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Math Coach Amy
Amy Ferguson Moncure

H Algebra 2/Trig · Unit 2

Complex Number Operations

Add, multiply, and divide complex numbers. Replace i² with −1. Rationalize denominators so there are no i’s in the denominator.

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Key Ideas Add and Subtract FOIL Divide Quiz

i = √(−1)

i² = −1

Add and Subtract
Distribute, then combine like terms.

2i + 8 + 4i − 7 = 6i + 1

Multiply
FOIL.

(6 + 2i)(5 − i)

30 − 6i + 10i − 2i²

30 − 6i + 10i − 2(−1)

32 + 4i

Conjugates
(a + bi)(a − bi) = a² + b². The i terms cancel and you get a real number.

(3 + 2i)(3 − 2i)

9 − 6i + 6i − 4i²

9 − 4(−1)

9 + 4

13

Divide
Rationalize the Denominator (no i’s in the denominator)

32i

i is the problem.

32i × ii

3i2i²

3i2(−1)

−3i2

53 + i

Multiply by the conjugate.

53 + i × 3 − i3 − i

15 − 5i9 − 3i + 3i − i²

15 − 5i9 − (−1)

15 − 5i10

Level 1

Add and Subtract

Distribute, then combine like terms.

Example
Simplify. −5(2 − 3i) + 2(4 + i)

−10 + 15i + 8 + 2i

−2 + 17i

Answer: −2 + 17i

1
Simplify. 4(1 + 2i) − 3(2 − i)
Show Answer

4 + 8i − 6 + 3i

−2 + 11i

Answer: −2 + 11i

2
Simplify. 6(3 − i) + 2(1 + 5i)
Show Answer

18 − 6i + 2 + 10i

20 + 4i

Answer: 20 + 4i

3
Simplify. −2(4 + 3i) + 5(1 − 2i)
Show Answer

−8 − 6i + 5 − 10i

−3 − 16i

Answer: −3 − 16i

4
Simplify. 3(6 − 2i) − 4(1 + i) + (5 − 3i)
Show Answer

18 − 6i − 4 − 4i + 5 − 3i

19 − 13i

Answer: 19 − 13i

5
Simplify. 2(5 + 4i) − 7(2 − i) + 3i
Show Answer

10 + 8i − 14 + 7i + 3i

−4 + 18i

Answer: −4 + 18i

Level 2

FOIL

FOIL. Replace i² with −1, then combine like terms.

Example
Simplify. (2 − 5i)(3 + 4i)

6 + 8i − 15i − 20i²

6 + 8i − 15i − 20(−1)

26 − 7i

Answer: 26 − 7i

1
Simplify. (4 − i)²
Show Answer

16 − 4i − 4i + i²

16 − 4i − 4i + (−1)

15 − 8i

Answer: 15 − 8i

2
Simplify. (2 − 3i)(2 + 3i)
Show Answer

4 + 6i − 6i − 9i²

4 − 9(−1)

13

Answer: 13

3
Simplify. (1 + 6i)(5 − 2i)
Show Answer

5 − 2i + 30i − 12i²

5 − 2i + 30i − 12(−1)

17 + 28i

Answer: 17 + 28i

4
Simplify. (2 + 5i)²
Show Answer

4 + 10i + 10i + 25i²

4 + 10i + 10i + 25(−1)

−21 + 20i

Answer: −21 + 20i

5
Simplify. (5 − i)(1 + 4i)
Show Answer

5 + 20i − i − 4i²

5 + 20i − i − 4(−1)

9 + 19i

Answer: 9 + 19i

Level 3

Divide

Rationalize the denominator. No i’s in the denominator.

Example
Simplify. 43i

i is the problem.

43i × ii

4i3i²

4i3(−1)

−43i

Answer: −43i

Example
Simplify. 4 − 2i3i

i is the problem.

4 − 2i3i × ii

4i − 2i²3i²

4i − 2(−1)3(−1)

−23 − 43i

Answer: −23 − 43i

Example
Simplify. 62 + i

Multiply by the conjugate.

62 + i × 2 − i2 − i

12 − 6i4 − 2i + 2i − i²

12 − 6i4 − (−1)

12 − 6i5

125 − 65i

Answer: 125 − 65i

Example
Simplify. 3 + i2 − i

Multiply by the conjugate.

3 + i2 − i × 2 + i2 + i

6 + 3i + 2i + i²4 + 2i − 2i − i²

6 + 3i + 2i + (−1)4 − (−1)

5 + 5i5

1 + i

Answer: 1 + i

1
Simplify. 52i
Show Answer

i is the problem.

52i × ii

5i2i²

5i2(−1)

−52i

Answer: −52i

2
Simplify. 74i
Show Answer

i is the problem.

