# Appendix E Complex Numbers E1 E Complex Numbers - Middlebury College

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9781285057095 AppE.qxp 2/18/13 8:25 AM Page E1 Appendix E E Complex Numbers E1 Complex Numbers Use the imaginary unit i to write complex numbers, and add, subtract, and multiply complex numbers. Find complex solutions of quadratic equations. Write the trigonometric forms of complex numbers. Find powers and nth roots of complex numbers. Operations with Complex Numbers Some equations have no real solutions. For instance, the quadratic equation x2 1 0 Equation with no real solution has no real solution because there is no real number x that can be squared to produce 1. To overcome this deficiency, mathematicians created an expanded system of numbers using the imaginary unit i, defined as i 冪 1 Imaginary unit where i 1. By adding real numbers to real multiples of this imaginary unit, you obtain the set of complex numbers. Each complex number can be written in the standard form a bi. The real number a is called the real part of the complex number a bi, and the number bi (where b is a real number) is called the imaginary part of the complex number. 2 Definition of a Complex Number For real numbers a and b, the number a bi is a complex number. If b 0, then a bi is called an imaginary number. A number of the form bi, where b 0, is called a pure imaginary number. To add (or subtract) two complex numbers, you add (or subtract) the real and imaginary parts of the numbers separately. Addition and Subtraction of Complex Numbers If a bi and c di are two complex numbers written in standard form, then their sum and difference are defined as follows. Sum: 共a bi兲 共c di兲 共a c兲 共b d兲i Difference: 共a bi兲 共c di兲 共a c兲 共b d兲i

9781285057095 AppE.qxp 2/18/13 8:25 AM Page E3 Appendix E Complex Numbers E3 Multiplying Complex Numbers a. 共3 2i兲共3 2i兲 3共3 2i兲 2i共3 2i兲 9 6i 6i 4i 2 9 6i 6i 4共 1兲 9 4 13 b. 共3 2i兲2 共3 2i兲共3 2i兲 3共3 2i兲 2i共3 2i兲 9 6i 6i 4i 2 9 6i 6i 4共 1兲 9 12i 4 5 12i Distributive Property Distributive Property i 2 1 Simplify. Write in standard form. Square of a binomial Distributive Property Distributive Property i 2 1 Simplify. Write in standard form. In Example 2(a), notice that the product of two complex numbers can be a real number. This occurs with pairs of complex numbers of the form a bi and a bi, called complex conjugates. 共a bi兲共a bi兲 a2 abi abi b2i 2 a2 b2共 1兲 a2 b2 To write the quotient of a bi and c di in standard form, where c and d are not both zero, multiply the numerator and denominator by the complex conjugate of the denominator to obtain a bi a bi c di c di c di c di 共ac bd 兲 共bc ad 兲i . c2 d 2 冢 冣 Multiply numerator and denominator by complex conjugate of denominator. Write in standard form. Writing Complex Numbers in Standard Form 2 3i 2 3i 4 2i 4 2i 4 2i 4 2i 冢 冣 8 4i 12i 6i 2 16 4i 2 8 6 16i 16 4 2 16i 20 4 1 i 10 5 Multiply numerator and denominator by complex conjugate of denominator. Expand. i 2 1 Simplify. Write in standard form.

