Lesson 4.1: Introduction to Imaginary

Learning Goals: 1) What is a real ? A rational number? An irrational number? A ? 2) How do we simplify radicals of negative numbers? 3) How do we simplify powers of 푖?

A is any number that is not imaginary; it includes rational or irrational number. 푎 A rational number is any number that can be expressed as a fraction ; it 푏 includes fractions, , and certain decimals An irrational number is any number that cannot be expressed as a fraction.

What irrationals do you know?

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Getting ready for today’s lesson: Solve each of the following for 푥: a) 푥2 = 4 or 푥2 − 4 = 0

√푥2 = √4 (푥 − 2)(푥 + 2) = 0 푥 = ±2 푥 = 2 & − 2 b) 푥2 − 3 = 0 c) 푥2 + 1 = 0 푥2 = 3 푥2 = −1

√푥2 = √3 √푥2 = √−1 error in calc 푥 = ±√3 푥 = ±√−1 = ±푖

The imaginary unit is defined as 푖 = √−1.

Calculator must be in “푎 + 푏푖” mode in order to compute imaginary numbers without error. Imaginary numbers can only be found with even indexes!

√−64 = 8푖 vs. 3√−64 = −4

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Simplifying radicals with negative radicands Let’s try some together! Express each of the following in simplest radical form.

1. √−121 = √−1 ∙ √121 = 푖 ∙ 11 = 11푖

2. 5√−8 + √−72 You cannot add until you have like terms! 5√−1 ∙ √4 ∙ √2 + √−1 ∙ √36√2 5 ∙ 푖 ∙ 2 ∙ √2 + 푖 ∙ 6 ∙ √2 10푖√2 + 6푖√2 16푖√2 2nd → decimal = 푖

Now you try! Express each of the following in simplest radical form.

3: −√−49 = −√−1 ∙ √49 = −푖 ∙ 7 = −7푖

4. 4√−18 − √−50 = 4 ∙ √−9 ∙ √2 − √−25 ∙ √2 = 12푖√2 − 5푖√2 = 7푖√2

5. −√−225 = −√−1 ∙ √225 = −푖 ∙ 15 = −15푖

Simplifying Powers of 풊 What happens when you raise 푖 to a power? The powers of 푖 repeat in a definite pattern (1, 푖, −1, −푖) 푖0 = 1 푖3 = −푖 푖6 = −1 푖1 = 푖 푖4 = 1 푖7 = −3퐸 − 13 − 푖 푖2 = −1 푖5 = 푖 푖8 = 1 − 2퐸 − 13푖

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To simplify powers of 풊: By Hand: Divide exponent by 4 and use the remainder! Clock: ÷ 4

With Calculator: 푎 + 푏푖 mode look at first or last or do “ipart” 푖22 = −1 − 2퐸−3 Math → NUM → 3: ipart( 푖part(푖22) = −1

Let’s try simplifying the following: 27 6. 푖27 = −푖 − 3퐸 − 13 − 푖 = 6.75 = −푖 4 Math → NUM → 3: ipart(푖27) = −푖

7. 2푖10 + 푖25 − 7푖21 = 2(−1) + (푖) − 7(푖) = −2 + 푖 − 7푖 = −2 − 6푖 Cannot just plug entire expression into calculator!

8. 푖32 ∙ 푖45 = (1)(푖) = 푖 표푟 푖77 = 푖

9. 5푖101 + 2푖14 = 5푖 + 2(−1) = 5푖 − 2

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Practice! Complete at least one problem from each row. The stars tell you the level of difficulty of the question. Challenge yourself!

1 2 10a) 3√−20 b) √−48 c) − √−63 4 3 3 ∙ √−4 ∙ √5 1 2 ∙ √−16 ∙ √3 − ∙ √−9 ∙ √7 3 ∙ −2푖 ∙ √5 4 3 1 2 −6푖√5 ∙ −4푖 ∙ √3 − ∙ 3푖 ∙ √7 4 3 −푖√3 −2푖√7 5 5 15 7 19 11a) 푖 = −푖 b) 2푖 + 7푖 c) 푖 −4푖 2(푖) + 7(−푖) −3푖4 −푖 − 4(−푖) 55 −5푖 = 13.75 4 −3(1) −푖 + 4푖

−3 3푖

−3 −푖 12a)√−49 + √−121 b) 14√−45 − 3√−125 2√−48 − 5√3 + 3√−75 7푖 + 11푖 14√−9√5 − 3√−25√5 2√−16 ∙ 3 − 5√3

18푖 14 ∙ 3푖√5 − 3 ∙ 5푖√5 + 3√−25 ∙ 3

42푖√5 − 15푖√5 2 ∙ 4푖√3 − 5√3 + 3 ∙ 5푖√3 27푖√5 8푖√3 − 5√3 + 15푖√3 23푖√3 + 5√3 5√3 + 23푖√3

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Homework 4.1: Intro to Imaginary Numbers

Directions: Express in simplest form in terms of 푖.

