Complex Numbers part12345678tttttttttttt

Daily Check
For each equation find the
discriminant and the number of
solutions.
2
1. 3 4 2 0
x x
  
2
2. 6 9 0
x x
  

Launched Object
h(t) = -16t2
+ 64t + 80
a) How many seconds until the max
height is reached?
b) What will be the max height?
c) How many seconds until the object
hits the ground?
2 sec.
144 ft.
5 sec.

Today’s Question:
How do we take the square root
of negative numbers?

-In the set of real numbers, negative numbers do
not have square roots.
-Imaginary numbers were invented so that negative
numbers would have square roots and certain
equations would have solutions.
-These numbers were devised using an imaginary
unit named i.
1
i  

-The imaginary numbers consist of all numbers bi,
where b is a real number and i is the imaginary unit,
with the property that i² = -1.
-The first four powers of i establish an important
pattern and should be memorized.
Powers of i
1 2 3 4
1 1
i i i i i i
   

Examples of how we use 1
i  
16 16 1
   
4 i
 
4i

81 81 1
   
9 i
 
9i


Examples of how we use 1
i  
45 45 1
   
3 3 5 1
    
3 5 1
  
3 5 i
 
3 5
i


2 2 2 5 5 1
      
200 200 1
   
2 5 2 1
   
10 2 i
 
10 2
i


Complex Numbers
A complex number has a real part
and imaginary part.
Standard form is:
bi
a
Real part Imaginary part
Example: 5+4i
Example: 5+4i

The Complex Plane
Imaginary Axis
Real Axis

Graphing in the complex plane
i
3
4 
.
i
5
2 

.
i
2
2 
.
i
3
4 

.

Adding and Subtracting
(add or subtract the real parts, then add or subtract
the imaginary parts)
Ex: )
3
3
(
)
2
1
( i
i 



)
3
2
(
)
3
1
( i
i 




i
5
2 

Ex: )
7
3
(
)
3
2
( i
i 


(2 3 ) ( 3 7 )
i i
    
i
4
1


Ex: )
3
2
(
)
3
(
2 i
i
i 



)
3
2
(
)
2
3
( i
i
i 





i
2
1



Absolute Value of a Complex
Number
The distance the complex number is
from the origin on the complex plane.
If you have a complex number
the absolute value can be found
using:
)
( bi
a
2
2
b
a 

Examples
1. i
5
2 

2
2
)
5
(
)
2
( 


25
4 

29

2. i
6

2
2
)
6
(
)
0
( 


36
0 

36

6

Which of these 2 complex numbers is
closest to the origin?
-2+5i

Try These!!!
1. 4 i

2 2
(4) ( 1)
  
16 1
 
17

2. 3i
2 2
(0) (3)
 
0 9
 
9

3

Which of these 2 complex numbers is
closest to the origin?
3i

Powers of i
1.) Find i23
2.) Find i2006
3.) Find i37
4.) Find i828
i


1


i

1


1.) 5
 1*5
  1 5
  5
i

1*7
  1 7
  7
i

1*99
  1 99
 
3 11
i

Simplify.
3.)
2.) 7
 
4.)
3.) 99

5.)
  
i 3 3 11
-Express these numbers in terms of i.

You try…
6.
7.
 7
  36
 160
8.
i 7
 6i
4 10
i

To multiply imaginary numbers or
an imaginary number by a real
number, it is important first to
express the imaginary numbers in
terms of i.

94i

2
2 5
i
 2 5

2
21
i

( 1) 21
  21

Multiplying
47 2
i 
2 5
i  
   
3 7
   
2 1 5
i  
2 5
i i
 
i i
3 7
9.
10.
11.

7.) 7 9
i i
 16i

8.) ( 5 6 ) (2 11 )
i i
    3
 5i

9.) (2 3 ) (4 2 )
i i
   2 3 4 2
i i
   
2 i
 
Add or Subtract
12.
13.
14.

Multiplying & Dividing
Complex Numbers
Part of 7.9 in your book

REMEMBER: i² = -1
12
 2
i 12( 1)
  12

2 2
7 i
 49( 1)
  49

Multiply
3 4
i i

 2
7i
1)
2)

You try…
3)
4)
 
7 12
i i
 2
11i

2
84i

 )
1
(
84 


84

   2
2
11 i

 )
1
(
121 

121



28
 8i
 21i
 2
6i

2
28 29 6
i i
  
28 29 6( 1)
i
   
28 29 6
i
  
22 29i
 
  
i
i 2
7
3
4 

5)
Multiply

You try…
  
i
i 10
3
2 

6)
2
10
3
20
6 i
i
i 



i
17
16 

2
10
17
6 i
i 


 
1
10
17
6 


 i
10
17
6 

 i

25
 35i
 35i
 2
49i

25 49( 1)
  
25 49
74
 

You try…
  
i
i 7
5
7
5 

7)

Conjugate
-The conjugate of a + bi is a – bi
-The conjugate of a – bi is a + bi

Find the conjugate of each
number…
3 4
 i 3 4
 i
 
4 7i  
4 7i
5i  5i
6 6
8)
9)
10)
11)
i
i
Because 0
6
as
same
the
is
0
6 


1
1
i
i


2
14 4
1
i
i
 


14 4
2
i
 
 7 2i
 
Divide…
 

5 9
1
i
i
12)

   
  
5 5 9 9
1
2
2
i i i
i i i

3 5
3 5
i
i


2
9 19
9 25
i
i
 


9 19
34
i
 

2 3
3 5


i
i
13)
You try…

  
  
6 10 9 15
9 15 15 25
2
2
i i i
i i i

Ex: Solve x2
+ 6x +10 = 0
a
ac
b
b
x
2
4
2




a =
b =
c =
2
6 6 4 1 10
2 1
    


1s
t
6 36 4 1 10
2 1
    


2nd
6 36 40
2
  

6 4
2
  

6 2
2
i
 

6 2 6 2
2 2
i i
and
   

3 3
i and i
   

Complex Numbers part12345678tttttttttttt

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