NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M3 PRECALCULUS AND ADVANCED TOPICS Lesson 2: Does Every Complex Number Have a Square Root? Student Outcomes . Students apply their understanding of polynomial identities that have been extended to the complex numbers to find the square roots of complex numbers. Lesson Notes In Precalculus Module 1, students used the polar form of a complex number to find powers and roots of complex numbers. However, nearly all of the examples used in those lessons involved complex numbers with arguments that 휋 휋 휋 were multiples of special angles ( , , and ). In this lesson, we return to the rectangular form of a complex number to 4 3 2 show algebraically that we can find the square roots of any complex number without having to express it first in polar form. Students use the properties of complex numbers and the fundamental theorem of algebra to find the square roots of any complex number by creating and solving polynomial equations (N-NC.C.8 and N-NC.C.9). While solving these equations, students see that we arrive at polynomial identities with two factors that guarantee two roots to the equation. Throughout the lesson, students use algebraic properties to justify their reasoning (MP.3) and examine the structure of expressions to support their solutions and make generalizations (MP.7 and MP.8). Classwork Opening (5 minutes) Scaffolding: Organize the class into groups of 3–5 students. Start by displaying the . Create an anchor chart with the question shown below. Give students time to consider an answer to this following formulas: question on their own, and then allow them to discuss it in small groups. For any complex number 푧, Have a representative from each group briefly summarize their small group 푧푛 = 푟푛(cos(푛휃) + 푖 sin(푛휃)). discussions. The 푛th roots of 푧 = 푟푒푖휃 are given by . Does every complex number have a square root? If yes, provide at 푛 휃 2휋푘 휃 2휋푘 least two examples. If no, explain why not. 푟 (cos ( + ) + 푖 sin ( + )) √ 푛 푛 푛 푛 If we think about transformations, then squaring a complex for integers 푘 and 푛 such that 푛 > 0 number dilates a complex number by the modulus and and 0 ≤ 푘 < 푛. rotates it by the argument. Thus, taking the square root of a complex number should divide the argument by 2 and . Ask students to explain the formulas have a modulus equal to the square root of the original above for given values of 푛, 푟, and 휃. 휋 modulus. For example, substitute for 휃, 1 for 푟, 2 For example, the complex number 푖 would have a square and 2 for 푛 into the formula for the 푛th root with modulus equal to 1 and argument equal to 45°, roots of 푧, and ask students what the √2 √2 expressions represent (the square root which is + 푖. 2 2 of 푖). Lesson 2: Does Every Complex Number Have a Square Root? 27 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from PreCal-M3-TE-1.3.0-08.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M3 PRECALCULUS AND ADVANCED TOPICS Student discussions and responses should reveal how much they recall from their work in Module 1 Lessons 18 and 19. In those lessons, students used the polar form of a complex number to find the 푛th roots of a complex number in polar form. Based on student responses to this question, the teacher may need to briefly review the polar form of a complex number and the formulas developed in Lessons 18 and 19 for finding powers of a complex number 푧 and roots of a complex number 푧. Exercises 1–6 (10 minutes) Students should work these exercises with their group. As students work, be sure to circulate around the classroom to monitor the groups’ progress. Pause as needed for whole-group discussion and debriefing, especially if a majority of the groups are struggling to make progress. Exercises 1–6 1. Use the geometric effect of complex multiplication to describe how to calculate a square root of 풛 = ퟏퟏퟗ + ퟏퟐퟎ풊. The square root of this number would have a modulus equal to the square root of |ퟏퟏퟗ + ퟏퟐퟎ풊| and an argument 퐚퐫퐠 (ퟏퟏퟗ+ퟏퟐퟎ풊) equal to . ퟐ 2. Calculate an estimate of a square root of ퟏퟏퟗ + ퟏퟐퟎ풊. |풛| = √ퟏퟏퟗퟐ + ퟏퟐퟎퟐ = ퟏퟔퟗ ퟏퟐퟎ 퐚퐫퐠(풛) = 퐚퐫퐜퐭퐚퐧 ( ) ퟏퟏퟗ ퟏ ퟏퟐퟎ The square root’s modulus is √ퟏퟔퟗ or ퟏퟑ, and the square root’s argument is 퐚퐫퐜퐭퐚퐧 ( ). The square root is ퟐ ퟏퟏퟗ close to ퟏퟑ(퐜퐨퐬(ퟐퟐ. ퟔ°) + 풊 퐬퐢퐧(ퟐퟐ. ퟔ°)) or ퟏퟐ + ퟒ. ퟗퟗ풊. If students get stuck on the next exercises, lead a short discussion to help set the stage for establishing that every complex number has two square roots that are opposites of each other. What are the square roots of 4? The square roots of 4 are 2 and −2 because (2)2 = 4 and (−2)2 = 4. What are the square roots of 5? 