1. √−36 2. √−12 3. √−18 − 2√−12

4. √−81 + 3푖 5. 3√−27 + 4√−48 6. √−49 + √−64 − √−25

7. √−64 + 2√−16 8. √−128

9. State if each of the following numbers is rational, irrational, or imaginary. a) √−25 b) √100 c) √20 d) 3√−8 e) 3√60

10. What is the value of 2푖8? 11. What is the value of 푖10?

12. 푖10 + 푖2 13. 푖10 + 푖25

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Lesson 4.2: Add, Subtract, and Multiply Complex Numbers

Learning Goals: 1) How do we add and subtract complex numbers? 2) How do we multiply complex numbers? 3) How do we graph complex numbers?

Do Now: In order to prepare for today’s lesson, answer the following questions below: a. What is the sum of 3 + 2푥 and 5 − 4푥? 8 − 2푥

b. Express in simplest form: (1 + 3푥) − (3 + 2푥) = 1 + 3푥 − 3 − 2푥 = −2 + 푥

A complex number is any number that can be expressed in the form 푎 + 푏푖; where 푎 and 푏 are real numbers and 푖 is the imaginary unit. Must be expressed in 푎 + 푏푖 form.

Examples: 2 + 5푖 − 4 − 푖 0 + 2푖 8 + 5푖 2푖 + 3 ∗∗ ퟑ + ퟐ풊 − 푖 + 7 ∗∗ ퟕ − 풊

Part I: Express complex numbers in 푎 + 푏푖 form. a) 3푖 + 2 b) −4푖 + 1 c) −푖 − 5 d) 8푖 2 + 3푖 1 − 4푖 −5 − 푖 0 + 8푖

Part II: Adding and Subtracting Complex Numbers. Answers in 푎 + 푏푖 form. 1. (2 + 3푖) + (5 + 푖) = 7 + 4푖

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2. (1 + 3푖) − (3 + 2푖) = 1 + 3푖 − 3 − 2푖 = −2 + 푖

3. Subtract 2 − 13푖 from 7 − 5푖 7 − 5푖 −(2 − 13푖)

5 + 8푖

4. Subtract 6 − 2푖√3 from 5 − 3푖√3 5 − 3푖√3 −(6 − 2푖√3)

−1 − 푖√3

5. (5 + √−36) − (3 − √−16) 5 + 6푖 − 3 + 4푖 2 + 10푖

6. (5 + √−12) + (8 + √−27)

(5 + √−4 ∙ 3) + (8 + √−9 ∙ 3)

(5 + 2푖√3) + (8 + 3푖√3)

13 + 5푖√3

Learning Goal #2: How to Multiply Complex Numbers

Is √푎 ∙ √푏 always equal to √푎푏? No, only with real numbers. Examine the following work and identify all mistakes. Then, resolve the problem correctly.

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Simplify: √−6 ∙ √−6 √36 푖√6 ∙ 푖√6 6푖 (no) 6푖2 6(−1) −6 (yes)

When multiplying like bases, what can you do to their exponents? ADD exponents 2푖(푖2 − 푖) = 2푖3 − 2푖2 = 2(−푖) − 2(−1) = −2푖 + 2 푎 + 푏푖 form 2 − 2푖

How do you reduce powers of "푖"? use 푖 rule 2푖3 − 2푖2 = 2(−푖) − 2(−1) = −2푖 + 2 푎 + 푏푖 form 2 − 2푖

