Hi Everyone!

On this page you will find some material about Lesson 9. Read through the material below, watch the videos, and follow up with your instructor if you have questions.

**Lesson 9: Complex Numbers**

**Learning Outcomes.**

- Define the imaginary number $i$.
- Identify the real part and the imaginary part of a complex number.
- Add, subtract, multiply and divide complex numbers.
- Find the complex conjugate of a complex number.
- Write a complex number in standard form.

**Topic**. This lesson covers Section 6.8: Complex Numbers.

**WeBWorK**. There is one WeBWorK assignment on today’s material:

ComplexNumbers

**Lesson Notes.**

**Video Lessons. **

Video Lesson 9 – part 1 (based on Lesson 9 Notes – part 1)

Video Lesson 9 – part 2 (based on Lesson 9 Notes – part 2)

### Warmup Questions

*These are questions on fundamental concepts that you need to know before you can embark on this lesson. Don’t skip them! Take your time to do them, and check your answer by clicking on the “Show Answer” tab.*

#### Warmup Question 1

Find a real solution for

$$x^2+1=0.$$

#### Show Answer 1

There is no real number $x$ satisfying $x^2+1=0$, so there is no real solution.

#### Warmup Question 2

Multiply

$$(a-2)(a+2).$$

#### Show Answer 2

We have a difference of squares.

$$(a-2)(a+2)=a^2-4$$

#### Warmup Question 3

Rationalize

$$\dfrac{1}{2-\sqrt{3}}.$$

#### Show Answer 3

$$\dfrac{1}{2-\sqrt{3}}=\dfrac{1(2+\sqrt 3)}{(2-\sqrt{3})(2+\sqrt 3)}=\dfrac{2+\sqrt 3}{4-3}=-2-\sqrt 3$$

### Review

*If you are not comfortable with the Warmup Questions, don’t give up! Click on the indicated lesson for a quick catchup. A brief review will help you boost your confidence to start the new lesson, and that’s perfectly fine.*

Need a review? Check

### Quick Intro

*This is like a mini-lesson with an overview of the main objects of study. It will often contain a list of key words, definitions and properties – all that is new in this lesson. We will use this opportunity to make connections with other concepts. It can be also used as a review of the lesson.*

**A Quick Intro to Complex Numbers**

**Key Words.** Complex number, imaginary number, real part, imaginary part, standard form, complex conjugate.

The **imaginary number** is a number defined as $i=\sqrt{-1}$.

A **complex number **is a number that can be written as $a+bi$ where $a$ and $b$ are real numbers. In this case, the **real part **of $a+bi$ is $a$, and the **imaginary part** of $a+bi$ is $b$.

$\bigstar$ $a+bi$ is the **standard form** of the complex number.

$\bigstar$ The **complex conjugate** of $a+bi$ is $a-bi$. Multiplying a complex number by its conjugates results in a real number:

$$(a+bi)(a-bi)=a^2+b^2$$

$\bigstar$ Adding complex numbers

$$(a+bi)+(c+di) = (a+c) +(b+d)i$$

$\bigstar$ Subtracting complex numbers

$$(a+bi)-(c+di) = (a-c) +(b-d)i$$

$\bigstar$ Multiplying complex numbers

$$(a+bi)(c+di) = ac+adi+bci+bdi^2= (ac-bd) +(ad+bc)i$$

$\bigstar$ Dividing complex numbers

$$\dfrac{a+bi}{c+di} = \dfrac{(a+bi)(c-di)}{(c+di)(c-di)}=\dfrac{ac+bd}{c^2+d^2} + \dfrac{bc-ad}{c^2+d^2}i$$

### Video Lesson I

*Many times the mini-lesson will not be enough for you to start working on the problems. You need to see someone explaining the material to you. In the video you will find a variety of examples, solved step-by-step – starting from a simple one to a more complex one. Feel free to play them as many times as you need. Pause, rewind, replay, stop… follow your pace!*

**Video Lesson** **1**

**A description of the video**

In this video you will see how to solve

- $x^2=-1$
- $x^2=-4$
- $x^2=-27$

### Try Questions I

*Now that you have read the material and watched the video, it is your turn to put in practice what you have learned. We encourage you to try the Try Questions on your own. When you are done, click on the “Show answer” tab to see if you got the correct answer.*

#### Try Question 1

Solve

$$x^2+8=0.$$

#### Show Answer 1

$$x^2+8=0$$

$$x^2=-8$$

$$x=\pm\sqrt{-8}$$

$$x=\pm\sqrt 8 i$$

$$x=\pm 2\sqrt 2 i$$

### Video Lesson II

*Many times the mini-lesson will not be enough for you to start working on the problems. You need to see someone explaining the material to you. In the video you will find a variety of examples, solved step-by-step – starting from a simple one to a more complex one. Feel free to play them as many times as you need. Pause, rewind, replay, stop… follow your pace!*

