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Lesson 8 NYS COMMON CORE MATHEMATICS CURRICULUM M1 PRECALCULUS AND ADVANCED TOPICS Lesson 8: Complex Number Division Classwork Opening Exercises Use the general formula to find the multiplicative inverse of each complex number. a. 2 + 3π b. β7 β 4π c. β4 + 5π Exercises 1β4 Find the conjugate, and plot the complex number and its conjugate in the complex plane. Label the conjugate with a prime symbol. 1. π΄: 3 + 4π 2. π΅: β2 β π 3. πΆ: 7 4. π·: 4π Lesson 8: Date: Complex Number Division 5/11/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.28 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 M1 PRECALCULUS AND ADVANCED TOPICS Exercises 5β8 Find the modulus. 5. 3 + 4π 6. β2 β π 7. 7 8. 4π Exercises 9β11 Given π§ = π + ππ. 9. Show that for all complex numbers π§, |ππ§| = |π§|. Lesson 8: Date: Complex Number Division 5/11/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.29 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 M1 PRECALCULUS AND ADVANCED TOPICS 10. Show that for all complex numbers π§, π§ β π§Μ = |π§|2 . 11. Explain the following: Every nonzero complex number π§ has a multiplicative inverse. It is given by 1 π§Μ = | |. π§ π§ Example 1 2 β 6π 2 + 5π Exercises 12β13 Divide. 12. 3 + 2π β2 β 7π 13. 3 3βπ Lesson 8: Date: Complex Number Division 5/11/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.30 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 8 NYS COMMON CORE MATHEMATICS CURRICULUM M1 PRECALCULUS AND ADVANCED TOPICS Problem Set 1. 2. Let π§ = 4 β 3π and π€ = 2 β π. Show that a. Μ | |π§| = |π§ b. | | = | Μ | c. If |π§| = 0, must it be that π§ = 0? d. Give a specific example to show that |π§ + π€| usually does not equal |π§| + |π€|. 1 π§ 1 π§ Divide. a. 1 β 2π 2π b. 5 β 2π 5 + 2π c. β3 β 2π β2 β β3π 3. Prove that |π§π€| = |π§| β |π€| for complex numbers π§ and π€. 4. Given π§ = 3 + π, π€ = 1 + 3. 5. a. Find π§ + π€, and graph π§, π€, and π§ + π€ on the same complex plane. Explain what you discover if you draw line segments from the origin to those points π§,π€, and π§ + π€. Then draw line segments to connect π€ to π§ + π€, and π§ + π€ to π§. b. Find βπ€, and graph π§, π€, and π§ β π€ on the same complex plane. Explain what you discover if you draw line segments from the origin to those points π§, π€, and π§ β π€. Then draw line segments to connect π€ to π§ β π€, and π§ β π€ to π§. Explain why |π§ + π€| β€ |π§| + |π€| and |π§ β π€| β€ |π§| + |π€| geometrically. (Hint: Triangle inequality theorem) Lesson 8: Date: Complex Number Division 5/11/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.31 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.