7. What is the product of 2 + √−9 and 3 − √−4, expressed in simplest 푎 + 푏푖 form?

(2 + √−9)(3 − √−4) = (2 + 3푖)(3 − 2푖) =

6−4푖 + 9푖 − 6푖2 = 6 + 5푖 − 6(−1) = 12 + 5푖

8. In an electrical circuit, the voltage, 퐸, in volts, the current, 퐼, in amps, and the opposition to the flow of current, called impedance, 푍, in ohms, are related by the equation 퐸 = 퐼푍. A circuit has a current of (5 + 푖) amps and an impedance of (−3 + 푖) ohms. Determine the voltage in 푎 + 푏푖 form. 퐸 = 퐼 ∗ 푍 퐸 = (5 + 푖)(−3 + 푖) −15 + 5푖 − 3푖 + 푖2 −15 + 2푖 − 1 퐸 = −16 + 2푖

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9. Express (푖3 − 1)(푖3 + 1) in simplest 푎 + 푏푖 form. These are conjugates! 푖6 + 푖3 − 푖3 − 1 −1 − 1 −2

10. Express ((5 − 푖) − 2(1 − 3푖)) in 푎 + 푏푖 form. 5 − 푖 − 2 + 6푖 3 + 5푖

Learning Goal #3: How to Graph Complex Numbers.

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11) Locate the point on the corresponding to the complex number given in parts (a) – (e). On one set of axes, label each point by its identifying letter. For example, the point corresponding to 5 + 2푖 should be labeled “a”. a) 5 + 2푖 b) −2 − 4푖 c) −푖 1 d) + 푖 2

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Homework 4.2: Add, Subtract, and Multiply Complex Numbers 1. Melissa and Joe are playing a game with complex numbers. If Melissa has a score of 5 − 4푖 and Joe has a score of 3 + 2푖, what is their total score? (1) 8 + 6푖 (2) 8 + 2푖 (3) 8 − 6푖 (4) 8 − 2푖

2. What is the sum of 2 − √−4 and −3 + √−16 expressed in 푎 + 푏푖 form?

3. Simplify and express in terms of 푖: 2√−32 − 5√−8

4. What is the product of 5 + √−36 and 1 − √−49, expressed in simplest 푎 + 푏푖 form? (1) −37 + 41푖 (2) 5 − 71푖 (3) 47 + 41푖 (4) 47 − 29푖

5. The complex number 푐 + 푑푖 is equal to (2 + 푖)2. What is the value of 푐?

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6. Two complex numbers are graphed below. What is the sum of 푤 and 푢. Expressed in standard complex form? (1) 7 + 3푖 (2) 3 + 7푖 (3) 5 + 7푖 (4) −5 + 3푖

7) On a graph, if point 퐴 represents 2 − 3푖 and point 퐵 represents −2 − 5푖, which quadrant contains 3퐴 − 2퐵? (1) I (2) II (3) III (4) IV

8) Find the sum of −2 + 3푖 and −1 − 2푖. Graph the resultant on the accompanying set of axes.

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Lesson 4.3: Structure Questions with Complex Numbers

Learning Goals: 1) What are complex conjugates? 2) How do you solve for missing variables in a complex number equation?

What are complex conjugates? Complex conjugates are two complex numbers that have the form 푎 + 푏푖 and 푎 − 푏푖. Identify the complex conjugates of the following complex numbers:

5 + 3푖 6 − 2푖 − 4 + 푖 − 1 − √2푖

5 − 3푖 6 + 2푖 − 4 − 푖 − 1 + √2푖

What’s true about the product of two complex conjugates? Middle terms cancel out and you always get a real number. Example 1: (5 − 7푖)(5 + 7푖) Example 2: (−3 + 8푖)(−3 − 8푖) 25 + 35푖 − 35푖 − 49푖2 9 + 24푖 − 24푖 − 64푖2 25 − 49(−1) 9 − 64(−1) 25 + 49 9 + 64 74 73 1. Perform the following complex calculation. Express your answer in simplest form. (4 + 2푖)2(4 − 2푖)2 (4 + 2푖)(4 + 2푖)(4 − 2푖)(4 − 2푖) expand it! (4 + 2푖)(4 − 2푖)(4 + 2푖)(4 − 2푖) conjugates! (16 − 4푖2)(16 − 4푖2) (16 − 4(−1))(16 − 4(−1)) (20)(20) 400 14

