1

Lesson 4.1: Introduction to Imaginary Numbers

Learning Goals:

1) What is a real number? A rational number? An irrational number? A

complex number?

2) How do we simplify radicals of negative numbers?

3) How do we simplify powers of 𝑖?

A real number is any number that is not imaginary; it includes rational or

irrational number.

A rational number is any number that can be expressed as a fraction 𝑎

𝑏; it

includes fractions, integers, certain decimals

An irrational number is any number that cannot be expressed as a fraction.

What irrationals do you know?

2

Getting ready for today’s lesson:

Solve each of the following for 𝑥:

a) 𝑥2 = 4 or 𝑥2 − 4 = 0

√𝑥2 = √4 (𝑥 − 2)(𝑥 + 2) = 0

𝑥 = ±2 𝑥 = 2 & − 2

b) 𝑥2 − 3 = 0 c) 𝑥2 + 1 = 0

𝑥2 = 3 𝑥2 = −1

√𝑥2 = √3 √𝑥2 = √−1 error in calc

𝑥 = ±√3 𝑥 = ±√−1 = ±𝑖

Calculator must be in “𝑎 + 𝑏𝑖” mode in order to compute imaginary numbers

without error.

Imaginary numbers can only be found with even indexes!

√−64 = 8𝑖 vs. √−643

= −4

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

3

Simplifying radicals with negative radicands

Let’s try some together!

Express each of the following in simplest radical form.

1. √−121 = √−1 ∙ √121 = 𝑖 ∙ 11 = 11𝑖

2. 5√−8 + √−72 You cannot add until you have like terms!

5√−1 ∙ √4 ∙ √2 + √−1 ∙ √36√2

5 ∙ 𝑖 ∙ 2 ∙ √2 + 𝑖 ∙ 6 ∙ √2

10𝑖√2 + 6𝑖√2

16𝑖√2 2nd → decimal = 𝑖

Now you try!

Express each of the following in simplest radical form.

3: −√−49 = −√−1 ∙ √49 = −𝑖 ∙ 7 = −7𝑖

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

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

Simplifying Powers of 𝒊

What happens when you raise 𝑖 to a power? The powers of 𝑖 repeat in a definite

pattern (1, 𝑖, −1, −𝑖)

𝑖0 = 1 𝑖3 = −𝑖 𝑖6 = −1

𝑖1 = 𝑖 𝑖4 = 1 𝑖7 = −3𝐸 − 13 − 𝑖

𝑖2 = −1 𝑖5 = 𝑖 𝑖8 = 1 − 2𝐸 − 13𝑖

4

To simplify powers of 𝒊:

By Hand: Divide exponent by 4 and use the remainder!

Clock: ÷ 4

With Calculator: 𝑎 + 𝑏𝑖 mode

look at first or last or do “ipart”

𝑖22 = −1 − 2𝐸−3

Math → NUM → 3: ipart(

𝑖part(𝑖22) = −1

Let’s try simplifying the following:

6. 𝑖27 = −𝑖 − 3𝐸 − 13 − 𝑖 27

4= 6.75 = −𝑖

Math → NUM → 3: ipart(𝑖27) = −𝑖

7. 2𝑖10 + 𝑖25 − 7𝑖21 = 2(−1) + (𝑖) − 7(𝑖) = −2 + 𝑖 − 7𝑖 = −2 − 6𝑖

Cannot just plug entire expression into calculator!

8. 𝑖32 ∙ 𝑖45 = (1)(𝑖) = 𝑖 𝑜𝑟 𝑖77 = 𝑖

9. 5𝑖101 + 2𝑖14 = 5𝑖 + 2(−1) = 5𝑖 − 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!

10a) 3√−20

3 ∙ √−4 ∙ √5

3 ∙ −2𝑖 ∙ √5

−6𝑖√5

b) 1

4√−48

1

4∙ √−16 ∙ √3

1

4∙ −4𝑖 ∙ √3

−𝑖√3

c) −2

3√−63

−2

3∙ √−9 ∙ √7

−2

3∙ 3𝑖 ∙ √7

−2𝑖√7

11a) 𝑖5 = −𝑖

55

4= 13.75

b) 2𝑖5 + 7𝑖15 2(𝑖) + 7(−𝑖)

−5𝑖

c) 𝑖7−4𝑖19

−3𝑖4

−𝑖 − 4(−𝑖)

−3(1)

−𝑖 + 4𝑖

−3

3𝑖

−3

−𝑖

12a)√−49 + √−121 7𝑖 + 11𝑖

18𝑖

b) 14√−45 − 3√−125

14√−9√5 − 3√−25√5

14 ∙ 3𝑖√5 − 3 ∙ 5𝑖√5

42𝑖√5 − 15𝑖√5

27𝑖√5

2√−48 − 5√3 + 3√−75

2√−16 ∙ 3 − 5√3

+ 3√−25 ∙ 3

2 ∙ 4𝑖√3 − 5√3 + 3 ∙ 5𝑖√3

8𝑖√3 − 5√3 + 15𝑖√3

23𝑖√3 + 5√3

5√3 + 23𝑖√3

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Homework 4.1: Intro to Imaginary Numbers

Directions: Express in simplest form in terms of 𝑖.

