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Imaginary and Complex Numbers
Negative numbers do not have square roots in the real-number system. However, a larger number system that contains the real-number system is designed so that negative numbers do have square roots. That system is called the complex-number system and it will allow us to solve equations like x2 + 1 = 0. The complex-number system makes use of i, a number that is, by definition, a square root of –1.
Example
Solution
Express in terms of i:
a) 9
b) 48
b) 48 1 16 3
a) 9 1 9
1 9
3, or 3 .i i
1 16 3
4 3
4 3
i
i
Addition and Subtraction
The complex numbers obey the commutative, associative, and distributive laws. Thus we can add and subtract them as we do binomials.
Example
Solution
a) (3 2 ) (7 8 )
b) (10 2 ) (9 )
i i
i i
a) (3 2 ) (7 8 ) (3 7) (2 8 )i i i i
b) (10 2 ) (9 ) (10 9) ( 2 )i i i i
10 (2 8)
10 10
i
i
1 3i
Multiplication
To multiply square roots of negative real numbers, we first express them in terms of i. For example,
6 5 1 6 1 5
6 5i i
2 30i
1 30
30
Multiply and simplify. When possible, write answers in the form a + bi.
Example
Solution
a) 2 10
b) 2 5 3
c) 2 4 3
i i
i i
a) 2 10 1 2 1 10
2 10i i
2 20
1 2 5
2 5
i
Solution continued
b) 2 5 3 2 5 2 3i i i i i
2c) 2 4 3 8 6 4 3i i i i i
210 6i i
10 6
6 10
i
i
8 2 3i
11 2i
Conjugates and Division
Conjugates are used when dividing complex numbers. The procedure is much like that used to rationalize denominators.
2 2( )( )a bi a bi a bi 2 2a b
Solution
Example Find the conjugate of each number.
a) 4 3
b) 6 9
c)
i
i
i
The conjugate is 4 – 3i.
The conjugate is –6 + 9i.
The conjugate is –i.
Solution
Example Divide and simplify to the form a + bi.
3 2a)
4b)
2 3
i
ii
i
3 2 3 2a)
i i
i i
i
i
2
23 2
3 2
1
i i
ii
3 2
2 3
i
i
4 4b)
2 3 2 3
i i
i i
2 3
2 3
i
i
2
2
(4 )(2 3 )
(2 3 )(2 3 )
8 12 2 3
4 9
i i
i i
i i i
i
8 14 3
4 95 14
13
i
i
5 14
13 13i
Powers of i
1i i2 1i 3 2i i i i 4 2 2 1i i i
5 4i i i i 6 4 2 1i i i 7 4 3i i i i 8 4 4 1i i i
Example
Simplify:
33
67
a)
b)
i
i
