Transcript File

Complex Numbers
Math is about to get imaginary!
Exercise
Simplify the following square roots:
1)
12 
4 3  2 3
2)
32 
16  2  4 2
3)
45 
9 5  3 5
4) 1000 
100  10  10 10
 Consider the quadratic equations:
x2-1 = 0 and x2+1= 0
 Solve the equations using square roots.
 Notice something weird?
Let’s look at their graphs to see what is going
on…
f(x) = x2 - 1
How many x-intercepts does this graph
have? What are they?
f(x) = x2 + 1
How many x-intercepts does this graph
have? What are they?
Identify the number and type of
solutions for each graph.
Quadratic Formula
 Do we remember it?

b
b
4
a
c
x

2
a
2
• What does it do?
It solves quadratic equations!
Using the Discriminant
Quadratic Equations can have two, one, or no
solutions.
Discriminant: The expression under the radical in the
quadratic formula that allows you to determine how
many solutions you will have before solving it.

b
b
4
a
c
x

2
a
2
Discriminant
Why is knowing the discriminant
important?
Find the discriminant of the functions below:
y  x2  5x  4
y  x2  4x  4
y  x2  4x  7
Put the functions into your graphing calculator:
Do you notice something about the discriminant and the graph?
Properties of the Discriminant
b

4
a
c

0
2
2 Solutions
Discriminant is a positive number
b

4
a
c

0
2
1 Solutions
Discriminant is zero
b

4
a
c

0
2
Discriminant is a negative number
No Solutions
Ex. 1 Find the number of solutions of the following.
a
)3
xx

5

1
2
35
x

x

1

0
a 3
b  5
(

5
)

4
(
3
)
(

1
)
c  1
2
2
2
51
2
37  0
2
s
o
l
u
t
i
o
n
s
b
.
) x

2
x

3
2
x

2
x

3

0
2
(

2
)
4
(
1
)
(
3
)
2
a 1
b  2
c3
41
2
8  0
N
O
s
o
l
u
t
i
o
n
s
c
.
) 4
xx

4

1
2
44
x

x

1

0
2
(

4
)
4
(
4
)
(
1
)
2
1
61
6
0 0
1
s
o
l
u
t
i
o
n
a  4
b  4
c1
Now it’s your turn!
Imaginary Numbers
Simplify imaginary numbers
Ex:  4i  7i 
2
28 i 
28 1 
28
2
Remember i  1
Ex# 2:
8  5 
i 8i 5 
Remember that
2
1  i
i  40  1 2 10 
2 10
Ex# 3: i
19
18
i  i i
i i  i
18
19
 i
9
2
i   i  1  i
2 9
9
Answer: -i
Complex Numbers:
A little real, A little imaginary…
A complex number has the form a + bi, where
a and b are real numbers.
a + bi
Real part
Imaginary part
Adding/Subtracting Complex Numbers
 When adding or subtracting complex numbers,
combine like terms.
Ex: 8  3i  2  5i 
8  2  3i  5i
10  2i
Try these on your own
1. (16  4i)  (12  3i)
2. (3i  7)  (34  6i)
3. (4  7i)  (8  9i)
4. (1 i)  (3  4i)
ANSWERS:
1. 4  i
2.  3i  41
3. 4  2i
4. 2 + 3i
Multiplying Complex Numbers
 To multiply complex numbers, you use the same
procedure as multiplying polynomials.
3  12 
36 
6i
1  36
Lets do another example.
3  2i 5  3i 
F
O
I
L
15  9i 10i  6i
15  9i 10i  6
2
2
i  1
Next
Answer:
21-i
Now try these:
1.
5  20
2. 4  5i 4  5i 
3. (3  2i)
2
Next
Answers:
1. 10i
2. 41
3. 5  12i
Now it’s your turn!
Do Now
①What is an imaginary number?
①What is i7 equal to?
②Simplify:
① √-32 *√2
② (5 + 2i)(5 – 2i)
The Conjugate
 Let z = a + bi be a complex number. Then,
the conjugate of z is a – bi
 Why are conjugates so helpful? Let’s find out!
The Conjugate
What happens when we multiply conjugates
(a + bi)(a – bi)
F
O
I
L
= a2 + abi – abi –(bi)2
= a2 –
(bi)2
= a2 – b2i2 = a2 – b2(-1)
= a2 + b 2
Lets do an example:
8i
Ex:
1  3i
8i 1  3i

1  3i 1  3i
Rationalize using
the conjugate
Next
8i  24i
8i  24

19
10
2
4i  12
5
Reduce the fraction
Lets do another example
4i
Ex:
2i
4  i i 4i  i
 
2
2i
2i i
2
Next
4i  1
4i  i

2
2
2i
2
Try these problems.
3
1.
2  5i
3-i
2.
2-i
1.
2  5i
9
7i
2.
5
So why are we learning all this
complex numbers stuff anyway?
Exit Slip!
① Simplify: (-4 + 2i) (3 - 9i)
① What is the conjugate of 2 – 3i?
② What type and how many solutions does the
equations x2 + 2x + 5 =0 have?
③ What are the solution(s) to the equation in
#3?