1. Unit 3 Imaginary Numbers

2. Warm-up: Explain why the − 4 can not be simplified.
Answer:
Since the index is 2 that means the radicand − 𝟒 has to be written as the product of 2 equal factors. Knowing that the product of 2 positive or 2 negative numbers is a positive real number makes the simplifying − 𝟒 impossible.

3. Objectives:
Identify pure imaginary numbers.
Identify the real number coefficient of a pure imaginary number.
Simplify powers of 𝑖 .
Simplify square roots of negative numbers and add/subtract imaginary numbers.

4. Imaginary Number 𝒊 : It is necessary to expand the number system to be able to solve equations and simplify radical expressions whose index is even and radicand is negative, such as, − 4 .
𝑖= − 1
− 𝑟
= − 1 ∙ 𝑟
=𝑖 𝑟
Pure Imaginary Number 𝒃𝒊 : The coefficient 𝑏 is a real number.
Example: 7𝑖
The coefficient is 7

5. Class Exercises: Simplify.
− 4 =__________ − 5 =__________
− 27 =__________ − 49 =__________
− 54 =__________ − 63 =__________ 𝟐𝒊
𝒊 𝟓
− 1 ∙ 4
− 1 ∙ 5 𝑖 ∙ 2 𝑖 ∙ 5
𝟑𝒊 𝟑
𝟑𝟓𝒊
− 1 ∙ 9 ∙ 3
5∙ − 1 ∙ 49 𝑖 ∙ 3 ∙ 3 5 ∙ 𝑖 ∙ 7
𝟑𝒊 𝟔
𝟗𝒊 𝟕
− 1 ∙ 9 ∙ 6
3∙ − 1 ∙ 9 ∙ 7 𝑖 ∙ 3 ∙ 6 3 ∙ 𝑖 ∙ 3 ∙ 7

6. Simplifying Powers of 𝒊 : Use the properties of exponents to rewrite the expression as a product of powers and simplify. 𝒊
− 𝟏
𝑖 1 =__________
𝑖 2 =__________
= −
𝑖∙𝑖
= − 1 ∙ − 1
𝑖 3 =__________
− 𝒊
𝑖 4 =__________ 𝟏
𝑖 2 ∙𝑖
= − 1 ∙𝑖
𝑖 2 2
= − 1 2
𝑖 5 =__________ 𝒊
𝑖 6 =__________
− 𝟏
𝑖 4 ∙𝑖
= 𝑖 ∙𝑖
= − ∙𝑖
𝑖 2 3
= − 1 3
=1∙𝑖
𝑖 7 =__________
− 𝒊
𝑖 8 =__________ 𝟏
𝑖 6 ∙𝑖
= 𝑖 ∙𝑖
= − ∙𝑖
𝑖 2 4
= − 1 4
= − 1 ∙𝑖

7. Class Exercises: Simplify.
7. 𝑖 31 =________ 𝑖 40 =________
9. 𝑖 50 =________ 𝑖 2125 =________
− 𝒊 𝟏 𝑖
= −
𝑖 30 ∙𝑖
= 𝑖 ∙𝑖
= − ∙𝑖
= − 1 ∙𝑖
− 𝟏 𝒊 𝑖
= −
𝑖 2124 ∙𝑖
𝑖 ∙𝑖
− ∙𝑖
1∙𝑖

8. Adding/Subtracting Square Roots of Negative Numbers:
1. Write the numbers in terms of 𝒊.
2. Simplify the radical expressions.
3. Add/subtract the coefficient.
The commutative, associative, and distributive properties hold for pure imaginary numbers.

9. Class Exercises: Add/Subtract.
− − 9 =__________ − 6 − − 12 =__________
− − 27 −4𝑖=__________ 𝟓𝒊
𝒊 𝟔 −𝟐𝒊 𝟑
− 1 ∙ 4
+ − 1 ∙ 9
− 1 ∙ 6
− − 1 ∙ 12 𝑖 ∙ 2 + 𝑖 ∙ 3 𝑖 ∙ 6
− − 1 ∙ 4 ∙ 3 2𝑖 + 3𝑖
𝑖 6 − 𝑖 ∙ 2 ∙ 3 −
2𝑖 3
𝟓𝒊 𝟑 −𝟒𝒊
2∙ − 1 ∙ 3
+ − 1 ∙ 27 2 ∙ 𝑖 ∙ 3
+ − 1 ∙ 9 ∙ 3
2𝑖 3 + 𝑖 ∙ 3 ∙ 3 +
3𝑖 3
−4𝑖
5𝑖 3
−4𝑖

