Imaginary numbers

3. 2
2
2

8. i1
i  

9. • You can't take the square root of a negative
number, right?
• When we were young and still in Coordinate
Algebra, no numbers that, when multiplied
by themselves, gave us a negative answer.
• Squaring a negative number always gives
you a positive. (-1)² = 1. (-2)² = 4 (-3)² = 9

10. So here’s what the math people
did: They used the letter “i” to
represent the square root of (-1).
“i” stands for “imaginary.”
1
i  
So, does 1

really exist?

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

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

13. 200

10 2 i
 
10 2
i

200 1
  

14. The first four powers of i establish an
important pattern and should be
memorized.
Powers of i
1 2
1
i i i
  
3 4
1
i i i
  

15. Divide the exponent by 4
No remainder: answer is 1.
Remainder of 1: answer is i.
Remainder of 2: answer is –1.
Remainder of 3: answer is –i.
i4
1

i i
1

i2
1
 
i i
3
 

16. Powers of i
Find i23
Find i2006
Find i37
Find i828
i


1


i

1


17. Complex Number System
Reals
Rationals
(fractions, decimals)
Integers
(…, -1, -2, 0, 1, 2, …)
Whole
(0, 1, 2, …)
Natural
(1, 2, …)
Irrationals
(no fractions)
pi, e
Imaginary
i, 2i, -3-7i, etc.

18. 1.) 5
 1 5
   1 5
  5
i

1 7
    1 7
  
7
i
 
1 99
   1 99
 
3 11
i

Express these numbers in terms of i.
2.) 7
 
3.) 99

11
9
 i

19. You try…
4.
5.
7
 36
160
6.
 i 7
 6i
 4 10
i

20. 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
7.
8.
9.

21. To mult. imaginary
numbers or an
imaginary number by a
real number, it’s
important to 1st express
the imaginary numbers
in terms of i.

22. a + bi
Complex Numbers
real imaginary
The complex numbers consist of all sums
a + bi, where a and b are real numbers and i
is the imaginary unit. The real part is a, and
the imaginary part is bi.

23. 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
10.
11.
12.

24. Examples
2
)
3
(
1. i
2
2
)
3
(
i

1( 3 3)
  
)
3
(
1


3


26
10
3
Solve
2. 2



x
36
3 2


x
12
2


x
12
2


x
12
i
x 

3
2i
x 


25. Multiplying
Treat the i’s like variables, then change
any that are not to the first power
Ex: )
3
( i
i 

2
3 i
i 


)
1
(
3 


 i
i
3
1

Ex: )
2
6
)(
3
2
( i
i 


2
6
18
4
12 i
i
i 




)
1
(
6
22
12 



 i
6
22
12 


 i
i
22
6



26. 3 11
:
1 2
i
Ex
i

 
)
2
1
)(
2
1
(
)
2
1
)(
11
3
(
i
i
i
i








2
2
4
2
2
1
22
11
6
3
i
i
i
i
i
i








)
1
(
4
1
)
1
(
22
5
3







i
4
1
22
5
3





i
5
5
25 i



5
5
5
25 i



i


 5