# 1.3 Complex Numbers; Quadratic Equations in the Complex Number System*

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2 SECTION.3 Complex Numbers; Quadrati Equations in the Complex Number System 05 In the progression outlined, eah time we enountered a situation that was unsuitable, we introdued a new number system to remedy this situation. The number system that results from introduing the number i is alled the omplex number system. DEFINITION Complex numbers are numbers of the form a bi, where a and b are real numbers. The real number a is alled the real part of the number a + bi; the real number b is alled the imaginary part of a + bi; and i is the imaginary unit, so i 2 -. For example, the omplex number i has the real part -5 and the imaginary part 6. When a omplex number is written in the form a + bi, where a and b are real numbers, we say it is in standard form. However, if the imaginary part of a omplex number is negative, suh as in the omplex number i, we agree to write it instead in the form 3-2i. Also, the omplex number a + 0i is usually written merely as a. This serves to remind us that the real numbers are a subset of the omplex numbers. The omplex number 0 + bi is usually written as bi. Sometimes the omplex number bi is alled a pure imaginary number. Add, Subtrat, Multiply, and Divide Complex Numbers Equality, addition, subtration, and multipliation of omplex numbers are defined so as to preserve the familiar rules of algebra for real numbers. Two omplex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. That is, Equality of Complex Numbers a + bi + di if and only if a and b d () Two omplex numbers are added by forming the omplex number whose real part is the sum of the real parts and whose imaginary part is the sum of the imaginary parts. That is, Sum of Complex Numbers a + bi2 + + di2 a b + d2i (2) To subtrat two omplex numbers, use this rule: Differene of Complex Numbers a + bi2 - + di2 a b - d2i (3) EXAMPLE Adding and Subtrating Complex Numbers (b) 3 + 5i i i + 8i 6 + 4i i i i 3-2i Now Work PROBLEM 3

3 06 CHAPTER Equations and Inequalities Produts of omplex numbers are alulated as illustrated in Example 2. EXAMPLE 2 Multiplying Complex Numbers 5 + 3i2 # 2 + 7i2 5 # 2 + 7i2 + 3i2 + 7i i + 6i + 2i 2 Distributive Property Distributive Property 0 + 4i i i Based on the proedure of Example 2, the produt of two omplex numbers is defined as follows: Produt of Complex Numbers a + bi2 # + di2 a - bd2 + ad + b2i (4) Do not bother to memorize formula (4). Instead, whenever it is neessary to multiply two omplex numbers, follow the usual rules for multiplying two binomials, as in Example 2, remembering that i 2 -. For example, 2 + i2 - i2 2-2i + i - i i Now Work PROBLEM 9 2i22i2 4i 2-4 Algebrai properties for addition and multipliation, suh as the ommutative, assoiative, and distributive properties, hold for omplex numbers. The property that every nonzero omplex number has a multipliative inverse, or reiproal, requires a loser look. DEFINITION If z a + bi is a omplex number, then its onjugate, denoted by z, is defined as z a + bi a - bi For example, 2 + 3i 2-3i and -6-2i i. EXAMPLE 3 Multiplying a Complex Number by Its Conjugate Find the produt of the omplex number z 3 + 4i and its onjugate z. Sine z 3-4i, we have zz 3 + 4i23-4i2 9-2i + 2i - 6i The result obtained in Example 3 has an important generalization. THEOREM The produt of a omplex number and its onjugate is a nonnegative real number. That is, if z a + bi, then zz a 2 + b 2 (5)

4 SECTION.3 Complex Numbers; Quadrati Equations in the Complex Number System 07 Proof If z a + bi, then zz a + bi2a - bi2 a 2 - bi2 2 a 2 - b 2 i 2 a 2 + b 2 To express the reiproal of a nonzero omplex number z in standard form, multiply the numerator and denominator of by z. That is, if z a + bi is a z nonzero omplex number, then a + bi z # z z z z zz a - bi a 2 + b 2 Use (5). a a 2 + b 2 - b a 2 + b 2 i EXAMPLE 4 Writing the Reiproal of a Complex Number in Standard Form Write in standard form a + bi; that is, find the reiproal of 3 + 4i i The idea is to multiply the numerator and denominator by the onjugate of 3 + 4i, that is, by the omplex number 3-4i. The result is 3 + 4i # 3-4i 3 + 4i 3-4i 3-4i i To express the quotient of two omplex numbers in standard form, multiply the numerator and denominator of the quotient by the onjugate of the denominator. EXAMPLE 5 Writing the Quotient of Two Complex Numbers in Standard Form Write eah of the following in standard form. + 4i 5-2i (b) 2-3i 4-3i (b) + 4i 5-2i + 4i # 5 + 2i 5-2i 5 + 2i 2-3i 4-3i 2-3i # 4 + 3i 4-3i 4 + 3i i i i 5 + 2i + 20i + 48i i 8 + 6i - 2i - 9i Now Work PROBLEM 27 EXAMPLE 6 Writing Other Expressions in Standard Form If z 2-3i and w 5 + 2i, write eah of the following expressions in standard form. z (b) z + w () z + z w

