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Operations with Complex Numbers

Operations with Complex Numbers. Unit 1 Lesson 2. Make Copies of:. Comparing Polynomials and Complex Numbers Graphic Organizer Kuta-Operations with Complex Numbers WS. GPS Standard. MM2N1b- Write complex numbers in the form a + bi MM2N1c- Add, subtract, multiply, and divide complex numbers

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Operations with Complex Numbers

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  1. Operations with Complex Numbers Unit 1 Lesson 2

  2. Make Copies of: • Comparing Polynomials and Complex Numbers Graphic Organizer • Kuta-Operations with Complex Numbers WS

  3. GPS Standard • MM2N1b- Write complex numbers in the form a + bi • MM2N1c- Add, subtract, multiply, and divide complex numbers • MM2N1d- Simplify expressions involving complex numbers

  4. Essential Questions • How do I add and subtract complex numbers? • How do I multiply complex numbers?

  5. COMPLEX NUMBERS Real Numbers Imaginary Numbers Rational Numbers Irrational Numbers

  6. Standard Form of a Complex Number a + bi IMAGINARY PART REAL PART

  7. Adding/Subtracting Complex Numbers • Adding and subtracting complex numbers is just like any adding/subtracting you have ever done with variables. • Simply combine like terms. • (6 + 8i) + (2 – 12i) = 8 – 4i • (7 + 4i) – (10 + 9i) = 7 + 4i – 10 – 9i = -3 – 5i

  8. To Add Complex Numbers • (a + bi) + (a + bi) • Drop the parentheses • Combine like terms • Remember: the real number comes first, then the imaginary number

  9. Examples • (3 + 5i) + (2 – 7i) • (12 – 3i) + (2 + 4i) • (13 +24i) + (17+ 5i) • (3 – 6i) + (5 – 2i) • (8 – 3i) + (4 – 11i)

  10. Test Prep Example • What is (5 – 2i) + (6 + 4i)? • -3i • 3i • 11 + 2i • 11 + 6i

  11. Test Prep Example • Perform the indicated operation. (2 + 3i) +(13 – 2i) = • A) 15 + 5iB) 15 + iC) 11 – 5iD) -11 – i

  12. To Subtract Complex Numbers: • (a + bi) – (a + bi) • Change the minus sign to plus • Change the sign of each term in the second set of parentheses • Drop parentheses • Combine like terms • Remember: real number comes first, then imaginary number

  13. Examples • (6 + 7i) – (4 + 3i) • (8 + 2i) – (3 – 7i) • (12 – 7i) – (2 + 6i) • (3 – 8i) – (7 – 11i)

  14. Test Prep Example • Perform the indicated operation. (-9 + 2i) – (-12 + 4i) = • -21 – 6i • -3 + 6i • 3 – 2i • 21 + 2i

  15. Multiplying Complex Numbers • This will be FOIL method with a slight twist at the end. • An i2 will ALWAYS show up. You will have to adjust for this. • (4 + 9i)(2 + 3i) = 8 + 12i + 18i + 27i2 = 8 + 30i – 27 = -19 + 30i • (7 – 3i)(6 + 8i) = 42 + 56i – 18i – 24i2 = 42 + 38i + 24 = 66 + 38i

  16. Examples

  17. Test Prep Examples • 1. (5 – 3i)(6 + 2i)Multiply and simplify. • A) 24 – 8i • B) 36 – 8i • C) 36 + 8i • D) 24 + 8i

  18. Binomial Squares and Complex Numbers • You can still do the five-step shortcut, or you can continue to do FOIL. • You will still have to adjust for the i2 that will show up. • (7 + 3i)2 = 49 + 42i + 9i2 = 49 + 42i – 9 = 40 + 21i • (8 – 9i)2 = 64 – 144i + 81i2 = 64 – 144i – 81= -17 – 144i

  19. Example

  20. Test Prep Example • Which has the same value as (4 + 3i)2 ? • 7 • 7 + 24i • 25 • 25 + 24i

  21. D2S and Complex Numbers • Situations that in the real numbers would have been differences of two squares (D2S) demonstrate in the complex numbers what are known as conjugates. • (3 + 4i)(3 – 4i) = (3)2 – (4i)2 = 9 – 16i2 = 9 + 16 = 25 • When conjugates are used, there will be no i in the answer.

  22. Examples

  23. Test Prep Example • 2.) Perform the indicated operation. (4 – 7i)(4 + 7i) = • -33 • 16 – 49i • 16 – 105i • 65

  24. Test Prep Example • What is the square of 4 – 7i? • A) 33 – 56i • B) -33 – 56i • C) -33 + 56i • D) 33 + 56i

  25. Test Prep Example • Which is equivalent to (3 + 2i)(2 + 5i)? • A) -4 + 19i • B) 16 + 19i • C) 6 + 29i

  26. Test Prep Example • What is a if a + bi = (2 – i)2 • A) a = 3 • B) a = 5 • C) a = 2 • D) a = 1

  27. Test Prep Example • Simplify: -10 + √-16 2 • A) -5 + 2i • B) -5 – 4i • C) 20 + 4i • D) 30 + 2i

  28. Test Prep Example • Perform the indicated operation. • (3 – 8i)(4 + i) = • A) 4 • B) 20 – 29i • C) 12 – 8i • D) 5 + 35i

  29. Test Prep Example • Multiply 2i(i – 2) over the set of complex numbers. • A) 0 • B) 2 – 4i • C) -2 – 4i • D) 2 + 4i

  30. Graphic Organizer • Comparing Polynomials and Complex Numbers.doc

  31. Assignment • Kuta-Operations with Complex Numbers.pdf

  32. Support Assignment • Pg 8: 1-27 • Pg. 13: 1 - 26

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