10. Assignment
WS 1

11. Warm-up: Simplify −
Answer:
4 ∙ −
2𝑖 23
2 23 ∙ 𝑖 23
8,388,608∙ 𝑖 22 ∙𝑖
8,388,608∙ 𝑖 ∙𝑖
8,388,608∙ − ∙𝑖
8,388,608∙ − 1 ∙𝑖
− 𝟖,𝟑𝟖𝟖,𝟔𝟎𝟖 𝒊

12. Objectives:
Multiply/divide imaginary numbers.

13. Multiplying/Dividing Imaginary Numbers: 1
Multiplying/Dividing Imaginary Numbers: 1. Always express square roots of negative numbers in terms of 𝑖 before multiplying or dividing. 2. Follow procedure to multiply/divide radical and pure imaginary expressions. 3. Rationalize the denominator from radical/pure imaginary expressions, if needed.

14. Class Exercises: Simplify.
− 4 ∙ − 9 =__________ − 2 ∙ − 8 =__________
− 6 ∙ − 12 =__________ − 3 ∙3 − 15 =_________
− 𝟔
− 𝟒
− 1 ∙ 4
∙ − 1 ∙ 9
− 1 ∙ 2
∙ − 1 ∙ 8 𝑖 ∙ 𝑖 ∙ 2
∙ − 1 ∙ 4 ∙ 2 2 ∙ 𝑖 ∙ 3
6 𝑖 2
𝑖 2 ∙ 𝑖 ∙ 2 ∙ 2
6∙ − 1
2 𝑖 2 4
2∙ − 1 ∙2
− 𝟔 𝟐
− 𝟏𝟖 𝟓
− 1 ∙ 6
∙ − 1 ∙ 12
2∙ − 1 ∙ 3
∙3∙ − 1 ∙ 15 15 𝑖 ∙ 6
∙ − 1 ∙ 4 ∙ 3 2 ∙ 𝑖 ∙ 3 ∙ 3 ∙ 𝑖 ∙
𝑖 6 ∙ 𝑖 ∙ 2 ∙ 3
6 𝑖
2 𝑖
6∙ − 1 ∙ 9 ∙ 5
2∙ − 1 ∙ 9 ∙ 2
6∙ − 1 ∙3∙ 5
2∙ − 1 ∙3∙ 2

15. Class Exercises: Simplify.
5. 8𝑖 2𝑖 =__________ 𝑖 13 5𝑖 =__________
− 3 =__________ − 6 =__________ 𝟒 𝟓
5 𝑖 12
5 𝑖 2 6
5 − 1 6
=5∙1
− 𝟒𝒊 𝟑 𝟑
− 𝟑𝒊 𝟑𝟎 𝟔
− 1 ∙ 6
4 − 1 ∙ 3
3 5 𝑖 6
∙ 𝑖 6 𝑖 6
∙ 𝑖 3 𝑖 3
4 𝑖 3
4𝑖 3 𝑖 2 9
= 4𝑖 − 1 ∙3
= 4𝑖 − 3
3𝑖 𝑖
= 3𝑖 − 1 ∙6
= 3𝑖 − 6
∙ − 1 − 1
∙ − 1 − 1

16. Class Exercises: Simplify. 9. − 𝑖 3 2 − 8 =__________
− 𝑖 − 8 =__________
𝟐 𝟖
− 𝑖 3 2∙ − 1 ∙ 4 ∙ 2
− 𝑖 3 2∙𝑖∙2∙ 2
− 𝑖 3 4𝑖 2 ∙ =
= ∙2
− 𝑖
= − − =

17. Assignment
WS 2

18. Warm-up: Simplify. 𝑖 4𝑛+1
𝑖 4𝑛 ∙ 𝑖 1
𝑖 𝑛 4 ∙ 𝑖
Answer: _____ 𝒊 1 ∙ 𝑖
𝑖 =
1 4 = 1
𝑖 =
𝑖 4 =
𝑖 4 =
𝑖 =
− = 1
𝑖 =
− = 1
𝑖 =
𝑖 12 =
𝑖 =
− = 1
𝑖 =
𝑖 16 =
𝑖 =
− = 1
𝑖 =
𝑖 20 =
𝑖 =
− = 1

19. Objectives:
Add/Subtract Complex Numbers.

20. Complex Number 𝒂+𝒃𝒊 : Is a combination of a real number and a pure imaginary number.
𝑎 is the real number.
𝑏𝑖 is the pure imaginary number.
3+4𝑖
𝑎 =3
𝑏𝑖 =4𝑖
What is the coefficient of 4𝑖? 4

