In mathematics, a complex variable opens a doorway

Complex Numbers
10.7
1. Write imaginary numbers using i.
2. Perform arithmetic operations with complex numbers.
3. Raise i to powers.

Imaginary unit:
Imaginary number: A number that can be expressed
in the form bi, where b is a real number and i is the
imaginary unit.
i
3 i
9
 i
5
2
1


i
1
2


i

16

21

32

1 16
   4
i
  4i

1 21
   21
i

1 32
   16 2
i
  4 2
i

1
1
2




i
i

Complex number: A number that can be expressed in
the form a + bi, where a and b are real numbers and i
is the imaginary unit.
i
3
4  i
5
7  i
5
4
3
2

Examples:

Complex Numbers: a + bi
b = 0: Real numbers
a = 0: Imaginary numbers
real imaginary

Add Complex Numbers
   
i
i 5
4
3
5 


1
1
2




i
i
i
i 5
4
3
5 


Add the real parts – add the imaginary parts
i
8
9 

Subtract Complex Numbers
   
i
i 2
1
3
8 



1
1
2




i
i
i
i 2
1
3
8 


i

9

9
Copyright © 2011 Pearson Education, Inc.
Simplify. (4 + 7i) – (2 + i)
a) 2 + 7i2
b) 2 + 8i
c) 6 + 6i
d) 6 + 8i

10
Simplify. (4 + 7i) – (2 + i)
a) 2 + 7i2
b) 2 + 8i
c) 6 + 6i
d) 6 + 8i

Multiply Complex Numbers
  
i
i 7
4 
1
1
2




i
i
2
28i

 
1
28 

28

 
i
i 8
5
7 

1
1
2




i
i
2
56
35 i
i 

 
1
56
35 

 i
56
35 
 i
i
35
56 

standard a + bi form

  
i
i 
 4
2
5
1
1
2




i
i
2
2
8
5
20 i
i
i 


2
2
3
20 i
i 

 
1
2
3
20 

 i
2
3
20 
 i
i
3
22 

 2
3
5 i

1
1
2




i
i
2
9
15
15
25 i
i
i 


 
1
9
30
25 

 i
9
30
25 
 i
i
30
16 
Rewrite & Foil
  
i
i 3
5
3
5 


15
Multiply. (4 + 7i)(2 + i)
a) 15 + 10i
b) 1 + 10i
c) 15 + 18i
d) 15 + 18i

16
Multiply. (4 + 7i)(2 + i)
a) 15 + 10i
b) 1 + 10i
c) 15 + 18i
d) 15 + 18i

Divide Complex Numbers
i
7
6
1
1
2




i
i
i
i
i

7
6
2
7
6
i
i

 
1
7
6


i
7
6


i
7
6i


7
6i



Binomial denominator conjugate
Divide Complex Numbers
i

6
5
1
1
2




i
i
 
 
 
i
i
i 


 6
6
6
5
2
36
5
30
i
i



 
1
36
5
30




i
i
37
5
37
30

standard a + bi form
37
5
30 i



19
Write in standard form.
a)
b)
c)
d)
4
2 3
i
i


5 14
13 13
i

5 14
13 13
i

11 14
13 13
i

11 14
13 13
i


20
Write in standard form.
a)
b)
c)
d)
4
2 3
i
i


5 14
13 13
i

5 14
13 13
i

11 14
13 13
i

11 14
13 13
i


Powers of i:
1
1
2




i
i
i
   1
1
1
2
2
4





 i
i
i
  i
i
i
i
i 




 1
2
3
1
2


i
 
  
  
   1
1
1
1
1
1
1
1
4
4
8
3
4
7
2
4
6
4
5




















i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i


41
i   i
i 
10
4
i
i 

 1

15
i   3
3
4
i
i  i
i 



 1
i
   1
1
1
2
2
4





 i
i
i
  i
i
i
i
i 




 1
2
3
1
2


i
Powers of i:

4
3
2
i
i
i
i 


Simplify:

In mathematics, a complex variable opens a doorway

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