1. √−36 2. √−12 3. √−18 − 2√−12

4. √−81 + 3𝑖 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) √−83

e) √603

10. What is the value of 2𝑖8? 11. What is the value of 𝑖10?

12. 𝑖10 + 𝑖2 13. 𝑖10 + 𝑖25

7

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 + 2𝑥 and 5 − 4𝑥? 8 − 2𝑥

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

Examples:

2 + 5𝑖 − 4 − 𝑖 0 + 2𝑖 8 + 5𝑖 2𝑖 + 3 ∗∗ 𝟑 + 𝟐𝒊 − 𝑖 + 7 ∗∗ 𝟕 − 𝒊

Part I: Express complex numbers in 𝑎 + 𝑏𝑖 form.

a) 3𝑖 + 2 b) −4𝑖 + 1 c) −𝑖 − 5 d) 8𝑖

2 + 3𝑖 1 − 4𝑖 −5 − 𝑖 0 + 8𝑖

Part II: Adding and Subtracting Complex Numbers. Answers in 𝑎 + 𝑏𝑖 form.

1. (2 + 3𝑖) + (5 + 𝑖) = 7 + 4𝑖

A complex number is any number that can be expressed in the form

𝑎 + 𝑏𝑖; where 𝑎 and 𝑏 are real numbers and 𝑖 is the imaginary unit. Must

be expressed in 𝑎 + 𝑏𝑖 form.

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

3. Subtract 2 − 13𝑖 from 7 − 5𝑖

7 − 5𝑖

−(2 − 13𝑖)

5 + 8𝑖

4. Subtract 6 − 2𝑖√3 from 5 − 3𝑖√3

5 − 3𝑖√3

−(6 − 2𝑖√3)

−1 − 𝑖√3

5. (5 + √−36) − (3 − √−16)

5 + 6𝑖 − 3 + 4𝑖

2 + 10𝑖

6. (5 + √−12) + (8 + √−27)

(5 + √−4 ∙ 3) + (8 + √−9 ∙ 3)

(5 + 2𝑖√3) + (8 + 3𝑖√3)

13 + 5𝑖√3

Learning Goal #2: How to Multiply Complex Numbers

Is √𝑎 ∙ √𝑏 always equal to √𝑎𝑏? No, only with real numbers.

Examine the following work and identify all mistakes. Then, resolve the problem

correctly.

9

Simplify: √−6 ∙ √−6

√36 𝑖√6 ∙ 𝑖√6

6𝑖 (no) 6𝑖2

6(−1)

−6 (yes)

When multiplying like bases, what can you do to their exponents? ADD

exponents

2𝑖(𝑖2 − 𝑖) = 2𝑖3 − 2𝑖2 = 2(−𝑖) − 2(−1) = −2𝑖 + 2 𝑎 + 𝑏𝑖 form 2 − 2𝑖

How do you reduce powers of "𝑖"? use 𝑖 rule

2𝑖3 − 2𝑖2 = 2(−𝑖) − 2(−1) = −2𝑖 + 2 𝑎 + 𝑏𝑖 form 2 − 2𝑖

7. What is the product of 2 + √−9 and 3 − √−4, expressed in

simplest 𝑎 + 𝑏𝑖 form?

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

6−4𝑖 + 9𝑖 − 6𝑖2 = 6 + 5𝑖 − 6(−1) = 12 + 5𝑖

8. In an electrical circuit, the voltage, 𝐸, in volts, the current, 𝐼, in amps, and the

opposition to the flow of current, called impedance, 𝑍, in ohms, are related by the

equation 𝐸 = 𝐼𝑍. A circuit has a current of (5 + 𝑖) amps and an impedance of

(−3 + 𝑖) ohms. Determine the voltage in 𝑎 + 𝑏𝑖 form.

