5-9. Complex Numbers. Key Concept. Square Root of a Negative Real Number. Key Concept. Complex Numbers VOCABULARY TEKS FOCUS ESSENTIAL UNDERSTANDING

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1 TEKS FOCUS 5-9 Complex Numbers VOCABULARY TEKS (7)(A) Add, subtract, and multiply complex TEKS (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas. Additional TEKS (1)(D), ()(F), (7)(B) Absolute value of a complex number The absolute value of a complex number is its distance from the origin on the complex number plane. Complex conjugates number pairs of the form a + bi and a - bi Complex number Complex numbers are the real numbers and imaginary Complex number plane The complex number plane is identical to the coordinate plane, except each ordered pair (a, b) represents the complex number a + bi. Imaginary number any number of the form a + bi, where a and b are real numbers and b 0 Imaginary unit The imaginary unit i is the complex number whose square is -1. Pure imaginary number If a = 0 and b 0, the number a + bi is a pure imaginary number. Analyze closely examine objects, ideas, or relationships to learn more about their nature ESSENTIAL UNDERSTANDING The complex numbers are based on a number whose square is -1. Every quadratic equation has complex number solutions (that sometimes are real numbers). You can define operations on the set of complex numbers so that when you restrict the operations to the subset of real numbers, you get the familiar operations on the real Key Concept Square Root of a Negative Real Number Algebra Example Note For any positive number a, 1-5 = i15 ( 1-5) 2 = (i15) 2 = i 2 (15) 2 1-a = 1-1 # a =-1 # 5 =-5 (not 5) = 1-1 # 1a = i1a. Key Concept Complex Numbers You can write a complex number in the form a + bi, where a and b are real 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. 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) 202 Lesson 5-9 Complex Numbers

2 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. The absolute value of a complex number is its distance from the origin in the complex plane. 0 a + bi 0 = 2a 2 + b 2 1i imaginary axis 2 2i 1 2i real axis Problem 1 Simplifying a Number Using i TEKS Process Standard (1)(E) 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 How do you write 1 18 by using the imaginary unit i? 1-18 = 1-1 # 18 = 1-1 # 118 Multiplication Property of Square Roots = i # 118 Definition of i = i # 12 Simplify. = i12 Problem 2 TEKS Process Standard (1)(D) Graphing in the Complex Number Plane What are the graph and absolute value of each number? 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. A 5 + i i 0 = 2(-5) = 1 B 6i 0 6i 0 = i 0 = = 16 = 6 5 units left, units up 5 i 66 8i i 2i 2 imaginary axis 6i real axis 2 6 units up PearsonTEXAS.com 20

3 Problem 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? A ( i) + ( + i) + (- ) + (- i) + i Use the commutative and associative properties = 0 - i and - + i are additive inverses. B (5 i) ( 2 + i) 5 - i i To subtract, add the opposite i - i Use the commutative and associative properties. 7-7i Simplify. Problem Multiplying Complex Numbers What is each product? How do you multiply two binomials? Multiply each term of one binomial by each term of the other binomial. A (i)( 5 + 2i) -15i + 6i 2 Distributive Property -15i + 6(-1) Substitute -1 for i i Simplify. B ( + i)( 1 2i) C ( 6 + i)( 6 i) - - 8i - i - 6i i - 6i - i i - i - 6(-1) Substitute 6 + 6i - 6i - (-1) 2-11i 1 for i 2. 7 Problem 5 Dividing Complex Numbers What is the goal? Write the quotient in the form a + bi. What is each quotient? A i i i i # - i - i - 27i - 6i 2-9i 2-27i - 6(-1) - 9(-1) 6-27i 9 Multiply numerator and denominator by the complex conjugate of the denominator. Substitute 1 for i 2. B 2 + i 1 i 2 + i # 1 + i 1 - i 1 + i 2 + 8i + i + 12i i - i - 16i i + i + 12(-1) 1 + i - i - 16(-1) i 17 - i i 20 Lesson 5-9 Complex Numbers

