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5-9 Complex Numbers TEKS FOCUS VOCABULARY ĚAbsolute value of a complex number – TEKS (7)(A) Add, subtract, and multiply complex numbers. The absolute value of a complex number is its distance from the origin on the complex number plane. TEKS (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas. ĚComplex conjugates – number pairs of the form a + bi and a - bi ĚComplex number – Complex numbers are the real numbers and imaginary numbers. Additional TEKS (1)(D), (4)(F), (7)(B) ĚComplex number plane – The complex number plane is identical to the coordinate plane, except each ordered pair (a, b) represents the complex number a + bi. ĚImaginary number – any number of the form a + bi, where a and b are real numbers and b ≠ 0 ĚImaginary unit – The imaginary unit i is the complex number whose square is -1. ĚPure imaginary number – If a = 0 and b ≠ 0, the number a + bi is a pure imaginary number. ĚAnalyze – closely examine objects, ideas, or relationships to learn more about their nature ESSENTIAL UNDERSTANDING r The complex numbers are based on a number whose square is -1. r Every quadratic equation has complex number solutions (that sometimes are real numbers). Key Concept Example 1-5 = i15 # # Lesson 5-9 Complex Numbers Note ( 1-5)2 = (i15)2 = i 2( 15)2 = -1 5 = -5 (not 5) # Complex Numbers You can write a complex number in the form a + bi, where a and b are real numbers. If b = 0, the number a + bi is a real number. If a = 0 and b ≠ 0, the number a + bi is a pure imaginary number. 202 so that when you restrict the operations to the subset of real numbers, you get the familiar operations on the real numbers. Square Root of a Negative Real Number Algebra For any positive number a, 1-a = 1-1 a = 1-1 1a = i1a. Key Concept r You can define operations on the set of complex numbers a Real part bi Imaginary part Complex Numbers (a bi ) Real Numbers (a 0i) Imaginary Numbers (a bi, b 0) Pure Imaginary Numbers (0 bi, b 0) Key Concept Complex Number Plane In the complex number plane, the point (a, b) represents the complex number a + bi. To graph a complex number, locate the real part on the horizontal axis and the imaginary part on the vertical axis. imaginary axis 3 real axis 4 2i 2 1i 1i The absolute value of a complex number is its distance from the origin in the complex plane. 2i 2i 0 a + bi 0 = 2a2 + b2 3 2i 3 2i 13 Problem 1 P TEKS Process Standard (1)(E) Simplifying a Number Using i Is 1−18 a real number? No. There is no real number that when multiplied by itself gives -18. You must use the imaginary unit i to write 1-18. How do you write 1−18 by using the imaginary unit i? H # 18 = 1-1 # 118 = i # 118 = i # 312 1-18 = 1-1 Multiplication Property of Square Roots Definition of i Simplify. = 3i12 Problem bl 2 TEKS Process Standard (1)(D) Graphing in the Complex Number Plane What are the graph and absolute value of each number? A −5 + 3i 8i imaginary axis 0 -5 + 3i 0 = 2( -5)2 + 32 6ii 6 = 134 Where is a pure imaginary number in the complex plane? The real part of a pure imaginary number is 0. The number must be on the imaginary axis. B 6i 0 6i 0 = 0 0 + 6i 0 = 202 + 62 = 136 =6 6 units up 4i 5 5 3i 5 units left, 3 units up 2i 66 4 2 2 real axis 4 2i PearsonTEXAS.com 203 Problem 3 Adding and Subtracting Complex Numbers How is adding complex numbers similar to adding algebraic expressions? Adding the real parts and imaginary parts separately is like adding like terms. What is each sum or difference? W A (4 − 3i) + (−4 + 3i) 4 + ( -4) + ( -3i) + 3i Use the commutative and associative properties. 0+0=0 4 - 3i and - 4 + 3i are additive inverses. B (5 − 3i) − (−2 + 4i) 5 - 3i + 2 - 4i To subtract, add the opposite. 5 + 2 - 3i - 4i Use the commutative and associative properties. 7 - 7i Simplify. Problem bl 4 Multiplying Complex Numbers What is each product? A (3i)(−5 + 2i) How do you multiply two binomials? Multiply each term of one binomial by each term of the other binomial. -15i + 6i 2 Distributive Property -15i + 6( -1) Substitute - 1 for i 2. -6 - 15i Simplify. B (4 + 3i)(−1 − 2i) -4 - 8i - 3i - C (−6 + i)(−6 − i) 36 + 6i - 6i - i 2 6i 2 -4 - 8i - 3i - 6( -1) 2 - 11i Substitute 1 for i 2. 36 + 6i - 6i - ( -1) 37 Problem bl 5 Dividing Complex Numbers What is each quotient? W What is the goal? Write the quotient in the form a + bi. A 9 + 12i 3i 9 + 12i 3i # -- 3i3i - 27i - 36i 2 - 9i 2 - 27i - 36( - 1) - 9( - 1) 36 - 27i 9 4 - 3i 204 Lesson 5-9 Complex Numbers Multiply numerator and denominator by the complex conjugate of the denominator. Substitute 1 for i 2. 