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MATH 1100
SECTION 3.4 Notes
Complex Numbers – Text Pages 164-169
Complex Numbers:
A complex number is a number written in the form
a  bi
where a, b are real numbers, and i 2  1 (i.e, i   1 ).
a is called the real part of the number, and b is called the
imaginary part of the number.
Examples: 2  3i ,
1 3
 i , 9i ,  11
2 4
Adding, Subtracting and Multiplying Complex Numbers:
a  bi   c  di   a  c   b  d i
a  bi   c  di   a  c   b  d i
a  bi c  di   ac  bd   ad  bc i
Example 1: Express the following in the form a  bi , by simplifying.
(a)
7  6i    3  7i 
(b)
 4  i   2  5i 
(c)
2
 1

  12i   24i 
3
 6

Example 2:
Simplify.
(a)
i 24
(b)
i 103
(Remember: i 2  1 )
Dividing Complex Numbers:
a  bi  a  bi  c  di  ac  bd   bc  ad i



c  di  c  di  c  di 
c2  d 2
Example 3: Simplify.
(a)
(b)
5i
3  4i
1  2i 3  i 
2i
Square Root of a Negative Number:
If –r is negative, then the principle square root of –r is,
r i r
The two square roots of –r are, i r and  i r .
Example 4: Simplify.
(a)
(b)

9
4
1 1
1 1
Example 5: Find all solutions of the equation and express them in the
form a  bi , by simplifying.
x 2  2x  2  0
(a)
(b)
z4
12
0
z