83. 31x + 2x + 9 = 3. Review Exercises. 85. Divide using synthetic division: 86. Divide: 90. Rationalize the denominator: Complex Numbers

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1 718 CHAPTER 10 Radicals, Radical Functions, and Rational Exponents 76. Now that I know how to solve radical equations, I can use models that are radical functions to determine the value of the independent variable when a function value is known. In Exercises 77 80, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. 77. The first step in solving 2x + 6 x + 2 is to square both sides, obtaining x + 6 x The equations 2x and x have the same solution set. 79. The equation - 1x 9 has no solution. 80. The equation 2x 2 + 9x + 3 -x has no solution because a principal square root is always nonnegative. 81. Find the length of the three sides of the right 1 + x x 7 triangle shown in the figure. x In Exercises 82 84, solve each equation x1x x + 2x x Review Exercises 85. Divide using synthetic division: (Section 5.6, Example 5) 86. Divide: (Section 7.2, Example 6) 87. Factor: 2x xy + 18y 2. (Section 6.5, Example 8) Preview Exercises 14x 4-3x 3 + 2x 2 - x - 12, 1x x 2-12 x 2 + 2x - 8 Exercises will help you prepare for the material covered in the next section. 88. Simplify: x x Multiply: 17-3x x Rationalize the denominator:, 6x + 18 x SECTION 10.7 Complex Numbers Objectives 1 Express square roots of negative numbers in terms of i. 2 Add and subtract complex numbers. 3 Multiply complex numbers. 4 Divide complex numbers. 5 Simplify powers of i Roz Chast from Cartoonbank. com. All rights reserved. Who is this kid warning us about our eyeballs turning black if we attempt to find the square root of -9? Don t believe what you hear on the street. Although square roots of negative numbers are not real numbers, they do play a significant role in algebra. In this section, we move beyond the real numbers and discuss square roots with negative radicands.

2 SECTION 10.7 Complex Numbers Express square roots of negative numbers in terms of i. The Imaginary Unit i In Chapter 11, we will study equations whose solutions involve the square roots of negative numbers. Because the square of a real number is never negative, there is no real number x such that x 2-1. To provide a setting in which such equations have solutions, mathematicians invented an expanded system of numbers, the complex numbers. The imaginary number i, defined to be a solution of the equation x 2-1, is the basis of this new set. The Imaginary Unit i The imaginary unit i is defined as i 2-1, where i 2-1. Using the imaginary unit i, we can express the square root of any negative number as a real multiple of i. For example, i. We can check that i by squaring 5i and obtaining i i The Square Root of a Negative Number If b is a positive real number, then 2-b 2b1-12 2b2-1 2bi or i2 EXAMPLE 1 Write as a multiple of i: 2-9 a. c. Expressing Square Roots of Negative Numbers as Multiples of i Study Tip We allow the use of the product rule 1ab 1a1b when a is positive and b is -1. However, you cannot use 1ab 1a1b when both a and b are negative. Solution a i c. 3 3A 1B i Be sure not to write i under the radical A 1B i In order to avoid writing i under a radical, let s agree to write i before any radical. Consequently, we express the multiple of i in part (b) as i23 and the multiple of i in part (c) as 4i25. CHECK POINT 1 Write as a multiple of i: a c

3 720 CHAPTER 10 Radicals, Radical Functions, and Rational Exponents A new system of numbers, called complex numbers, is based on adding multiples of i, such as 5i, to the real numbers. Real numbers a + bi with b 0 Complex numbers a + bi Imaginary numbers a + bi with b 0 FIGURE 10.9 The complex number system Complex Numbers and Imaginary Numbers The set of all numbers in the form a + bi, with real numbers a and b, and i, the imaginary unit, is called the set of complex numbers. The real number a is called the real part, and the real number b is called the imaginary part of the complex number a + bi. If b Z 0, then the complex number is called an imaginary number (Figure 10.9). Here are some examples of complex numbers. Each number can be written in the form a + bi. 4+6i 2i0+2i 33+0i a, the real part, is 4. b, the imaginary part, is 6. a, the real part, is 0. b, the imaginary part, is 2. a, the real part, is 3. b, the imaginary part, is 0. Can you see that b, the imaginary part, is not zero in the first two complex numbers? Because b Z 0, these complex numbers are imaginary numbers. By contrast, the imaginary part of the complex number on the right is zero. This complex number is not an imaginary number. The number 3, or 3 + 0i, is a real number. 2 Add and subtract complex numbers. Adding and Subtracting Complex Numbers The form of a complex number a + bi is like the binomial a + bx. Consequently, we can add, subtract, and multiply complex numbers using the same methods we used for binomials, remembering that i 2-1. Adding and Subtracting Complex Numbers 1. (a + bi) + (c + di) 1a + c2 + 1b + d2i In words, this says that you add complex numbers by adding their real parts, adding their imaginary parts, and expressing the sum as a complex number. 2. 1a + bi2-1c + di2 1a - c2 + 1b - d2i In words, this says that you subtract complex numbers by subtracting their real parts, subtracting their imaginary parts, and expressing the difference as a complex number. EXAMPLE 2 Adding and Subtracting Complex Numbers Perform the indicated operations, writing the result in the form a + bi: 15-11i i2 a i i2. Solution a i i2 5-11i i i + 4i i 12-7i Remove the parentheses. Group real and imaginary terms. Add real parts and add imaginary parts. Simplify.