**Video Lesson 2**

**A description of the video**

In this video you will see

- what a complex number is
- the standard form of a complex number
- the complex conjugate
- how to perform arithmetic operations with complex numbers, leaving the answer in standard form:
- $3i+5+3$
- $3(3i+8)$
- $\dfrac{3(i+2)}{2}$
- $(3i+2)+(-2i+3)$
- $(3i+2)-(-2i+3)$
- $i(2i-1)$
- $(3i+1)(-2i-1)$
- $(5-2i)(5+2i)$
- $\dfrac{5i+1}{2i}$
- $\dfrac{1+3i}{-3+2i}$

### Try Questions II

*Now that you have read the material and watched the video, it is your turn to put in practice what you have learned. We encourage you to try the Try Questions on your own. When you are done, click on the “Show answer” tab to see if you got the correct answer.*

#### Try Question 2

Multiply

$$(8-2i)(4+i).$$

#### Show Answer 2

$$(8-2i)(4+i) = 8\cdot 4 + 8i -8i -2i^2 = 32+2 = 34$$

#### Try Question 3

Divide

$$\dfrac{-2+i}{-1+3i}.$$

#### Show Answer 3

$$\dfrac{-2+i}{-1-3i} = \dfrac{(-2+i)(-1+3i)}{(-1-3i)(-1+3i)} = \dfrac{2-6i-i+3i^2}{1+9}$$

$$=\dfrac{-1-7i}{10} = -\dfrac{1}{10} -\dfrac{7}{10}i$$

#### Try Question 4

Write in standard form

$$\dfrac{1-3i}{2-5i}-\dfrac{1+6i}{5-i}.$$

#### Show Answer 4

$$\dfrac{1-3i}{2-5i}-\dfrac{1+6i}{5-i}= \dfrac{(1-3i)(2+5i)}{(2-5i)(2+5i)}-\dfrac{(1+6i)(5+i)}{(5-i)(5+i)}$$

$$ =\dfrac{2+5i -6i-15i^2}{4+25}-\dfrac{5 + i + 30 i +6i^2}{25+1} $$

$$= \dfrac{17-i}{29}-\dfrac{-1 + 31i}{26} = \dfrac{17}{29}-\dfrac{1}{29}i + \dfrac{1}{26}- \dfrac{31}{26}i $$

$$=\dfrac{17}{29} + \dfrac{1}{26} -\dfrac{1}{29}i – \dfrac{31}{26}i = \dfrac{471}{754}-\dfrac{925}{754}i$$

### WeBWorK

*You should now be ready to start working on the WeBWorK problems. Doing the homework is an essential part of learning. It will help you practice the lesson and reinforce your knowledge.*

**WeBWork**

It is time to do the homework on WeBWork:

ComplexNumbers

When you are done, come back to this page for the Exit Questions.

### Exit Questions

*After doing the WeBWorK problems, come back to this page. The Exit Questions include vocabulary checking and conceptual questions. Knowing the vocabulary accurately is important for us to communicate. You will also find one last problem. All these questions will give you an idea as to whether or not you have mastered the material. Remember: the “Show Answer” tab is there for you to check your work!*

#### Exit Questions

- What is an imaginary number?
- What does it mean for a complex number to be written in standard form?
- What is the complex conjugate?
- How is the arithmetic with complex numbers similar to arithmetic with square roots?
- Why is there a similarity?

$\bigstar$ Write the expressions in standard form.

(a) $i^{83}$

(b) $\dfrac{3+2i}{4+6i}$

#### Show Answer

(a) $$i^{83} = i^{4\cdot 20+3} = (i^4)^{20}\cdot i^3 = 1^{20}\cdot i^3 = i^3 = -i$$

(b) $$\dfrac{3+2i}{4+6i}= \dfrac{3+2i}{4+6i}\cdot\dfrac{4-6i}{4-6i}= \dfrac{(3+2i)(4-6i)}{(4+6i)(4-6i)}$$

$$=\dfrac{12-18i+8i-12i^2}{4^2+6^2}= \dfrac{(12+12)+(-18+8)i}{52} $$

$$= \dfrac{24-10i}{52} = \dfrac{24}{52}-\dfrac{10}{52}i= \dfrac{6}{13}-\dfrac{5}{26}i$$

### Need more help?

*Don’t wait too long to do the following.*

- Watch the additional video resources.

- Talk to your instructor.
- Form a study group.
- Visit a tutor. For more information, check the tutoring page.