2 + 3i B 1 − 4i 2 + 3i 1 - 4i # 11 ++ 4i4i 2 + 8i + 3i + 12i 2 1 + 4i - 4i - 16i 2 2 + 8i + 3i + 12( - 1) 1 + 4i - 4i - 16( - 1) - 10 + 11i 17 10 11 - 17 + 17i Problem 6 TEKS Process Standard (1)(F) Factoring Using Complex Conjugates Is the expression factorable using real numbers? No. Look for factors using complex numbers. What is the factored form of 2x2 + 32? W 2x2 + 32 2(x2 + 16) Factor out the GCF. 2(x + 4i )(x - 4i ) Use a2 + b2 = (a + bi )(a - bi ) to factor (x 2 + 16). Check 2(x2 + 4xi - 4xi - 16i 2) 2(x2 - 16( -1)) Multiply the binomials. i2 = -1 2(x2 + 16) Simplify within the binomial. 2x2 + 32 Multiply. Problem bl 7 Finding Imaginary Solutions What are the solutions of 2x2 − 3x + 5 = 0? Use the Quadratic Formula with a = 2, b = -3, and c = 5. Simplify. x= − b t 2b2 − 4ac 2a = − ( − 3) t 2( − 3)2 − 4(2)(5) 2(2) = 3 t 29 − 40 4 = 3 t 2 − 31 4 = 231 3 t i 4 4 PearsonTEXAS.com 205 HO ME RK O NLINE WO PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd™ tutorial video. Simplify each number by using the imaginary number i. 1. 1-4 For additional support when completing your homework, go to PearsonTEXAS.com. 2. 1-7 3. 1-15 4. 1-50 Plot each complex number and find its absolute value. 5. 2i 6. 5 + 12i 7. 2 - 2i 8. 3 - 6i Simplify each expression. 9. (2 + 4i) + (4 - i) 10. ( -3 - 5i) + (4 - 2i) 11. (7 + 9i) + ( -5i) 12. (12 + 5i) - (2 - i) 13. ( -6 - 7i) - (1 + 3i) 14. (8 + i)(2 + 7i) 15. ( -6 - 5i)(1 + 3i) 16. ( -6i)2 17. (9 + 4i)2 Write each quotient as a complex number. 3 - 2i 5i i+2 21. i - 2 18. - 2i 4 - 3i 19. 1 + i 20. - 1 - 4i 22. 2 -4 3i 23. 3 + 2i (1 + i)2 Find the factored forms of each expression. Check your answer. 24. x2 + 25 25. x2 + 1 26. 3s2 + 75 1 27. x2 + 4 28. 4b2 + 1 29. -9x2 - 100 Find all solutions to each quadratic equation. 30. x2 + 2x + 3 = 0 31. -3x2 + x - 3 = 0 32. 2x2 - 4x + 7 = 0 33. x2 - 2x + 2 = 0 34. x 2 + 5 = 4x 35. 2x(x - 3) = -5 36. a. Name the complex number represented by each point on the graph at the right. imaginary axis F 4i b. Find the additive inverse of each number. 2i c. Find the complex conjugate of each number. d. Find the absolute value of each number. 37. Connect Mathematical Ideas (1)(F) In the complex number plane, what geometric figure describes the complex numbers with absolute value 10? A 2i E 38. Solve (x + 3i)(x - 3i) = 34. Simplify each expression. 206 Lesson 5-9 39. (8i)(4i)( -9i) 40. (2 + 1-1) + ( -3 + 1-16) 41. (8 - 1-1) - ( -3 + 1-16) 42. 2i(5 - 3i) 43. -5(1 + 2i) + 3i(3 - 4i) 44. (3 + 1-4)(4 + 1-1) Complex Numbers 3 4i B D C 3 real axis 45. Analyze Mathematical Relationships (1)(F) In the equation x2 - 6x + c = 0, find values of c that will give: a. two real solutions b. two imaginary solutions c. one real solution 46. A student wrote the numbers 1, 5, 1 + 3i, and 4 + 3i to represent the vertices of a quadrilateral in the complex number plane. What type of quadrilateral has these vertices? The multiplicative inverse of a complex number z is 1z where z ≠ 0. Find the multiplicative inverse, or reciprocal, of each complex number. Then use complex conjugates to simplify. Check each answer by multiplying it by the original number. 47. 2 + 5i 48. 8 - 12i 49. a + bi Find the sum and product of the solutions of each equation. 50. x2 - 2x + 3 = 0 51. 5x2 + 2x + 1 = 0 52. -2x2 + 3x - 3 = 0 b For ax2 + bx + c = 0, the sum of the solutions is − a and the product of the c solutions is a. Find a quadratic equation for each pair of solutions. Assume a = 1. 53. -6i and 6i 54. 2 + 5i and 2 - 5i 55. 4 - 3i and 4 + 3i Two complex numbers a + bi and c + di are equal when a = c and b = d. Solve each equation for x and y. 56. 2x + 3yi = -14 + 9i 57. 3x + 19i = 16 - 8yi 58. -14 - 3i = 2x + yi 59. Show that the product of any complex number a + bi and its complex conjugate is a real number. 60. For what real values of x and y is (x + yi)2 an imaginary number? 61. Explain Mathematical Ideas (1)(G) True or false: The conjugate of the additive inverse of a complex number is equal to the additive inverse of the conjugate of that complex number. Explain your answer. TEXAS Test Practice T 62. How can you rewrite the expression (8 - 5i)2 in the form a + bi? A. 39 + 80i B. 39 - 80i C. 69 + 80i D. 69 - 80i 63. How many solutions does the quadratic equation 4x2 - 12x + 9 = 0 have? F. two real solutions H. two imaginary solutions G. one real solution J. one imaginary solution 64. What are the solutions of 3x2 - 2x - 4 = 0? A. 1 { 113 3 B. 1 { i 111 3 C. - 1 { 113 3 D. - 1 { i 111 3 65. Using factoring, what are all four solutions to x4 - 16 = 0? Show your work. PearsonTEXAS.com 207