4 SECTION 10.7 Complex Numbers 721 Study Tip The following examples, using the same integers as in Example 2, show how operations with complex numbers are just like operations with polynomials. a x x2 12-7x x x x x x a i i i i i + 6i i i CHECK POINT i i i i2. Remove the parentheses. Change signs of real and imaginary parts in the complex number being subtracted. Group real and imaginary terms. Add real parts and add imaginary parts. Simplify. Add or subtract as indicated: 3 Multiply complex numbers. Multiplying Complex Numbers Multiplication of complex numbers is performed the same way as multiplication of polynomials, using the distributive property and the FOIL method. After completing the multiplication, we replace any occurrences of i 2 with -1. This idea is illustrated in the next example. EXAMPLE 3 Find the products: a. Solution a. 4i13-5i2 4i13-5i2 4i # 3-4i # 5i Multiplying Complex Numbers 17-3i i2. Distribute 4i throughout the parentheses. 12i - 20i 2 Multiply. 12i Replace with i Simplify to 12i + 20 and write in a + bi form. 17-3i i2 F O I L i i+6i+15i i + 6i i + 6i i Use the FOIL method. i 2-1 Group real and imaginary terms. Combine real and imaginary terms. CHECK POINT 3 Find the products: 7i12-9i2 a i216-7i2. Consider the multiplication problem 5i # 2i 10i The problem 5i # 2i can also be given in terms of square roots of negative numbers: 2-25 # 2-4. Because the product rule for radicals only applies to real numbers, multiplying radicands is incorrect. When performing operations with square roots of negative numbers, begin by expressing all square roots in terms of i. Then perform the indicated operation.

5 722 CHAPTER 10 Radicals, Radical Functions, and Rational Exponents CORRECT: i 2i 10i 2 10( 1) 10 INCORRECT! 25 4 ( 25)( 4) EXAMPLE 4 Multiply: 2-3 # 2-5. Solution Multiplying Square Roots of Negative Numbers 2-3 # # i23 # i25 Express square roots in terms of i. i # and i # i i i CHECK POINT 4 Multiply: 2-5 # Divide complex numbers. Conjugates and Division It is possible to multiply imaginary numbers and obtain a real number. Here is an example: F O I L (4+7i)(4-7i)16-28i+28i-49i i ( 1)65. Replace i 2 with 1. You can also perform i214-7i2 using the formula 1A + B21A - B2 A 2 - B 2. A real number is obtained even faster: i214-7i i i The conjugate of the complex number a + bi is a - bi. The conjugate of the complex number a - bi is a + bi. The multiplication problem that we just performed involved conjugates. The multiplication of conjugates always results in a real number: (a+bi)(a-bi)a 2 -(bi) 2 a 2 -b 2 i 2 a 2 -b 2 ( 1)a 2 +b 2. The product eliminates i. Conjugates are used to divide complex numbers. The goal of the division procedure is to obtain a real number in the denominator. This real number becomes the denominator of a and b in the quotient a + bi. By multiplying the numerator and

6 SECTION 10.7 Complex Numbers 723 the denominator of the division by the conjugate of the denominator, you will obtain this real number in the denominator. Here are two examples of such divisions: 7+4i 2-5i The conjugate of the denominator is 2 + 5i. 5i-4 5i-4 or. 3i 0+3i The conjugate of the denominator is 0 3i, or 3i. The procedure for dividing complex numbers, illustrated in Examples 5 and 6, should remind you of rationalizing denominators. EXAMPLE 5 Divide and simplify to the form a + bi: Using Conjugates to Divide Complex Numbers 7 + 4i 2-5i. Solution The conjugate of the denominator is 2 + 5i. Multiplication of both the numerator and the denominator by 2 + 5i will eliminate i from the denominator while maintaining the value of the expression i 2-5i 7 + 4i # 2 + 5i 2-5i 2 + 5i F O I L Multiply by i+8i+20i (5i) 2 Use FOIL in the numerator and 1A - B21A + B2 A 2 - B 2 in the denominator i + 20i2 4-25i i i i 29 Simplify. i 2-1 Perform the multiplications involving -1. Combine real terms in the numerator and denominator i Express the answer in the form a + bi. CHECK POINT 5 Divide and simplify to the form a + bi: 6 + 2i 4-3i. EXAMPLE 6 Divide and simplify to the form a + bi: Using Conjugates to Divide Complex Numbers 5i i

