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Curriculum 2.0 Algebra 2: Unit 2-Topic 1, SLT 6 Name: Operations with Complex Numbers Sample Answers Date: Period: Adding Complex Numbers Enter #1–5 on your calculator. Record the solution below and discuss with a partner how you think the calculator is adding the two complex numbers. 1. (3 + 4i) + (2 + 5i) = 5 + 9i 2. (1 – 6i) + (3 – 2i) = 4 – 8i 3. (2 + 5i) + (6 – 8i) = 8– 3i 4. (4 – 3i) + (–5 – 7i) = –1 – 10i 5. (–2 + 3i) + (1 – 2i) = –1 + i 6. Based on the equivalent expressions produced by the calculator, what process is used to add two complex numbers? Be sure to think about the Commutative, Associative, and Distributive Properties. Grouping symbols are removed and the commutative property allows me to reorder the terms. I can use the associative property to regroup terms so that the real terms are grouped together and the imaginary terms are grouped together. I then add the real terms and the imaginary terms to result in one complex number. Subtracting Complex Numbers Enter #7–12 on your calculator. Record the solution below and discuss with a partner how you think the calculator is subtracting the two complex numbers. 7. (3 + 4i) – (2 + 5i) = 1–i 8. (1 – 6i) – (3 – 2i) = –2 – 4i 9. (2 + 5i) – (6 – 8i) = –4 + 13i 10. (4 – 3i) – (–5 – 7i) = 9 + 4i 11. (–2 + 3i) – (1 – 2i) = –3 + 5i 12. Based on the equivalent expressions produced by the calculator, what process is used to subtract two complex numbers? Be sure to think about the Commutative, Associative, and Distributive Properties. The subtraction sign can be viewed as adding –1 times the second quantity. I then distribute the –1 to all terms in the second quantity. Then I follow the same process as for adding using commutative and associative properties to regroup real and imaginary terms together. Adapted from: Texas Instruments: Complex Numbers Page 1 of 3 Curriculum 2.0 Algebra 2: Unit 2-Topic 1, SLT 6 Name: Operations with Complex Numbers Sample Answers Date: Period: Multiplying Complex Numbers Enter #13–19 on your calculator. Record the solution below and discuss with a partner how you think the calculator is multiplying the two complex numbers. 13. (3 + 4i)(2 + 5i) = –14 + 23i 14. (1 – 6i)(3 – 2i) = –9 – 20i 15. Why is there not a term containing i2 in the solutions to #13 and #14? When distributing the imaginary term in the first complex number to the imaginary term in the second complex number, the product contains i 2 . But i 2 = –1, a real number, so this is regrouped with the other real number terms. 16. (2 + 5i)(6 – 8i) = 52 + 14i 17. (–2 + 3i)(1 – 2i) = 4 + 7i 18. (4 – 3i)(–5 – 7i) = –41 – 13i 19. Based on the equivalent expressions produced by the calculator, what process is used to multiply two complex numbers? Be sure to think about the Commutative, Associative, and Distributive Properties. I distribute each term in the first complex number to both of the terms in the second complex number. This results in 4 terms, two imaginary and two real that can then be regrouped with the commutative and associative properties. Use the strategies determined in #1 – 19 to add, subtract, or multiply the complex numbers below. 20. (2 4i ) (2 4i) 4 21. (2 4i ) (2 4i) –8i 22. (2 4i)(2 4i) 20 23. (7 2i ) (7 2i) 14 24. (7 2i ) (7 2i) 4i 25. (7 2i)(7 2i) 53 26. (4 4i ) (4 4i) –8 27. (4 4i ) (4 4i) 8i 28. (4 4i)(4 4i) 32 29. What do you notice? Adding and multiplying complex conjugate pairs always result in real numbers. Subtracting always results in a pure imaginary number. Adapted from: Texas Instruments: Complex Numbers Page 2 of 3 Curriculum 2.0 Algebra 2: Unit 2-Topic 1, SLT 6 Name: Operations with Complex Numbers Sample Answers Date: 30. Period: Why does this happen? Adding: Because the imaginary terms of complex conjugate pairs are always opposite, by definition, their sum is 0. This eliminates the imaginary part of the complex number, leaving only the sum of the real parts. The sum of the real parts will always be double the real part in the original complex numbers. Multiplying: Using the distributive property, multiplying complex conjugates will always result in a number of the form a2 abi abi b2i 2 . The sum of the two imaginary terms will always be 0, eliminating the imaginary part of the number. Subtracting: The real parts of complex conjugates are the same number that when subtracted results in a value of 0, leaving only the imaginary part. Adapted from: Texas Instruments: Complex Numbers Page 3 of 3