𝐸 = 𝐼 ∗ 𝑍

𝐸 = (5 + 𝑖)(−3 + 𝑖)

−15 + 5𝑖 − 3𝑖 + 𝑖2

−15 + 2𝑖 − 1

𝐸 = −16 + 2𝑖

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9. Express (𝑖3 − 1)(𝑖3 + 1) is simplest 𝑎 + 𝑏𝑖 form. These are conjugates!

𝑖6 + 𝑖3 − 𝑖3 − 1

−1 − 1

−2

10. Express ((5 − 𝑖) − 2(1 − 3𝑖)) in 𝑎 + 𝑏𝑖 form.

5 − 𝑖 − 2 + 6𝑖

3 + 5𝑖

Learning Goal #3: How to Graph Complex Numbers.

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11) Locate the point on

the complex plane

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 + 2𝑖 should be labeled

“a”.

a) 5 + 2𝑖

b) −2 − 4𝑖

c) −𝑖

d) 1

2+ 𝑖

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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 − 4𝑖 and Joe has a score of 3 + 2𝑖, what is their total score?

(1) 8 + 6𝑖 (2) 8 + 2𝑖 (3) 8 − 6𝑖 (4) 8 − 2𝑖

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

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

4. What is the product of 5 + √−36 and 1 − √−49, expressed in simplest 𝑎 + 𝑏𝑖 form?

(1) −37 + 41𝑖 (2) 5 − 71𝑖 (3) 47 + 41𝑖 (4) 47 − 29𝑖

5. The complex number 𝑐 + 𝑑𝑖 is equal to (2 + 𝑖)2. What is the value of 𝑐?

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6. Two complex numbers are graphed

below. What is the sum of 𝑤 and 𝑢.

Expressed in standard complex form?

(1) 7 + 3𝑖

(2) 3 + 7𝑖

(3) 5 + 7𝑖

(4) −5 + 3𝑖

7) On a graph, if point 𝐴 represents 2 − 3𝑖 and point 𝐵 represents −2 − 5𝑖, which

quadrant contains 3𝐴 − 2𝐵? (1) I (2) II (3) III (4) IV

8) Find the sum of −2 + 3𝑖 and −1 − 2𝑖. Graph the resultant on the

accompanying set of axes.

14

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 𝑎 + 𝑏𝑖 and 𝑎 − 𝑏𝑖.

Identify the complex conjugates of the following complex numbers:

5 + 3𝑖 6 − 2𝑖 − 4 + 𝑖 − 1 − √2𝑖

5 − 3𝑖 6 + 2𝑖 − 4 − 𝑖 − 1 + √2𝑖

What’s true about the product of two complex conjugates? Middle terms cancel

out and you always get a real number.

Example 1: (5 − 7𝑖)(5 + 7𝑖) Example 2: (−3 + 8𝑖)(−3 − 8𝑖)

25 + 35𝑖 − 35𝑖 − 49𝑖2 9 + 24𝑖 − 24𝑖 − 64𝑖2

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 + 2𝑖)2(4 − 2𝑖)2

(4 + 2𝑖)(4 + 2𝑖)(4 − 2𝑖)(4 − 2𝑖) expand it!

(4 + 2𝑖)(4 − 2𝑖)(4 + 2𝑖)(4 − 2𝑖) conjugates!

(16 − 4𝑖2)(16 − 4𝑖2)

(16 − 4(−1))(16 − 4(−1))

(20)(20)

400

15

Express each of the following in simplest 𝑎 + 𝑏𝑖 form.

2. ((5 − 𝑖) − 2(1 − 3𝑖))2 3. (3 − 𝑖)(4 + 7𝑖) − ((5 − 𝑖) − 2(1 − 3𝑖))

(5 − 𝑖 − 2 + 6𝑖)2 12 + 21𝑖 − 4𝑖 − 7𝑖2 − (5 − 𝑖 − 2 + 6𝑖)

(3 + 5𝑖)2 12 + 17𝑖 + 7 − 5 + 𝑖 + 2 − 6𝑖

(3 + 5𝑖)(3 + 5𝑖) 16 + 12𝑖

9 + 15𝑖 + 15𝑖 + 25𝑖2

9 + 30𝑖 − 25

−16 + 30𝑖

Two complex numbers 𝒂 + 𝒃𝒊 and 𝒄 + 𝒅𝒊 are equal if and only if 𝒂 = 𝒄 and 𝒃 = 𝒅.

For example: Find the real values of 𝑎 and 𝑏 in each of the following equations.