Express each of the following in simplest 푎 + 푏푖 form. 2 2. ((5 − 푖) − 2(1 − 3푖)) 3. (3 − 푖)(4 + 7푖) − ((5 − 푖) − 2(1 − 3푖)) (5 − 푖 − 2 + 6푖)2 12 + 21푖 − 4푖 − 7푖2 − (5 − 푖 − 2 + 6푖) (3 + 5푖)2 12 + 17푖 + 7 − 5 + 푖 + 2 − 6푖 (3 + 5푖)(3 + 5푖) 16 + 12푖 9 + 15푖 + 15푖 + 25푖2 9 + 30푖 − 25 −16 + 30푖

Two complex numbers 풂 + 풃풊 and 풄 + 풅풊 are equal if and only if 풂 = 풄 and 풃 = 풅. For example: Find the real values of 푎 and 푏 in each of the following equations.

4. 7 + 2푖 = 푎 + 푏푖 5. −3 + 푏푖 = 푎 + 8푖 7 = 푎 2푖 = 푏푖 −3 = 푎 푏푖 = 8푖 2 = 푏 푏 = 8

6. 4푖 − 6 = 푎 + 푏푖 7. −푖 + 푎 = 5 + 푏푖 4푖 = 푏푖 − 6 = 푎 −푖 = 푏푖 푎 = 5 4 = 푏 −1 = 푏

8. −2 − 푎 = 5푖 + 푏푖 9. 3푎 + 6 = 8푖 − 2푏푖 −2 − 5푖 = 푎 + 푏푖 3푎 + 2푏푖 = −6 + 8푖 −2 = 푎 − 5푖 = 푏푖 3푎 = −6 2푏푖 = 8푖 −5 = 푏 푎 = −2 푏 = 4

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10. 2(푎 + 4) = (3 + 6푏)푖 11. 8 − 2푖 + 푎 − 푏푖 = 5 + 3푖 2푎 + 8 = 3푖 + 6푏푖 3 − 푏푖 = −푎 + 5푖 2푎 − 6푏푖 = −8 + 3푖 3 = −푎 − 푏푖 = 5푖 2푎 = −8 − 6푏푖 = 3푖 푎 = −3 푏 = −5 1 푎 = −4 푏 = − 3

COMMON CORE QUESTIONS Directions: Find the real values of 푥 and 푦 in each of the following equations using the fact that if 풂 + 풃풊 = 풄 + 풅풊, then 풂 = 풄 and 풃 = 풅. 12. 5푥 + 3푦푖 = 20 + 9푖 13. 2(5푥 + 9) = (10 − 3푦)푖 5푥 = 20 3푦푖 = 9푖 10푥 + 18 = 10푖 − 3푦푖 푥 = 4 푦 = 3 10푥 + 3푦푖 = −18 + 10푖 10푥 = −18 3푦푖 = 10푖 −18 −9 10 푥 = = 푦 = 10 5 3

14. 3 + 5푖 + 푥 − 푦푖 = 6 − 2푖 15. 푥 + 푦푖 = (1 − 푖)(2 + 8푖) 푥 − 푦푖 = 3 − 7푖 푥 + 푦푖 = 2 + 8푖 − 2푖 − 8푖2 푥 = 3 − 푦푖 = −7푖 푥 + 푦푖 = 10 + 6푖 푦 = 7 푥 = 10 푦 = 6

16. 3(7 − 2푥) − 5(4푦 − 3)푖 = 푥 − 2(1 + 푦)푖 21 − 6푥 − 20푦푖 + 15푖 = 푥 − 2푖 − 2푦푖 21 − 18푦푖 = 7푥 − 17푖 21 = 7푥 − 18푦푖 = −2푖 1 푥 = 3 푦 = 9

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Homework 4.3: Structure Questions with Complex Numbers 1. Perform the following complex calculations. Express each answer in simplest 푎 + 푏푖 form. a. (13 + 4푖) + (7 + 5푖) b. (4 + 푖) + (2 − 푖) − (1 − 푖)

c. −푖(2 − 푖)(5 + 6푖)

2. Find the real values of 푥 and 푦 in each of the following equations using the fact that if 풂 + 풃풊 = 풄 + 풅풊, then 풂 = 풄 and 풃 = 풅. a. −10푥 + 12푖 = 20 + 3푦푖 b. 3(4푥 + 2) = (8 − 푦)푖

3. Express in 푎 + 푏푖 form: (3 − 2푦푖)(2 + 7푖) − [(6 + 5푦푖) + 2(3 + 4푖)]

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