4 Problem 6 TEKS Process Standard (1)(F) Factoring Using Complex Conjugates Is the expression factorable using real numbers? No. Look for factors using complex What is the factored form of 2x 2 + 2? 2x (x ) Factor out the GCF. 2(x + i )(x - i ) Use a 2 + b 2 = (a + bi )(a - bi ) to factor (x ). Check Problem 7 2(x 2 + xi - xi - 16i 2 ) Multiply the binomials. 2(x 2-16(-1)) i 2 =-1 2(x ) Simplify within the binomial. 2x Multiply. Finding Imaginary Solutions What are the solutions of 2x 2 x + 5 = 0? Use the Quadratic Formula with a = 2, b =-, and c = 5. Simplify. = b t 2b2 ac x 2a = ( ) t 2( )2 (2)(5) 2(2) = t 29 0 = t 2 1 = t 21 i PearsonTEXAS.com 205

5 ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. Simplify each number by using the imaginary number i. For additional support when completing your homework, go to PearsonTEXAS.com Plot each complex number and find its absolute value. 5. 2i i i i Simplify each expression. 9. (2 + i) + ( - i) 10. (- - 5i) + ( - 2i) 11. (7 + 9i) + (-5i) 12. (12 + 5i) - (2 - i) 1. (-6-7i) - (1 + i) 1. (8 + i)(2 + 7i) 15. (-6-5i)(1 + i) 16. (-6i) (9 + i) 2 Write each quotient as a complex number i 5i i 1 + i 21. i + 2 i i Find the factored forms of each expression. Check your answer i i i (1 + i) 2 2. x x s x b x Find all solutions to each quadratic equation. 0. x 2 + 2x + = x 2 + x - = x 2 - x + 7 = 0. x 2-2x + 2 = 0. x = x 5. 2x(x - ) =-5 6. a. Name the complex number represented by each point on the graph at the right. b. Find the additive inverse of each number. c. Find the complex conjugate of each number. F i 2i D imaginary axis B d. Find the absolute value of each number. 7. Connect Mathematical Ideas (1)(F) In the complex number plane, what geometric figure describes the complex numbers with absolute value 10? A E i C real axis 8. Solve (x + i)(x - i) =. Simplify each expression. 9. (8i)(i)(-9i) 0. ( ) + ( ) 1. (8-1-1) - ( ) 2. 2i(5 - i). -5(1 + 2i) + i( - i). ( + 1-)( + 1-1) 206 Lesson 5-9 Complex Numbers

6 5. Analyze Mathematical Relationships (1)(F) In the equation x 2-6x + c = 0, find values of c that will give: a. two real solutions b. two imaginary solutions c. one real solution 6. A student wrote the numbers 1, 5, 1 + i, and + i 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 1 z 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 i i 9. a + bi Find the sum and product of the solutions of each equation. 50. x 2-2x + = x 2 + 2x + 1 = x 2 + x - = 0 For ax 2 + bx + c = 0, the sum of the solutions is a b and the product of the solutions is a c. Find a quadratic equation for each pair of solutions. Assume a = i and 6i i and 2-5i i and + i Two complex numbers a + bi and c + di are equal when a = c and b = d. Solve each equation for x and y x + yi =-1 + 9i 57. x + 19i = 16-8yi i = 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 62. How can you rewrite the expression (8-5i) 2 in the form a + bi? A i B. 9-80i C i D i 6. How many solutions does the quadratic equation x 2-12x + 9 = 0 have? F. two real solutions H. two imaginary solutions G. one real solution J. one imaginary solution 6. What are the solutions of x 2-2x - = 0? A. 1 { 11 B. 1 { i111 C. - 1 { 11 D. - 1 { i Using factoring, what are all four solutions to x - 16 = 0? Show your work. PearsonTEXAS.com 207