Complex Numbers P. 1

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6-8 The Complex Numbers Objective:

To add, subtract, multiply, and divide complex numbers.

Vocabulary

Imaginary number A number of the form a + bi, where a and b are real numbers and b & 0. Examples: 3 + 7* 2 4- i'V5 Complex number A number of the form a 4- bi, where a and b are real numbers. The number a is called the real part of a + bi, and b (not W) is called the imaginary part, Notice that when b ~ 0, the complex number a + bi becomes the real number a. "This shows that a complex.number can be either real or imaginary. Complex conjugates The numbers a 4- bi and a — bi are complex conjugates. Their product is the real number #2 4- b2. Equality of complex numbers a + bi - c + di if and only if a = c and b = d. Sum of complex numbers (a + bi) + (c 4- di) = (a 4- c) 4- (b + d)i Product of complex numbers (a 4- bi)(c 4- di) = (ae — bd) 24- (ad 4- bc)i This results from using the FOIL method and the fact that i — — 1.

Example 1

Simplify

Solution

a. (4 4- 50 4- (8 - 70 = ( 4 4 - 8 ) 4- (5 - 7)/ * 12 - 2i

a. (4 4- 50 4- (8 - 70

b. 3(2 4- 40 - 2(4 - 50

b. 3(2 4- 40 ~ 2(4 - 50 = (64- 120 - (*I - 100 = (6 - 8) 4- (12 4- 10)/ = - 2 4 - 22i

example 2

Simplify :

Solution

a. (2 4- 30(3 4- 40 - 6 4- 8/ 4- 9i 4- 12*2

a. (2 4- 30(3 + 4i)

b. (2 4- 502

c. (3 4- 20(3 - 20

Use the FOIL method.

"

* 6 4- 17/ 4- 12(-1) = - 6 4 - I7i b. (2 4- Si)2 = = 4 4 - 20* 4- 25/2 « 4 4- 20i + 25(-l) * -21 4- 20i c. (3 4- 20(3 - 20 = 9 - 4i2 = 9 - 4< - 1) - 13

Simplify. 1. (7 4- 30 4- (2 - 50 4. 2i(4 4- 70 7, (4 4- 90(4 - 90 10. (2 - 50(3 4- 40

2. 5. 8. 11.

13. (-2

14. (1 4- 402d - 402

4 j'V5)2

(4 - 70 - (5 - 30 -4i(~5 - 0 (-3 + 20(4 4- SO (4 - /V7)(4 4- IN/?)

Use the pattern: (a 4- b)2 » «2 4- 2a6 4- f*2. Use the pattern:

(a 4- b)(a ~ b) = a2 - b2.

3. 2(-3 4- 0 ~ 5(2 - 20

6. (2 4- 0(2 - 0 9, (-3 4- 0(4 4- 30 12. (3 - 502 15. (V5 - V^XN/S 4 V^S)

Study Guide, ALGEBRA AND TRIGONOMETRY, Structure and Method, Book 2 Copyright © by Hougrttori Mlfflin Company. All rights reserved.

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