 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 + isinj,  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   
Exponentiation and root extraction of complex numbers examples
Example: Compute    
Solution:
or in the polar form,  
and  
since exponentiation with integer exponent  
then  
Example: Compute    
Solution:  As square root of a complex number is a complex number, then
and, two complex numbers are equal if their real parts are equal and their imaginary parts are equal, that is
Intermediate algebra contents
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