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Complex Numbers
OBJECTIVES
•Use the imaginary unit i to write complex
numbers
•Add, subtract, and multiply complex numbers
•Use quadratic formula to find complex solutions
of quadratic equations
Consider the quadratic equation x2 + 1 = 0.
What is the discriminant ?
a = 1 , b = 0 , c = 1 therefore the discriminant is
02 – 4 (1)(1) = – 4
If the discriminant is negative, then the quadratic
equation has no real solution. (p. 114)
Solving for x , gives x2 = – 1
2
x  1
x  1
We make the following definition:
i  1
Note that squaring both sides yields i 2  1
Real numbers and imaginary numbers are
subsets of the set of complex numbers.
Real Numbers
Imaginary
Numbers
Complex Numbers
Definition of a Complex Number
If a and b are real numbers, the number a + bi is a
complex number written in standard form.
If b = 0, the number a + bi = a is a real number.
If b  0, the number a + bi is called an imaginary
number.
A number of the form bi, where b  0, is called a
pure imaginary number.
Write the complex number in standard form
1   8  1  i 8  1  i 4  2  1  2i 2
Equality of Complex Numbers
Two complex numbers a + bi and c + di, are equal
to each other if and only if a = c and b = d
a  bi  c  di
Find real numbers a and b such that the equation
( a + 6 ) + 2bi = 6 –5i .
a+6=6
2b = – 5
a=0
b = –5/2
Addition and Subtraction of Complex Numbers,
If a + bi and c +di are two complex numbers written
in standard form, their sum and difference are
defined as follows.
Sum:
(a  bi)  (c  di)  (a  c)  (b  d )i
Difference: (a  bi )  (c  di )  (a  c)  (b  d )i
Perform the subtraction and write the answer
in standard form.
Ex. 1 ( 3 + 2i ) – ( 6 + 13i )
3 + 2i – 6 – 13i
–3 – 11i
Ex. 2
8   18  4  3i 2 
8  i 9  2  4  3i 2 
8  3i 2  4  3i 2
4
Properties for Complex Numbers
• Associative Properties of Addition and Multiplication
• Commutative Properties of Addition and Multiplication
• Distributive Property of Multiplication
Multiplying complex numbers is similar to multiplying
polynomials and combining like terms.
#28 Perform the operation and write the result in standard
form. ( 6 – 2i )( 2 – 3i )
F O I L
12 – 18i – 4i + 6i2
12 – 22i + 6 ( -1 )
6 – 22i
Consider ( 3 + 2i )( 3 – 2i )
9 – 6i + 6i – 4i2
9 – 4( -1 )
9+4
13
This is a real number. The product of two
complex numbers can be a real number.
Complex Conjugates and Division
Complex conjugates-a pair of complex numbers of
the form a + bi and a – bi where a and b are
real numbers.
( a + bi )( a – bi )
a 2 – abi + abi – b 2 i 2
a 2 – b 2( -1 )
a2+b2
The product of a complex conjugate pair is a
positive real number.
To find the quotient of two complex numbers
multiply the numerator and denominator
by the conjugate of the denominator.
a  bi 
c  di 

a  bi  c  di 


c  di  c  di 
ac  adi  bci  bdi

2
2
c d
2
ac  bd  bc  ad i

2
2
c d
Perform the operation and write the result in
standard form.
(Try p.131 #45-54)
6  7i 
1  2i 

6  7i  1  2i 


1  2i  1  2i 
6  14  5i
6  12i  7i  14i


2
2
1 4
1 2
2
20  5i

5
20 5i


5
5
 4 i
Principle Square Root of a Negative Number,
If a is a positive number, the principle square root
of the negative number –a is defined as
a 
ai  i a .
Use the Quadratic Formula to solve the quadratic
equation.
9x2 – 6x + 37 = 0
a = 9 , b = - 6 , c = 37
What is the discriminant?
( - 6 ) 2 – 4 ( 9 )( 37 )
36 – 1332
-1296
Therefore, the equation has no real solution.
9x2 – 6x + 37 = 0
a = 9 , b = - 6 , c = 37
x
  6 
 6  4937
29
2
6  36  1332
x
18
6   1296
6
1296
x


i
18
18
18
1 36i
x 
3 18
1
  2i
3