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
L  Lasts
Examples:
1. Simplify: (2+3i)(34i)
Solution: = [2x3]+[(3i)(4i)]+[2x(4i)]+[(3i)x3]
= 6 + (12i.i) 8i +9i
= 6  12(1) +i
= 6+12+i
= 18 +i
2. Simplify: (23i)^{2}
Solution: = (23i)(23i)
= [2x2]+[(3i)(3i)]+[2(3i)]+[2(3i)]
= 4+9i^{2}6i6i
As i^{2} = 1
= 4+9(1)12i
= 512i
= (5+12i)
There is another easiest way to do the multiplication;
(a+ib)(c+id) = (acbd)+i(ad+bc)
Let us do the the second example using the above rule;
(23i)(23i) = [(2x2)(3)x(3)]+i[2x(3)+2x(3)]
= [49]+i[66]
= 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 = 32i
Conjugate of 47i = 4+7i
Conjugate of 3+9i = 39i
Division:
To divide two numbers we have to use conjugate of the denominator.
Example:
Divide 3+4i/23i
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: 32i/52i
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. (25i)(7+4i)
3427i.

2. (5+2i)(23i) 1611i.

3. (5+2i)/(25i)
1i.

4. (7+i)/(1+i)
43i

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: