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Name
Class
Date
Reteaching
4-8
Complex Numbers
• A complex number consists of a real part and an imaginary part. It is written in
the form a 1 bi, where a and b are real numbers.
• i 5 !21 and i2 5 (!21)(!21) 5 21
• When adding or subtracting complex numbers, combine the real parts and
then combine the imaginary parts.
• When multiplying complex numbers, use the Distributive Property or FOIL.
Problem
What is (3 2 i) 1 (2 1 3i)?
(3 2 i) 1 (2 1 3i)
5 3 2 i 1 2 1 3i
Circle real parts. Put a square around imaginary parts.
5 (3 1 2) 1 (21 1 3)i
Combine.
5 5 1 2i
Simplify.
Problem
What is the product (7 2 3i)(24 1 9i)?
Use FOIL to multiply:
(7 2 3i)(24 1 9i) 5 7(24) 1 7(9i) 1 (23i)(24) 1 (23i)(9i)
(7 2 3i)(24 1 9i)
5 228 1 63i 1 12i 2 27i2
First 5 7(24)
5 228 1 75i 2 27i2
Outer 5 7(9i)
You can simplify the expression by substituting 21 for i2.
Inner 5 (23i)(24)
(7 2 3i)(24 1 9i) 5 228 1 75i 2 27(21)
5 21 1 75i
Last 5 (23i)(9i)
Exercises
Simplify each expression.
1. 2i 1 (24 2 2i)
24
2. (3 1 i)(2 1 i)
5 1 5i
3. (4 1 3i)(1 1 2i)
22 1 11i
4. 3i(1 2 2i)
6 1 3i
5. 3i(4 2 i)
3 1 12i
6. 3 2 (22 1 3i) 1 (25 1 i)
22i
7. 4i(6 2 2i)
8 1 24i
8. (5 1 6i) 1 (22 1 4i)
3 1 10i
9. 9(11 1 5i)
99 1 45i
Prentice Hall Algebra 2 • Teaching Resources
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79
Name
Class
Date
Reteaching (continued)
4-8
Complex Numbers
• The complex conjugate of a complex number a 1 bi is the complex number a 2 bi.
• (a 1 bi)(a 2 bi) 5 a2 1 b2
• To divide complex numbers, use complex conjugates to simplify the denominator.
Problem
4 1 5i
What is the quotient 2 2 i ?
4 1 5i
22i 5
4 1 5i
21i
5 22i ?21i
5
5
5
5
8 1 4i 1 10i 1 5i2
(2 2 i)(2 1 i)
8 1 4i 1 10i 1 5i2
22 1 12
8 1 14i 1 5(21)
4 1 1
3 1 14i
5
3
14
5 5 1 5i
The complex conjugate of 2 2 i is 2 1 i.
Multiply both numerator and denominator by 2 1 i.
Use FOIL to multiply the numerators.
Simplify the denominator. (a 1 bi)(a 2 bi) 5 a2 1 b2
Substitute 21 for i2.
Simplify.
Write as a complex number a 1 bi.
Exercises
Find the complex conjugate of each complex number.
10. 1 2 2i 1 1 2i
11. 3 1 5i 3 2 5i
12. i 2i
13. 3 2 i 3 1 i
14. 2 1 3i 2 2 3i
15. 25 2 2i 25 1 2i
Write each quotient as a complex number.
3i
16. 1 2 2i 265 1 53i
6
15
9
17. 3 1 5i 17
2 17
i
18.
2 1 5i 1
19. 3 2 i 10
1 17
10 i
24 2 i
10
20. 2 1 3i 211
13 1 13 i
61i
7
21. 25 2 2i 232
29 1 29 i
2 1 2i
2 2 2i
i
Prentice Hall Algebra 2 • Teaching Resources
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
80