7 724 CHAPTER 10 Radicals, Radical Functions, and Rational Exponents 5i - 4 Solution The denominator of is 3i, or 0 + 3i. The conjugate of the denominator 3i is 0-3i. Multiplication of both the numerator and the denominator by -3i will eliminate i from the denominator while maintaining the value of the expression. 5i - 4 5i - 4 # -3i Multiply by 1. 3i 3i -3i -15i2 + 12i -9i i i i i Multiply. Use the distributive property in the numerator. i 2-1 Perform the multiplications involving -1. Express the division in the form Simplify real and imaginary parts. a + bi. CHECK POINT 6 Divide and simplify to the form a + bi: 3-2i. 4i 5 Simplify powers of i. Powers of i Using the fact that i 2-1, any integral power of i greater than or equal to 2 can be simplified to either -i, i, -1, or 1. Here are some examples: i 3 i 2 # i 1-12i -i i 4 1i i 5 i 4 # i 1i # i # i i i 6 1i Here is a procedure for simplifying powers of i: Simplifying Powers of i 1. Express the given power of i in terms of i Replace i 2 with -1 and simplify. Use the fact that -1 to an even power is 1 and -1 to an odd power is -1. Simplify: a. i 12 i 39 c. Solution EXAMPLE 7 a. c. i 12 1i i 50 1i CHECK POINT 7 Simplifying Powers of i 50. i 39 i 38 i 1i i i 1-12i -i Simplify: a. i 16 i 25 c. i 35. i

8 SECTION 10.7 Complex Numbers 725 Blitzer Bonus The Patterns of Chaos One of the new frontiers of mathematics suggests that there is an underlying order in things that appear to be random, such as the hiss and crackle of background noises as you tune a radio. Irregularities in the heartbeat, some of them severe enough to cause a heart attack, or irregularities in our sleeping patterns, such as insomnia, are examples of chaotic behavior. Chaos in the mathematical sense does not mean a complete lack of form or arrangement. In mathematics, chaos is used to describe something that appears to be random, but actually contains underlying patterns that are far more intricate than previously assumed. The patterns of chaos appear in images like the one on the right and the one in the chapter opener, called the Mandelbrot set. Magnified portions of this image yield repetitions of the original structure, as well as new and unexpected patterns. The Mandelbrot set transforms the hidden structure of chaotic events into a source of wonder and inspiration. The Mandelbrot set is made possible by opening up graphing to include complex numbers in the form a + bi. Each complex number is plotted like an ordered pair in a coordinate system consisting of a real axis and an imaginary axis. Plot certain complex numbers in this system, add color to the magnified boundary of the graph, and the patterns of chaos begin to appear. NOT AVAILABLE FOR ELECTRONIC VIEWING 10.7 EXERCISE SET Practice Exercises In Exercises 1 16, express each number in terms of i and simplify, if possible In Exercises 17 32, add or subtract as indicated. Write the result in the form a + bi i i i i i i i i i i i i i i i i i i i i i i i i i i2 A5 - i23ba5 + i23b i i A2 + i23b + A7 + 4i23B A4 + i25b + A8 + 6i25B In Exercises 33 62, find each product. Write imaginary results in the form a + bi i15 + 3i i14 + 7i i17i i14i i12-5i i13-5i i i i i i212-3i i213-2i i i i i i213-5i i212-7i i i i i2 49. A3 - i22ba3 + i22b i i i i # # # # # # # # 2-5 In Exercises 63 84, divide and simplify to the form a + bi i 4 + i 2i i 5i 2 + i