5 08 CHAPTER Equations and Inequalities z w z # w w # w 2-3i25-2i i25-2i2 0-4i - 5i + 6i i i (b) z + w 2-3i i2 7 - i 7 + i () z + z 2-3i i2 4 The onjugate of a omplex number has ertain general properties that we shall find useful later. For a real number a a + 0i, the onjugate is a a + 0i a - 0i a. That is, THEOREM The onjugate of a real number is the real number itself. Other properties of the onjugate that are diret onsequenes of the definition are given next. In eah statement, z and w represent omplex numbers. THEOREM The onjugate of the onjugate of a omplex number is the omplex number itself. z2 z (6) The onjugate of the sum of two omplex numbers equals the sum of their onjugates. z + w z + w (7) The onjugate of the produt of two omplex numbers equals the produt of their onjugates. z # w z # w (8) We leave the proofs of equations (6), (7), and (8) as exerises. Powers of i The powers of i follow a pattern that is useful to know. i i i 2 - i 6 i 4 # i 2 - i 3 i 2 # i - # i -i i 7 i 4 # i 3 -i i 4 i 2 # i And so on. The powers of i repeat with every fourth power. i 5 i 4 # i # i i i 8 i 4 # i 4 EXAMPLE 7 Evaluating Powers of i i 27 i 24 # i 3 i # i 3 6 # i 3 -i (b) i 0 i 00 # i i # i 25 # i i

6 SECTION.3 Complex Numbers; Quadrati Equations in the Complex Number System 09 EXAMPLE 8 Writing the Power of a Complex Number in Standard Form Write 2 + i2 3 in standard form. Use the speial produt formula for x + a2 3. NOTE Another way to find (2 + i) 3 is to multiply out (2 + i) 2 (2 + i). x + a2 3 x 3 + 3ax 2 + 3a 2 x + a 3 Using this speial produt formula, 2 + i # i # # i 2 # 2 + i i i2 2 + i. Now Work PROBLEM 4 2 Solve Quadrati Equations in the Complex Number System Quadrati equations with a negative disriminant have no real number solution. However, if we extend our number system to allow omplex numbers, quadrati equations will always have a solution. Sine the solution to a quadrati equation involves the square root of the disriminant, we begin with a disussion of square roots of negative numbers. DEFINITION WARNING In writing -N N i be sure to plae i outside the symbol. If N is a positive real number, we define the prinipal square root of denoted by -N, as 2-N 2N i where i is the imaginary unit and i 2 -. N, EXAMPLE 9 Evaluating the Square Root of a Negative Number - i i (b) -4 4 i 2i () -8 8 i 22 i EXAMPLE 0 Solving Equations Solve eah equation in the omplex number system. x 2 4 (b) x 2-9 x 2 4 x ;4 ;2 The equation has two solutions, -2 and 2. The solution set is {-2, 2}. (b) x 2-9 x ;-9 ;9 i ;3i The equation has two solutions, -3i and 3i. The solution set is {-3i, 3i}. Now Work PROBLEMS 49 AND 53

7 0 CHAPTER Equations and Inequalities WARNING When working with square roots of negative numbers, do not set the square root of a produt equal to the produt of the square roots (whih an be done with positive numbers). To see why, look at this alulation: We know that However, it is also true that , so Z A 225 iba24 ib 5i22i2 0i 2-0 Here is the error. Beause we have defined the square root of a negative number, we an now restate the quadrati formula without restrition. THEOREM Quadrati Formula In the omplex number system, the solutions of the quadrati equation ax 2 + bx + 0, where a, b, and are real numbers and a Z 0, are given by the formula x -b ; 4 b2-4a (9) EXAMPLE Solving a Quadrati Equation in the Complex Number System Solve the equation x 2-4x in the omplex number system. Here a, b -4, 8, and b 2-4a Using equation (9), we find that x --42 ; ; 26 i 2 4 ; 4i 2 2 ; 2i The equation has two solutions 2-2i and 2 + 2i. The solution set is 52-2i, 2 + 2i6. Chek: 2 + 2i: 2 + 2i i i + 4i i i: 2-2i i i + 4i i Now Work PROBLEM 59 The disriminant b 2-4a of a quadrati equation still serves as a way to determine the harater of the solutions. Charater of the s of a Quadrati Equation In the omplex number system, onsider a quadrati equation ax 2 + bx + 0 with real oeffiients.. If b 2-4a 7 0, the equation has two unequal real solutions. 2. If b 2-4a 0, the equation has a repeated real solution, a double root. 3. If b 2-4a 6 0, the equation has two omplex solutions that are not real. The solutions are onjugates of eah other.

