Multiplying and Dividing Complex Numbers
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Multiplication:

Multiplying two complex numbers is accomplished in a manner similar to multiplying two binomials.  You can use the FOIL process of multiplication, distributive multiplication, or your personal favorite means of multiplication. 

Distributive Multiplication:
(2 + 3i) (4 + 5i) = 2(4 + 5i) + 3i(4 + 5i)
                           = 8 + 10i + 12i + 15
i2
                           = 8 + 22i + 15(-1)
                           = 8 + 22i -15
                           = -7 + 22   Answer
 Be sure to replace
i2 with (-1) and proceed with
  the simplification.  Answer should be in a + bi
                               
form.

 

     The product of two complex numbers is a complex number.

(a+bi)(c+di) = a(c+di) + bi(c+di)
                   = ac + adi + bci + bd
i2
             
        = ac + adi + bci + bd(-1)
                   = ac + adi + bci - bd
                   = (ac - bd) + (adi + bci)
                   = (ac -bd) + (ad + bc)i  
 answer in
                                                        a+bi
form

The conjugate of a complex number a + bi is the complex number a - bi.
  For example, the conjugate of  4 + 2i  is  4 - 2i.
  (Notice that only the sign of the bi term is changed.)

The product of a complex number and its conjugate
is a real number, and is always positive.


  (a + bi)(a - bi) = a2 + abi - abi - b2i2
                            
= a2 - b2 (-1)
(the middle terms drop out)
                       = a2 + b2   Answer
      This is a real number ( no i's ) and since both
         values are squared, the answer is positive.

 

Division:

When dividing two complex numbers,
1.    write the problem in fractional form,
2.    rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator.
  (Remember that a complex number times its conjugate will give a real number.
 This process will remove the i from the denominator.)

 Example: 

Dividing using the conjugate:

  

 

        Answer

 

Find out how to use your TI-83+/84+ graphing calculator for multiplying and dividing complex numbers. It will be very helpful!
Click here.