21. Class Exercises: Identify 𝑎 and 𝑏 for each of the following. 
1. 5−7𝑖 𝑖
𝑎=_______ 𝑎=_______
𝑏=_______ 𝑏=_______
− 9
5+ − 7𝑖
0+5𝑖 𝟓 𝟎
− 𝟕 𝟓
= − 1 ∙ 9
4+0𝑖 3𝑖 𝑖 ∙ 3
0+3𝑖 𝟒 𝟎 𝟎 𝟑

22. Adding/Subtracting Imaginary Numbers: Add the real with the real and imaginary with the imaginary.
3+4𝑖 𝑖
𝑖+5𝑖 𝑖
𝟏𝟎+𝟗𝒊

23. Class Exercises: Simplify.
𝑖 + − 2 +9𝑖 =_______________
6. − 1 +7𝑖 + 3−2𝑖 =_______________   
7. 6−11𝑖 − 6−4𝑖 =_______________
𝑖 − − 10 +8𝑖 =_______________
𝟑+𝟏𝟓𝒊
5+ − 𝑖+9𝑖
𝟐+𝟓𝒊
− 𝑖−2𝑖
− 𝟕𝒊
6−11𝑖 + − 6 +4𝑖
6+ − − 11𝑖 +4𝑖
0 + − 7𝑖
𝟏𝟒+ − 𝟑𝒊
4+5𝑖 + 10−8𝑖
𝑖−8𝑖

24. Class Exercises: Simplify.
− −2 − 3 =_______________
− − 4 − − 5 −2 − 4 =_______________
𝟏𝟏−𝒊 𝟑
− 3 −2 − 3
11 + − − 3
11 + − − 1 ∙ 3
− 𝑖 3
− 𝟏 +𝟔𝒊
− − − 4
− − − 4
− − 4
− − 1 ∙ 4
− 𝑖∙2

25. Absolute Value of a Complex Number: The absolute vale of a complex number is a real number.
To Evaluate the Absolute Value of a Complex Number:
1. Identify 𝑎 and 𝑏 .
2. Substitute the values for 𝑎 and 𝑏 into the expression 𝑎+𝑏𝑖 = 𝑎 2 + 𝑏 2 and simplify.

26. Class Exercises: Simplify.
𝑖 =_______________
12. 5−4𝑖 =_______________
− 5 =_______________
𝟑 𝟏𝟑
𝑎=6
𝑏=9 =
= 117
= 9 ∙ 13
=3∙ 13 𝟒𝟏
𝑎=5
𝑏= − 4
− 4 2 =
= 41 𝟔
𝑏= 5
𝑎=1
1+𝑖 5
= 1+5
= 6

27. Assignment
WS 3

28. Warm-up: Simplify 𝑖
Answer: 1 =𝟏

29. Objectives:
Identify Complex Conjugate Numbers.
Multiply Complex Numbers.

30. Complex Conjugate Numbers: Two complex numbers whose first terms are the same and whose second terms are opposites. The product of complex conjugates is a real number.
𝑎+𝑏𝑖
𝑎−𝑏𝑖
Multiplying Complex Numbers:
1. FOIL.
2. Simplify.

31. Class Exercises: Determine the complex conjugate for each of the given complex numbers.
1. 4+3𝑖 ____________ 2. 6−2𝑖 ____________
3. − 3 +4𝑖 ____________ 4. 1− − 25 ____________

32. Class Exercises: Multiply the complex numbers.
𝑖 3+7𝑖 =_______________
6. − 4 −6𝑖 2−4𝑖 =_______________
𝑖 2 =_______________
− − − 3 =_______________
𝑖 4−3𝑖 =_______________
10. 6−2𝑖 6+2𝑖 =_______________
− − − 3 6 =_______________
− 5 − − − − 8 =_______________

33. Assignment
WS 4

34. Warm-up: Simplify 𝑖
Answer: 1 =𝟏

35. Objectives:
Divide Complex Numbers.

36. Dividing Complex Numbers:
1. Rationalize the denominator by multiplying by 1 in the form of 𝑝 𝑝 , where 𝑝 is the complex conjugate of the denominator.
2. FOIL the numerator and denominator.
3. Simplify.

37. Class Exercises: Divide the complex numbers.
𝑖 2+3𝑖 =_______________ 𝑖 1−2𝑖 =_______________
3. 3− − − 25 =_______________ 𝑖 3 −7𝑖 =_______________

38. Assignment
WS 5