2 5 The square roots of 5 are √5 and −√5 because (√5) = 5 and (−√5) = 5. 3. Every real number has two square roots. Explain why. The fundamental theorem of algebra guarantees that a second-degree polynomial equation has two solutions. To find the square roots of a real number 풃, we need to solve the equation 풛ퟐ = 풃, which is a second-degree polynomial equation. Thus, the two solutions of 풛ퟐ = 풃 are the two square roots of 풃. If 풂 is one of the square roots, then – 풂 is MP.3 the other. 4. Provide a convincing argument that every complex number must also have two square roots. By the same reasoning, if 풘 is a complex number, then the polynomial equation 풛ퟐ = 풘 has two solutions. The two solutions to this quadratic equation are the square roots of 풘. If 풂 + 풃풊 is one square root, then 풂 − 풃풊 is the other. Lesson 2: Does Every Complex Number Have a Square Root? 28 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from PreCal-M3-TE-1.3.0-08.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M3 PRECALCULUS AND ADVANCED TOPICS 5. Explain how the polynomial identity 풙ퟐ − 풃 = (풙 − √풃)(풙 + √풃) relates to the argument that every number has two square roots. To solve 풙ퟐ = 풃, we can solve 풙ퟐ − 풃 = ퟎ. Since this quadratic equation has two distinct solutions, we can find two square roots of 풃. The two square roots are opposites of each other. MP.3 6. What is the other square root of ퟏퟏퟗ + ퟏퟐퟎ풊? It would be the opposite of ퟏퟐ + ퟓ풊, which is the complex number, −ퟏퟐ − ퟓ풊. Example 1 (10 minutes): Find the Square Roots of ퟏퟏퟗ + ퟏퟐퟎ풊 The problem with using the polar form of a complex number to find its square roots is Scaffolding: that the argument of these numbers is not an easily recognizable number unless we . Students may benefit from pick our values of 푎 and 푏 very carefully, such as 1 + √3푖. practice and review of Recall from the last lesson that we proved using complex numbers that the equation factoring fourth-degree 푥2 = 1 had exactly two solutions. Here is another approach to finding both square polynomials. See Algebra II roots of a complex number that involves creating and solving a system of equations. Module 1. Some sample The solutions to these equations provide a way to define the square roots of a problems are provided below. complex number. Students have to solve a fourth-degree polynomial equation that 푥4 − 2푥2 − 8 has both real and imaginary solutions by factoring using polynomial identities. 푥4 − 9푥2 − 112 푥4 − 푥2 − 12 Example: Find the Square Roots of ퟏퟏퟗ + ퟏퟐퟎ풊 . Post the following identities on Let 풘 = 풑 + 풒풊 be the square root of ퟏퟏퟗ + ퟏퟐퟎ풊. Then the board: 2 2 풘ퟐ = ퟏퟏퟗ + ퟏퟐퟎ풊 푎 − 푏 = (푎 + 푏)(푎 − 푏) 푎2 + 푏2 = (푎 + 푏푖)(푎 − 푏푖). and . Students could use technology ퟐ (풑 + 풒풊) = ퟏퟏퟗ + ퟏퟐퟎ풊. like Desmos to quickly find real a. Expand the left side of this equation. number solutions to equations like those in part (b) by 풑ퟐ − 풒ퟐ + ퟐ풑풒풊 = ퟏퟏퟗ + ퟏퟐퟎ풊 graphing each side and finding the 푥-coordinates of the b. Equate the real and imaginary parts, and solve for 풑 and 풒. intersection points. 풑ퟐ − 풒ퟐ = ퟏퟏퟗ and ퟐ풑풒 = ퟏퟐퟎ. Solving for 풒 and substituting gives ퟔퟎ ퟐ 풑ퟐ − ( ) = ퟏퟏퟗ. 풑 Multiplying by 풑ퟐ gives the equation 풑ퟒ − ퟏퟏퟗ풑ퟐ − ퟑퟔퟎퟎ = ퟎ. And this equation can be solved by factoring. (풑ퟐ + ퟐퟓ)(풑ퟐ − ퟏퟒퟒ) = ퟎ We now have two polynomial expressions that we know how to factor: the sum and difference of squares. (풑 + ퟓ풊)(풑 − ퟓ풊)(풑 − ퟏퟐ)(풑 + ퟏퟐ) = ퟎ The solutions are ퟓ풊, −ퟓ풊, ퟏퟐ, and −ퟏퟐ. Since 풑 must be a real number by the definition of complex number, we can choose ퟏퟐ or −ퟏퟐ for 풑. Using the equation ퟐ풑풒 = ퟏퟐퟎ, when 풑 = ퟏퟐ, 풒 = ퟓ, and when 풑 = −ퟏퟐ, 풒 = −ퟓ. Lesson 2: Does Every Complex Number Have a Square Root? 29 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from PreCal-M3-TE-1.3.0-08.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.