8 SECTION.3 Complex Numbers; Quadrati Equations in the Complex Number System The third onlusion in the display is a onsequene of the fat that if b 2-4a -N 6 0 then, by the quadrati formula, the solutions are x -b + 4 b2-4a -b + 2-N -b + 2N i -b + 2N i and x -b - 4 b2-4a -b - 2-N -b - 2N i -b - 2N i whih are onjugates of eah other. EXAMPLE 2 Determining the Charater of the s of a Quadrati Equation Without solving, determine the harater of the solutions of eah equation. 3x 2 + 4x (b) 2x 2 + 4x + 0 () 9x 2-6x + 0 Conepts and Voabulary Here a 3, b 4, and 5, so b 2-4a The solutions are two omplex numbers that are not real and are onjugates of eah other. (b) Here a 2, b 4, and, so b 2-4a The solutions are two unequal real numbers. () Here a 9, b -6, and, so b 2-4a The solution is a repeated real number, that is, a double root. Now Work PROBLEM 73.3 Assess Your Understanding Are You Prepared? Answers are given at the end of these exerises. If you get a wrong answer, read the pages listed in red.. Name the integers and the rational numbers in the set b -3, 0, 2, 6 (pp. 4 5) 5, p r. 2. True or False Rational numbers and irrational numbers are in the set of real numbers. (pp. 4 5) 3 3. Rationalize the denominator of (p. 45) In the omplex number 5 + 2i, the number 5 is alled the 6. True or False The onjugate of 2 + 5i is -2-5i. part; the number 2 is alled the 7. True or False All real numbers are omplex numbers. part; the number i is alled the. 8. True or False If 2-3i is a solution of a quadrati equation 5. The equation x 2-4 has the solution set. with real oeffiients, then i is also a solution. Skill Building In Problems 9 46, write eah expression in the standard form a + bi i i i i i i i i i i i i i i2 7. 2i2-3i2 8. 3i-3 + 4i i22 + i i22 - i i2-6 - i i23 + i i i 29. a ib i -2i 30. a ib i 6 - i + i i 2 + 3i - i 3. + i i2 2

9 2 CHAPTER Equations and Inequalities 33. i i i i i i i 3-4i i 3-2i i i i 7 + i i 4 + i i 6 + i 4 + i i 7 + i 5 + i 3 + i In Problems 47 52, perform the indiated operations and express your answer in the form a + bi i24i i23i - 42 In Problems 53 72, solve eah equation in the omplex number system. 53. x x x x x 2-6x x 2 + 4x x 2-6x x 2-2x x 2-4x x 2 + 6x x 2 + 2x 64. 3x 2 + 6x 65. x 2 + x x 2 - x x x x x 4 7. x 4 + 3x x 4 + 3x In Problems 73 78, without solving, determine the harater of the solutions of eah equation in the omplex number system x 2-3x x 2-4x x 2 + 3x x x 77. 9x 2-2x x 2 + 2x i is a solution of a quadrati equation with real i is a solution of a quadrati equation with real oeffiients. Find the other solution. oeffiients. Find the other solution. In Problems 8 84, z 3-4i and w 8 + 3i. Write eah expression in the standard form a + bi. 8. z + z 82. w - w 83. zz 84. z - w Appliations and Extensions 85. Eletrial Ciruits The impedane Z, in ohms, of a iruit element is defined as the ratio of the phasor voltage V, in volts, aross the element to the phasor urrent I, in amperes, through the elements. That is, Z V. If the voltage aross a I iruit element is 8 + i volts and the urrent through the element is 3-4i amperes, determine the impedane. 86. Parallel Ciruits In an a iruit with two parallel pathways, the total impedane Z, in ohms, satisfies the formula where Z is the impedane of the first pathway Z +, Z Z 2 Explaining Conepts: Disussion and Writing 9. Explain to a friend how you would add two omplex numbers and how you would multiply two omplex numbers. Explain any differenes in the two explanations. 92. Write a brief paragraph that ompares the method used to rationalize the denominator of a radial expression and the method used to write the quotient of two omplex numbers in standard form. 93. Use an Internet searh engine to investigate the origins of omplex numbers. Write a paragraph desribing what you find and present it to the lass. Are You Prepared? Answers and Z 2 is the impedane of the seond pathway. Determine the total impedane if the impedanes of the two pathways are Z 2 + i ohms and Z 2 4-3i ohms. 87. Use z a + bi to show that z + z and z - z 2bi. 88. Use z a + bi to show that z z. 89. Use z a + bi and w + di to show that z + w z + w. 90. Use z a + bi and w + di to show that z # w z # w. 94. Explain how the method of multiplying two omplex numbers is related to multiplying two binomials. 95. What Went Wrong? A student multiplied 2-9 and 2-9 as follows: 2-9 # 2-9 2(-9)(-9) 28 9 The instrutor marked the problem inorret. Why?. Integers: 5-3, 06; rational numbers: b -3, 0, 6 2. True 3. 3A2-23B 5 r

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