Complex  Multiplication and division





  In this page we are going to see complex multiplication and division.

Multiplication:

    We are going to use FOIL method to multiply two numbers.

    FOIL:

            F  - Firsts

            O - Outers

            I  -  Inners

            - Lasts


Complex multiplication and division - Examples

Examples:

1.     Simplify: (2+3i)(3-4i)

  Solution:           =      [2x3]+[(3i)(-4i)]+[2x(-4i)]+[(3i)x3]

                         =         6   + (-12i.i)  -8i +9i

                       

                         =        6 - 12i2-8i+9i(we know i2=-1)

                         =       6 - 12(-1) +i

                         =       6+12+i

                         =       18 +i

2.            Simplify: (2-3i)2

Solution:             =   (2-3i)(2-3i)

                         =   [2x2]+[(-3i)(-3i)]+[2(-3i)]+[2(-3i)]

                         =   4+9i2-6i-6i

As i2 = -1

                         =   4+9(-1)-12i

                          =    -5-12i

                          =    -(5+12i)

There is another easiest way to do the multiplication;

           (a+ib)(c+id) = (ac-bd)+i(ad+bc)

Let us do the the second example using the above rule;

           (2-3i)(2-3i) =  [(2x2)-(-3)x(-3)]+i[2x(-3)+2x(-3)]

                           =      [4-9]+i[-6-6]

                           =        -5  -12i

We got the same answer. So it is easy to use the above rule to multiply two complex numbers.

Conjugate:

     Conjugate of a complex  is a number having the same real part but having the negative imaginary part.

Example:

        Conjugate of 3+2i   =  3-2i

        Conjugate of 4-7i   =   4+7i

        Conjugate of -3+9i =   -3-9i

Division:

       To divide two numbers we have to use conjugate of the denominator.


Example:

        Divide 3+4i/2-3i

Solution:

      To do the division first we have to multiply the numerator and denominator by the conjugate of the denominator. 

Now let us do one more problem for division.

Example:

Divide: 3-2i/5-2i

Solution:

      To do the division we have to multiply the numerator and denominator by the conjugate of the denominator.

Using the above methods we can do complex multiplication and division. Practice the above problems well to master in multiplication and division and try to do the problems given below on your own. Solutions for the practice problems are also given. If you have any doubts please contact us, we will help you to clear the doubts. 

Problems for practice:

Simplify:

1. (2-5i)(7+4i)

jQuery UI Accordion - Default functionality
34-27i.

2. (5+2i)(2-3i)

jQuery UI Accordion - Default functionality
16-11i.

3. (5+2i)/(2-5i)

jQuery UI Accordion - Default functionality
1i.

4. (7+i)/(1+i)

jQuery UI Accordion - Default functionality
4-3i

Mandelbrot set

We will see some interesting fact about complex numbers.  The beautiful Mandelbrot set is based on complex numbers. In this set, the sequence does not approach infinity. It is related to Julia sets.

The following image of Mandelbrot set is created by sampling complex numbers. This is named after the great mathematician Benoit Mandelbrot.

Uses of complex numbers:

  •        Complex numbers are used both in pure math and real world applications.
  •        Some functions which generate fractals includes complex numbers.
  •        Complex numbers have practical applications in technical fields.
  •         Eigen values and Eigen vectors involves complex numbers.
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