9 726 CHAPTER 10 Radicals, Radical Functions, and Rational Exponents i 6i i 1 + i i 2-3i i 5-2i i 4 + 5i 5 - i i 3-2i i 7i 8-5i 3 + 4i i 5i 4 + 7i 5 + i i -4i In Exercises , simplify each expression. i 85. i i 87. i i 89. i i 91. i i 93. i i i i i2 99. i 24 + i i 28 + i 30 Practice PLUS In Exercises , perform the indicated operation(s) and write the result in the form a + bi i211 - i i213 + i i212 - i i211 + i i i i i i 4 + i 12 i 8 + i i 8 - i 7 i 4 + i Let f1x2 x 2-2x + 2. Find f11 + i Let f1x2 x 2-2x + 5. Find f11-2i2. In Exercises , simplify each evaluation to the form a + bi Let f1x2 x - 3i and g1x2 4x + 2i. Find 1fg Let f1x2 12x - i and g1x2 6x + 3i. Find 1fg2a Let f1x2 x Find f13i x Let f1x2 x Find f14i x i 5i 2-3i 1 - i 1 + i 2 + 3i 3 - i 6-3i 4 + 2i Application Exercises Complex numbers are used in electronics to describe the current in an electric circuit. Ohm s law relates the current in a circuit, I, in amperes, the voltage of the circuit, E, in volts, and the resistance of the circuit, R, in ohms, by the formula E IR. Use this formula to solve Exercises Find E, the voltage of a circuit, if I 14-5i2 amperes and R i2 ohms Find E, the voltage of a circuit, if I 12-3i2 amperes and R i2 ohms The mathematician Girolamo Cardano is credited with the first use (in 1545) of negative square roots in solving the now-famous problem, Find two numbers whose sum is 10 and whose product is 40. Show that the complex numbers 5 + i215 and 5 - i215 satisfy the conditions of the problem. (Cardano did not use the symbolism i215 or even He wrote R.m 15 for 2-15, meaning radix minus 15. He regarded the numbers 5 + R.m 15 and 5 - R.m 15 as fictitious or ghost numbers, and considered the problem manifestly impossible. But in a mathematically adventurous spirit, he exclaimed, Nevertheless, we will operate. ) Writing in Mathematics 118. What is the imaginary unit i? 119. Explain how to write 2-64 as a multiple of i What is a complex number? Explain when a complex number is a real number and when it is an imaginary number. Provide examples with your explanation Explain how to add complex numbers. Give an example Explain how to subtract complex numbers. Give an example Explain how to find the product of 2i and 5 + 3i Explain how to find the product of 2i + 3 and 5 + 3i Explain how to find the product of 2i + 3 and 2i Explain how to find the product of 2-1 and 2-4. Describe a common error in the multiplication that needs to be avoided What is the conjugate of 2 + 3i? What happens when you multiply this complex number by its conjugate? 128. Explain how to divide complex numbers. Provide an example with your explanation Explain each of the three jokes in the cartoon on page A stand-up comedian uses algebra in some jokes, including one about a telephone recording that announces You have just reached an imaginary number. Please multiply by i and dial again. Explain the joke. Explain the error in Exercises i225 5i 132. A 2-9B #

10 SECTION 10.7 Complex Numbers 727 Critical Thinking Exercises Make Sense? In Exercises , determine whether each statement makes sense or does not make sense and explain your reasoning The joke in the cartoon below is based on the math teacher not realizing that the average of complex real numbers is sometimes a complex imaginary number. In Exercises , perform the indicated operations and write the result in the form a + bi i213 - i2 1 + i 1 + 2i i 1-2i i NOT AVAILABLE FOR ELECTRONIC VIEWING 134. The word imaginary in imaginary numbers tells me that these numbers are undefined By writing the imaginary number 5i, I can immediately see that 5 is the constant and i is the variable When I add or subtract complex numbers, I am basically combining like terms. In Exercises , determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement Some irrational numbers are not complex numbers i213-7i2 is an imaginary number i i In the complex number system, x 2 + y 2 (the sum of two squares) can be factored as 1x + yi21x - yi2. Review Exercises 144. Simplify: (Section 7.5, Example 3 or Example 6) 145. Solve for x: 1 x + 1 y 1 z. (Section 7.6, Example 7) 146. If f(x) 2x 2 - x and g(x) x - 6, find (g - f)(3). (Section 8.3, Example 4) Preview Exercises Exercises will help you prepare for the material covered in the first section of the next chapter Solve by factoring: 148. Solve by factoring: x y y. y x x 2x 2 + 7x x Use substitution to determine if - 26 is a solution of the quadratic equation 3x GROUP PROJECT CHAPTER 10 Group members should consult an almanac, newspaper, magazine, or the Internet and return to the group with as much data as possible that show phenomena that are continuing to grow over time, but whose growth is leveling off. Select the five data sets that you find most intriguing. Let x represent the number of years after the first year in each data set. Model the data by hand using f1x2 a1x + Use the first and last data points to find values for a and The first data point corresponds to x 0. Its second coordinate gives the value of To find a, substitute the second data point into f1x2 a1x + b, with the value that you obtained for Now solve the equation and find a. Substitute a and b into f1x2 a1x + b to obtain a square root function that models each data set. Then use the function to make predictions about what might occur in the future. Are there circumstances that might affect the accuracy of the predictions? List some of these circumstances.