74i × ii

7i4i²

7i4(−1)

−74i

Answer: −74i

3
Simplify. 85i
Show Answer

i is the problem.

85i × ii

8i5i²

8i5(−1)

−85i

Answer: −85i

4
Simplify. 5 + i2i
Show Answer

i is the problem.

5 + i2i × ii

5i + i²2i²

5i + (−1)2(−1)

12 − 52i

Answer: 12 − 52i

5
Simplify. 6 − 3ii
Show Answer

i is the problem.

6 − 3ii × ii

6i − 3i²i²

6i − 3(−1)(−1)

−3 − 6i

Answer: −3 − 6i

6
Simplify. 8 + 4i4i
Show Answer

i is the problem.

8 + 4i4i × ii

8i + 4i²4i²

8i + 4(−1)4(−1)

1 − 2i

Answer: 1 − 2i

7
Simplify. 81 + i
Show Answer

Multiply by the conjugate.

81 + i × 1 − i1 − i

8 − 8i1 − i + i − i²

8 − 8i1 − (−1)

8 − 8i2

4 − 4i

Answer: 4 − 4i

8
Simplify. 103 − i
Show Answer

Multiply by the conjugate.

103 − i × 3 + i3 + i

30 + 10i9 + 3i − 3i − i²

30 + 10i9 − (−1)

30 + 10i10

3 + i

Answer: 3 + i

9
Simplify. 42 + 5i
Show Answer

Multiply by the conjugate.

42 + 5i × 2 − 5i2 − 5i

8 − 20i4 − 10i + 10i − 25i²

8 − 20i4 − 25(−1)

8 − 20i29

829 − 2029i

Answer: 829 − 2029i

10
Simplify. 2 − i1 + 2i
Show Answer

Multiply by the conjugate.

2 − i1 + 2i × 1 − 2i1 − 2i

2 − 4i − i + 2i²1 − 2i + 2i − 4i²

2 − 4i − i + 2(−1)1 − 4(−1)

−5i5

−i

Answer: −i

11
Simplify. 4 + 2i1 − i
Show Answer

Multiply by the conjugate.

4 + 2i1 − i × 1 + i1 + i

4 + 4i + 2i + 2i²1 + i − i − i²

4 + 4i + 2i + 2(−1)1 − (−1)

2 + 6i2

1 + 3i

Answer: 1 + 3i

12
Simplify. 5 − 3i2 + 7i
Show Answer

Multiply by the conjugate.

5 − 3i2 + 7i × 2 − 7i2 − 7i

10 − 35i − 6i + 21i²4 − 14i + 14i − 49i²

10 − 35i − 6i + 21(−1)4 − 49(−1)

−11 − 41i53

−1153 − 4153i

Answer: −1153 − 4153i

Quiz

Quiz

Ten mixed problems. Each one is a different type.

1
Simplify. (1 + 3i)(4 − 2i)
Show Answer

4 − 2i + 12i − 6i²

4 − 2i + 12i − 6(−1)

10 + 10i

Answer: 10 + 10i

2
Simplify. 62i
Show Answer

i is the problem.

62i × ii

6i2i²

6i2(−1)

−3i

Answer: −3i

3
Simplify. 3i + 5 − 7i + 2
Show Answer

5 + 2 + 3i − 7i

7 − 4i

Answer: 7 − 4i

4
Simplify. 1 + 2i1 − i
Show Answer

Multiply by the conjugate.

1 + 2i1 − i × 1 + i1 + i

1 + i + 2i + 2i²1 + i − i − i²

1 + i + 2i + 2(−1)1 − (−1)

−1 + 3i2

−12 + 32i

Answer: −12 + 32i

5
Simplify. (5 − 2i)²
Show Answer

25 − 10i − 10i + 4i²

25 − 10i − 10i + 4(−1)

21 − 20i

Answer: 21 − 20i

6
Simplify. 3(2 − i) − 4(1 + 2i)
Show Answer

6 − 3i − 4 − 8i

2 − 11i

Answer: 2 − 11i

7
Simplify. 23 − 8i
Show Answer

Multiply by the conjugate.

23 − 8i × 3 + 8i3 + 8i

6 + 16i9 + 24i − 24i − 64i²

6 + 16i9 − 64(−1)

6 + 16i73

673 + 1673i

Answer: 673 + 1673i

8
Simplify. (4 + 3i)(4 − 3i)
Show Answer

16 − 12i + 12i − 9i²

16 − 9(−1)

25

Answer: 25

9
Simplify. 3i(2 − 5i)
Show Answer

6i − 15i²

6i − 15(−1)

15 + 6i

Answer: 15 + 6i

10
Simplify. 3 + 6i3i
Show Answer

i is the problem.

3 + 6i3i × ii

3i + 6i²3i²

3i + 6(−1)3(−1)

2 − i

Answer: 2 − i

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