Unit 3 Imaginary Numbers

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.

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.

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

Class Exercises: Simplify.   1. − 4 =__________ 2. − 5 =__________ 3. − 27 =__________ 4. 5 − 49 =__________ 5. − 54 =__________ 6. 3 − 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

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 =__________ = − 1 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 2 2 ∙𝑖 = − 1 2 ∙𝑖 𝑖 2 3 = − 1 3 =1∙𝑖 𝑖 7 =__________ − 𝒊 𝑖 8 =__________ 𝟏 𝑖 6 ∙𝑖 = 𝑖 2 3 ∙𝑖 = − 1 3 ∙𝑖 𝑖 2 4 = − 1 4 = − 1 ∙𝑖

Class Exercises: Simplify.   7. 𝑖 31 =________ 8. 𝑖 40 =________ 9. 𝑖 50 =________ 10. 𝑖 2125 =________ − 𝒊 𝟏 𝑖 2 20 = − 1 20 𝑖 30 ∙𝑖 = 𝑖 2 15 ∙𝑖 = − 1 15 ∙𝑖 = − 1 ∙𝑖 − 𝟏 𝒊 𝑖 2 25 = − 1 25 𝑖 2124 ∙𝑖 𝑖 2 1062 ∙𝑖 − 1 1062 ∙𝑖 1∙𝑖

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.

Class Exercises: Add/Subtract. 11. − 4 + − 9 =__________ 12. − 6 − − 12 =__________ 13. 2 − 3 + − 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

Assignment WS 1

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

Objectives: Multiply/divide imaginary numbers.

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.

Class Exercises: Simplify.   1. − 4 ∙ − 9 =__________ 2. − 2 ∙ − 8 =__________ 3. − 6 ∙ − 12 =__________ 4. 2 − 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 45 2 𝑖 2 18 6∙ − 1 ∙ 9 ∙ 5 2∙ − 1 ∙ 9 ∙ 2 6∙ − 1 ∙3∙ 5 2∙ − 1 ∙3∙ 2

Class Exercises: Simplify. 5. 8𝑖 2𝑖 =__________ 6. 25 𝑖 13 5𝑖 =__________ 7. 4 − 3 =__________ 8. 3 5 − 6 =__________ 𝟒 𝟓 5 𝑖 12 5 𝑖 2 6 5 − 1 6 =5∙1 − 𝟒𝒊 𝟑 𝟑 − 𝟑𝒊 𝟑𝟎 𝟔 3 5 − 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 3 − 1 ∙3 = 4𝑖 3 − 3 3𝑖 30 𝑖 2 36 = 3𝑖 30 − 1 ∙6 = 3𝑖 30 − 6 ∙ − 1 − 1 ∙ − 1 − 1

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

Assignment WS 2

Warm-up: Simplify. 𝑖 4𝑛+1 𝑖 4𝑛 ∙ 𝑖 1 𝑖 𝑛 4 ∙ 𝑖 Answer: _____ 𝒊 1 ∙ 𝑖 𝑖 0 4 = 1 4 = 1 𝑖 1 4 = 𝑖 4 = 𝑖 4 = 𝑖 2 2 = − 1 2 = 1 𝑖 2 4 = − 1 4 = 1 𝑖 3 4 = 𝑖 12 = 𝑖 2 6 = − 1 6 = 1 𝑖 4 4 = 𝑖 16 = 𝑖 2 8 = − 1 8 = 1 𝑖 5 4 = 𝑖 20 = 𝑖 2 10 = − 1 10 = 1

Objectives: Add/Subtract Complex Numbers.

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

Class Exercises: Identify 𝑎 and 𝑏 for each of the following.  1. 5−7𝑖 2. 5𝑖 𝑎=_______ 𝑎=_______ 𝑏=_______ 𝑏=_______ 3. 4 4. − 9 5+ − 7𝑖 0+5𝑖 𝟓 𝟎 − 𝟕 𝟓 = − 1 ∙ 9 4+0𝑖 3𝑖 𝑖 ∙ 3 0+3𝑖 𝟒 𝟎 𝟎 𝟑

Adding/Subtracting Imaginary Numbers: Add the real with the real and imaginary with the imaginary. 3+4𝑖 + 7+5𝑖 3+7 + 4𝑖+5𝑖 10 + 9𝑖 𝟏𝟎+𝟗𝒊

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

Class Exercises: Simplify. 9. 2+ − 3 + 9−2 − 3 =_______________ 10. − 6 + − 4 − − 5 −2 − 4 =_______________ 𝟏𝟏−𝒊 𝟑 2+9 + − 3 −2 − 3 11 + − − 3 11 + − − 1 ∙ 3 11 + − 𝑖 3 − 𝟏 +𝟔𝒊 − 6 + − 4 + 5+2 − 4 − 6 +5 + − 4 +2 − 4 − 1 + 3 − 4 − 1 + 3 − 1 ∙ 4 − 1 + 3𝑖∙2

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.

Class Exercises: Simplify.   11. 6+9𝑖 =_______________ 12. 5−4𝑖 =_______________ 13. 1+ − 5 =_______________ 𝟑 𝟏𝟑 𝑎=6 𝑏=9 6 2 + 9 2 = 36+81 = 117 = 9 ∙ 13 =3∙ 13 𝟒𝟏 𝑎=5 𝑏= − 4 5 2 + − 4 2 = 25+16 = 41 𝟔 𝑏= 5 𝑎=1 1+𝑖 5 1 2 + 5 2 = 1+5 = 6

Assignment WS 3

Warm-up: Simplify. 1 2 + 1 2 𝑖 Answer: 1 2 2 + 1 2 2 1 4 + 1 4 1 2 + 1 2 1 =𝟏

Objectives: Identify Complex Conjugate Numbers. Multiply Complex Numbers.

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.

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 ____________

Class Exercises: Multiply the complex numbers.   5. 2+5𝑖 3+7𝑖 =_______________ 6. − 4 −6𝑖 2−4𝑖 =_______________ 7. 6+2𝑖 2 =_______________ 8. 1+ − 3 4− − 3 =_______________ 9. 4+3𝑖 4−3𝑖 =_______________ 10. 6−2𝑖 6+2𝑖 =_______________ 11. 1+ − 3 6 1− − 3 6 =_______________ 12. − 5 − − 8 − 5 + − 8 =_______________

Assignment WS 4

Warm-up: Simplify. 1 3 + 2 3 𝑖 Answer: 1 3 2 + 2 3 2 1 9 + 4 9 1 3 + 3 3 1 =𝟏

Objectives: Divide Complex Numbers.

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.

Class Exercises: Divide the complex numbers.   1. 1+2𝑖 2+3𝑖 =_______________ 2. 4+3𝑖 1−2𝑖 =_______________ 3. 3− − 16 2+ − 25 =_______________ 4. 12 3 +7𝑖 3 −7𝑖 =_______________

Assignment WS 5