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2. Complex Numbers. ALGEBRA 2 LESSON 5-6. 1. Simplify – –169. 2. Find |–1 + i |. –13 i. 5-6. Pages 274–276 Exercises. 1. 2 i 2. i 7 3. i 15 4. 9 i 5. 5 i 2 6. 4 i 7. 4 i 2 8. 9 i. 9. –10 i 10. 6 i 2 11. 2 + i 3

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Complex Numbers

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### Complex Numbers

ALGEBRA 2 LESSON 5-6

1.Simplify – –169.

2.Find |–1 + i|

–13i

5-6

Pages 274–276 Exercises

1.2i

2.i 7

3.i 15

4.9i

5.5i 2

6.4i

7.4i 2

8.9i

9.–10i

10.6i 2

11.2 + i 3

12.8 + 2i 2

13.6 – 2i 7

14.3 + 2i

15.7 – 5i

16.2 + i

17.–2 – 5i 2

18.4 + 6i 2

19.2

20.13

21.2 2

22. 17

23.3 5

24.–4i

### Complex Numbers

ALGEBRA 2 LESSON 5-6

5-6

25.–5 + 3i

26.–9 – i

27.3 + 2i

28.4 – 7i

29.6 + 3i

30.1 – 7i

31.7 + 4i

32.–2 – 3i

33.10 + 6i

34.–7 – 10i

35.10

36.26 – 7i

37.9 + 58i

38.9 – 23i

39.–36

40.65 + 72i

41.± 5i

42.±

43.±

44.± i 7

45.±6i

46.±

47.–i, –1 – i, i

48.–2i, –4 – 2i, 12 + 14i

49.1 – i, 1 – 3i, –7 – 7i

50.± i 65

51.±7i

8i 3

3

i 15

3

i 2

2

### Complex Numbers

ALGEBRA 2 LESSON 5-6

5-6

52.± i

53.No; the test scores were real numbers. He added the scores and divided by the number of scores. The set of real numbers is closed with respect to addition and division so he should have gotten a real number.

54.a.A: –5, B: 3 + 2i, C: 2 – i, D: 3i, E: –6 – 4i, F: –1 + 5i

b.5, –3 – 2i, –2 + i, –3i, 6 + 4i, 1 – 5i

55.a.Check students’ work.

b.a circle with radius 10 and center at the origin

56.–5, 5

57.288i

58.–1 + 5i

59.10 – 4i

60.8 – 2i

61.11 – 5i

### Complex Numbers

ALGEBRA 2 LESSON 5-6

5-6

### Complex Numbers

• You can apply all the operations of real numbers to complex numbers

• Multiplication

### Complex Numbers

ALGEBRA 2 LESSON 5-6

Find the additive inverse of –7 – 9i.

–7 – 9i

–(–7 – 9i)Find the opposite.

7 + 9iSimplify.

5-6

### Complex Numbers

ALGEBRA 2 LESSON 5-6

Simplify (3 + 6i) – (4 – 8i).

(3 + 6i) – (4 – 8i) = 3 + (–4) + 6i + 8iUse commutative and associative properties.

= –1 + 14iSimplify.

5-6

### Complex Numbers

ALGEBRA 2 LESSON 5-6

Find each product.

a. (3i)(8i)

(3i)(8i) = 24i2Multiply the real numbers.

= 24(–1)Substitute –1 for i2.

= –24Multiply.

b.(3 – 7i)(2 – 4i)

(3 – 7i)(2 – 4i) = 6 – 14i – 12i + 28i2 Multiply the binomials.

= 6 – 26i + 28(–1)Substitute –1 for i2.

= –22 – 26iSimplify.

5-6

x= ±i 6Find the square root of each side.

Check:9x2 + 54 = 0

9x2 + 54 = 0

9(i 6)2 + 54 0

9(i(– 6))2 + 54 0

9(6)i2 + 54 0

9(6i2) + 54 0

54(–1)–54

54(–1) –54

54= 54

–54= –54

### Complex Numbers

ALGEBRA 2 LESSON 5-6

Solve 9x2 + 54 = 0.

9x2 + 54 = 0

9x2= –54Isolate x2.

x2= –6

5-6

### Complex Numbers

• Functions of the form f(z) = z2 + c where c is a complex number generate fractal graphs on the complex plane like the one at the left. To test if z belongs to the graph, use z as the first input value and repeatedly use each output as the next input. If the output values do not become infinitely large, then z is on the graph.

• f(z) = z2 + a+bi

Use z = 0 as the first input value.

f(0)= 02 – 4i

= –4i

f(–4i)= (–4i)2 – 4iFirst output becomes second input. Evaluate for z = –4i.

= –16 – 4i

= [(–16)2 + (–16)(–4i) + (–16)(–4i) + (–4i)2] – 4i

= (256 + 128i – 16) – 4i

= 240 + 124i

### Complex Numbers

ALGEBRA 2 LESSON 5-6

Find the first three output values for f(z) = z2 – 4i. Use z = 0 as the first input value.

f(–16 – 4i)= (–16 – 4i)2 – 4iSecond output becomes third input. Evaluate for z = –16 – 4i.

The first three output values are –4i, –16 – 4i, 240 + 124i.

5-6

### Assignment 43

• Page 274 34-48 even, 56-64 even