Download 5.4 Complex Numbers

5.7
Complex
Numbers
Objectives:
• Identify, operate with, and graph
complex numbers.
• Find the complex roots of quadratic
equations that model real-world
situations.
Complex Numbers
• A complex number has a real part &
an imaginary part.
• Standard form is:
a  bi
Real part
Example: 5+4i
Imaginary part
Example 1:
Name the real part and the imaginary part of each
complex number.
a) 2 + 4i
Real: 2 Imaginary: 4i
b) 0 – 3i
Real: 0 Imaginary: -3i
c) 0 + 6i
Real: 0 Imaginary: 6i
d) 7 + 0i
Real: 7 Imaginary: 0i
e) -1 – 0i
Real: -1 Imaginary: 0i
Definition of Equal Complex
Numbers
Two complex numbers are equal if their
real parts are equal and their imaginary
parts are equal.
If a + bi = c + di,
then a = c and b = d
Example 2:
Find the values for x and y such that
2x + 16i = 6 + 4yi
2x = 6
x=3
16i = 4yi
y=4
Operations of Complex
Numbers
• Operating with complex numbers is
very similar to operating with
binomials.
• Treat i like a variable.
When adding or subtracting complex
numbers, combine like terms.
Example 3:
a) Simplify:
Ex: 8  3i  2  5i 
10  2i
b) Simplify
8 7i 12 11i
4 18i
c) Simplify
9 6i 12 2i 
3 8i
Multiplying complex numbers
To multiply complex numbers, you use the
same procedure as multiplying binomials.
Example 4:
a) Simplify
−𝟑𝒊 𝟔 + 𝟓𝒊
= −𝟏𝟖𝒊 − 𝟏𝟓𝒊²
= 𝟏𝟓 − 𝟏𝟖𝒊
b) Simplify
8 5i2 3i
F
O
I
L
16 24i 10i 15i
16 14i 15
31 14i
2
c) Simplify
6 2i 5 3i 
F
O
I
L
3018i  10i  6i
30 28i  6
24 28i
2
How do we handle quadratic equations
with complex roots?
Example 5: Do Now:
Solve the following quadratic:
x2 – 8x + 17 = 0
Example 5: Complex Roots
x2 – 8x + 17 = 0
standard form y = ax2 + bx + c
Quadratic Formula
x
determine a, b, and c
substitute into
quadratic formula
evaluate and
simplify
b 
b2  4 ac
2a
a = 1, b = -8, c = 17
x
(8) 
(8) 2  4(1)(17)
2(1)
64  68
8  4
x

2
2
8
2
x   i  4 i
2
2
8
Example 6:
Solve the equation
and express its roots
in the form a + bi.
x
x2
 x5
2
b 
b2  4 ac
2a
put in standard form
x2 – 2x + 10 = 0
determine a, b, and c
substitute into
quadratic formula
evaluate and
simplify
a = 1, b = -2, c = 10
x
x
(2) 
2
(2) 2  4(1)(10)
2(1)
4  40
2  36

2
2
x  1  3i
Classwork:
Operations with Complex
Numbers Worksheet
Homework
Pg 299-300
Exercises: 1-4, 7-15, 18-21
Example 7:
A manufacturing company is selling a new product and
they want to know if it would be profitable to do so.
The variable x represents the number (in hundreds) of
items manufactured and sold.
The cost is C(x) = 3x + 40
The revenue is R(x) = -x² + 15x
a) Find the break even points, where the cost equals the
revenue. Use your graphing calculator to check your
answer.
b) Should the company launch their new product?
Explain.
C(x) = 3x + 40
R(x) = -x² + 15x
a) Find the break even points, where the cost equals the
revenue. Use your graphing calculator to check your
answer.
Solution:
3x + 40 = -x² + 15x
x² - 12x + 40 = 0
Example 7:
𝑥=
12± 12²−4(1)(40)
2(1)
𝑥=
12± 144−160
2
𝑥=
𝑥=
12± −16
2
12±4𝑖
2
𝑥 = 6 + 2𝑖 𝑎𝑛𝑑 6 − 2𝑖
The solutions are not
real, so there are no
break even points.
On the graph, they do
NOT intersect.
Example 7:
b) Should the company launch their new product?
Explain.
The cost function is always above the revenue function.
Therefore, cost always exceeds revenue.
They should NOT launch their new product.
Conjugates
In order to simplify a fractional complex
number, use a conjugate.
What is a conjugate?
a + bi and a – bi are conjugates of each other.
The complex conjugate of a + bi is denoted 𝑎 + 𝑏𝑖
Example 8: Simplify
a)
8i
Ex:
1  3i
8i 1  3i

1  3i 1  3i
Use the conjugate
8i  24i
8i  24

19
10
2
4i  12
5
𝟏𝟐 𝟒
=
+ 𝒊
𝟓 𝟓
Example 8 (Continued)
You try these!
3
b)
1.
2  5i
3-i
c)
2.
2-i
𝟐 𝟓
+ 𝒊
𝟗
𝟗
𝟕 𝟏
+ 𝒊
𝟓 𝟓
The Complex plane
Real Axis
Imaginary Axis
The Complex plane
To graph a + bi on
the complex
plane, plot the
point (a, b).
Real Axis
Example, the
point (3, 4)
represents the
complex number
3 + 4i.
Imaginary Axis
Graphing in the complex plane
.
 2  5i
2  2i
4  3i
.
.
.
 4  3i
Imaginary Axis
Real Axis
Homework:
5.7 Practice and Apply