4. 7 + 2𝑖 = 𝑎 + 𝑏𝑖 5. −3 + 𝑏𝑖 = 𝑎 + 8𝑖

7 = 𝑎 2𝑖 = 𝑏𝑖 −3 = 𝑎 𝑏𝑖 = 8𝑖

2 = 𝑏 𝑏 = 8

6. 4𝑖 − 6 = 𝑎 + 𝑏𝑖 7. −𝑖 + 𝑎 = 5 + 𝑏𝑖

4𝑖 = 𝑏𝑖 − 6 = 𝑎 −𝑖 = 𝑏𝑖 𝑎 = 5

4 = 𝑏 −1 = 𝑏

8. −2 − 𝑎 = 5𝑖 + 𝑏𝑖 9. 3𝑎 + 6 = 8𝑖 − 2𝑏𝑖

−2 − 5𝑖 = 𝑎 + 𝑏𝑖 3𝑎 + 2𝑏𝑖 = −6 + 8𝑖

−2 = 𝑎 − 5𝑖 = 𝑏𝑖 3𝑎 = −6 2𝑏𝑖 = 8𝑖

−5 = 𝑏 𝑎 = −2 𝑏 = 4

16

10. 2(𝑎 + 4) = (3 + 6𝑏)𝑖 11. 8 − 2𝑖 + 𝑎 − 𝑏𝑖 = 5 + 3𝑖

2𝑎 + 8 = 3𝑖 + 6𝑏𝑖 3 − 𝑏𝑖 = −𝑎 + 5𝑖

2𝑎 − 6𝑏𝑖 = −8 + 3𝑖 3 = −𝑎 − 𝑏𝑖 = 5𝑖

2𝑎 = −8 − 6𝑏𝑖 = 3𝑖 𝑎 = −3 𝑏 = −5

𝑎 = −4 𝑏 = −1

3

COMMON CORE QUESTIONS

Directions: Find the real values of 𝑥 and 𝑦 in each of the following equations

using the fact that if 𝒂 + 𝒃𝒊 = 𝒄 + 𝒅𝒊, then 𝒂 = 𝒄 and 𝒃 = 𝒅.

12. 5𝑥 + 3𝑦𝑖 = 20 + 9𝑖 13. 2(5𝑥 + 9) = (10 − 3𝑦)𝑖

5𝑥 = 20 3𝑦𝑖 = 9𝑖 10𝑥 + 18 = 10𝑖 − 3𝑦𝑖

𝑥 = 4 𝑦 = 3 10𝑥 + 3𝑦𝑖 = −18 + 10𝑖

10𝑥 = −18 3𝑦𝑖 = 10𝑖

𝑥 =−18

10=

−9

5 𝑦 =

10

3

14. 3 + 5𝑖 + 𝑥 − 𝑦𝑖 = 6 − 2𝑖 15. 𝑥 + 𝑦𝑖 = (1 − 𝑖)(2 + 8𝑖)

𝑥 − 𝑦𝑖 = 3 − 7𝑖 𝑥 + 𝑦𝑖 = 2 + 8𝑖 − 2𝑖 − 8𝑖2

𝑥 = 3 − 𝑦𝑖 = −7𝑖 𝑥 + 𝑦𝑖 = 10 + 6𝑖

𝑦 = 7 𝑥 = 10 𝑦 = 6

16. 3(7 − 2𝑥) − 5(4𝑦 − 3)𝑖 = 𝑥 − 2(1 + 𝑦)𝑖

21 − 6𝑥 − 20𝑦𝑖 + 15𝑖 = 𝑥 − 2𝑖 − 2𝑦𝑖

21 − 18𝑦𝑖 = 7𝑥 − 17𝑖

21 = 7𝑥 − 18𝑦𝑖 = −2𝑖

𝑥 = 3 𝑦 =1

9

17

Homework 4.3: Structure Questions with Complex Numbers

1. Perform the following complex calculations. Express each answer in simplest

𝑎 + 𝑏𝑖 form.

a. (13 + 4𝑖) + (7 + 5𝑖) b. (4 + 𝑖) + (2 − 𝑖) − (1 − 𝑖)

c. −𝑖(2 − 𝑖)(5 + 6𝑖)

2. Find the real values of 𝑥 and 𝑦 in each of the following equations using the

fact that if 𝒂 + 𝒃𝒊 = 𝒄 + 𝒅𝒊, then 𝒂 = 𝒄 and 𝒃 = 𝒅.

a. −10𝑥 + 12𝑖 = 20 + 3𝑦𝑖 b. 3(4𝑥 + 2) = (8 − 𝑦)𝑖

3. Express in 𝑎 + 𝑏𝑖 form: (3 − 2𝑦𝑖)(2 + 7𝑖) − [(6 + 5𝑦𝑖) + 2(3 + 4𝑖)]