Imaginary & Complex Numbers
Once upon a time…
 1  n o real so lu tio n
-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.
i
1
-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
i i
1
i  1
2
i  i
3
i 1
4
i 1
4
i  i
3
i
i  1
2
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.
Powers of i
1.) Find i23
 i
2.) Find i2006
 1
i
3.) Find i37
4.) Find i828
1
Complex Number System
Reals
Imaginary
i, 2i, -3-7i, etc.
Rationals
(fractions, decimals)
Integers
(…, -1, -2, 0, 1, 2, …)
Whole
(0, 1, 2, …)
Natural
(1, 2, …)
Irrationals
(no fractions)
pi, e
Simplify.
3.)
1.)
2.)
4.) 
35.).)
-Express these numbers in terms of i.
5 
1 * 5 
7  
99 
1 5  i
5
1 * 7   1 7  i 7
 1 * 99 
 1 99
 i 3  3 11
 3i 1 1
You try…
6.
7  i
7. 
8.
36
7
 6i
 160  4i 10
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.
Multiplying
9.
47 i  2  9 4i
10.
2i 
11.

 5  2i 
3 
 1  5  2i  i 5
2
 2i
5  2 5
7  i 3 i 7  i
  (  1) 21 
2
21
21
Complex Numbers
a + bi
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.
Add or Subtract
7.) 7 i  9 i  1 6 i
12.
8.) (  5  6 i )  (2  11i )   3  5 i
13.
14.
9.) (2  3 i )  (4  2 i )  2  3 i  4  2 i
 2  i
Multiplying & Dividing
Complex Numbers
Part of 7.9 in your book
REMEMBER: i² = -1
Multiply
1)
2)
3i  4 i  12 i  12(  1)   12
2
7 i 
2
 7 i
2
2
 49(  1)   4 9
You try…
3)
 7 i  12 i   84 i   84 (  1)
2
 84
4)
  11i     11  i   121 (  1)
2
2
2
  121
Multiply
5)
 4  3 i 7  2 i 
 2 8  8i  2 1i  6i
 28  29 i  6 i
2
 28  29 i  6(  1)
 28  29i  6
 2 2  2 9i
2
You try…
6)
 2  i 3  10 i 
 6  20 i  3 i  10 i
 6  17 i  10 i
 6  17 i  10   1 
 6  17 i  10
2
 16  17 i
2
You try…
7)
5  7 i 5  7 i 
 2 5  3 5i  3 5i  4 9i
 25  49(  1)
 25  49
 74
2
Conjugate
-The conjugate of a + bi is a – bi
-The conjugate of a – bi is a + bi
Find the conjugate of each
number…
8)
3  4i
9)
 4  7i
10)
11)
5i
3  4i
 4  7i
5i
6
6
Because 6  0 i is the same as 6  0 i
Divide…
12)
 5  9i 1  i
1 i
1 i

 14  4 i
1 i
2


 5  5i  9 i  9 i
1 i  i  i
 14  4 i
2
2
2
  7  2i
You try…
13)
2  3i 3  5 i
3  5i 3  5 i

 9  19 i
9  25 i
2


6  10i  9 i  15i
2
9  15i  15i  25i
 9  19 i
34
2