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 푖? A real number 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, integers, and certain decimals An irrational number is any number that cannot be expressed as a fraction. What irrationals do you know? 1 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 2 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 3 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 4 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 5 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 6 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 7 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. 8 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 9 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. 10 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 + 2푖 should be labeled “a”. a) 5 + 2푖 b) −2 − 4푖 c) −푖 1 d) + 푖 2 11 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? 12 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. 13 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.