9781285057095 AppE.qxp E4 2/18/13 Appendix E 8:25 AM Page E4 Complex Numbers Complex Solutions of Quadratic Equations When using the Quadratic Formula to solve a quadratic equation, you often obtain a result such as 冪 3, which you know is not a real number. By factoring out i 冪 1, you can write this number in standard form. 冪 3 冪3共 1兲 冪3冪 1 冪3i The number 冪3i is called the principal square root of 3. REMARK The definition of principal square root uses the rule 冪ab 冪a冪b for a 0 and b 0. This rule is not valid when both a and b are negative. For example, 冪 5冪 5 冪5共 1兲冪5共 1兲 冪5i冪5i 冪25i 2 5i 2 5 whereas 冪共 5兲共 5兲 冪25 5. To avoid problems with multiplying square roots of negative numbers, be sure to convert to standard form before multiplying. Principal Square Root of a Negative Number If a is a positive number, then the principal square root of the negative number a is defined as 冪 a 冪ai. Writing Complex Numbers in Standard Form a. 冪 3冪 12 冪3i冪12i 冪36i 2 6共 1兲 6 b. 冪 48 冪 27 冪48i 冪27i 4冪3i 3冪3i 冪3i 2 2 c. 共 1 冪 3 兲 共 1 冪3i 兲 2 共 1兲2 2冪3i 共冪3 兲 共i 2兲 1 2冪3i 3共 1兲 2 2冪3i Complex Solutions of a Quadratic Equation Solve 3x2 2x 5 0. Solution 共 2兲 冪共 2兲2 4共3兲共5兲 2共3兲 2 冪 56 6 2 2冪14i 6 1 冪14 i 3 3 x Quadratic Formula Simplify. Write 冪 56 in standard form. Write in standard form.

9781285057095 AppE.qxp 2/18/13 8:25 AM Page E5 Appendix E Just as real numbers can be represented by points on the real number line, you can represent a complex number (3, 2) or 3 2i 3 ( 1, 3) or 2 1 3i z a bi 1 1 E5 Trigonometric Form of a Complex Number Imaginary axis 2 Complex Numbers 1 2 3 Real axis ( 2, 1) or 2 i as the point 共a, b兲 in a coordinate plane (the complex plane). The horizontal axis is called the real axis and the vertical axis is called the imaginary axis, as shown in Figure E.1. The absolute value of a complex number a bi is defined as the distance between the origin 共0, 0兲 and the point 共a, b兲. The Absolute Value of a Complex Number The absolute value of the complex number z a bi is given by Figure E.1 ⱍa biⱍ 冪a2 b2. When the complex number a bi is a real number 共that is, b 0兲, this definition agrees with that given for the absolute value of a real number. ⱍa 0iⱍ 冪a2 02 ⱍaⱍ To work effectively with powers and roots of complex numbers, it is helpful to write complex numbers in trigonometric form. In Figure E.2, consider the nonzero complex number a bi. By letting be the angle from the positive real axis (measured counterclockwise) to the line segment connecting the origin and the point 共a, b兲, you can write Imaginary axis (a, b) a r cos r b θ a and b r sin where r 冪a2 b2. Consequently, you have Real axis a bi 共r cos 兲 共r sin 兲i from which you can obtain the trigonometric form of a complex number. Figure E.2 Trigonometric Form of a Complex Number The trigonometric form of the complex number z a bi is given by z r共cos i sin 兲 where a r cos , b r sin , r 冪a2 b2, and tan b兾a. The number r is the modulus of z, and is called an argument of z. The trigonometric form of a complex number is also called the polar form. Because there are infinitely many choices for , the trigonometric form of a complex number is not unique. Normally, is restricted to the interval 0 2 , although on occasion it is convenient to use 0.

9781285057095 AppE.qxp E6 2/18/13 Appendix E 8:25 AM Page E6 Complex Numbers Trigonometric Form of a Complex Number Write the complex number z 2 2冪3i in trigonometric form. Solution The absolute value of z is r 2 2冪3i 冪共 2兲2 共 2冪3 兲2 冪16 4 ⱍ ⱍ and the angle is given by Imaginary axis 3 2 4π 3 1 1 2 z 4 3 z 2 2 3i Figure E.3 4 b a 2冪3 2 冪 3. tan Real axis Because tan共 兾3兲 冪3 and because z 2 2冪3i lies in Quadrant III, choose to be 兾3 4 兾3. So, the trigonometric form is z r 共cos i sin 兲 4 4 4 cos i sin . 3 3 冢 冣 See Figure E.3. The trigonometric form adapts nicely to multiplication and division of complex numbers. Consider the two complex numbers z1 r1共cos 1 i sin 1 兲 and z2 r2共cos 2 i sin 2 兲. The product of z1 and z2 is z1z2 r1r2共cos 1 i sin 1 兲共cos 2 i sin 2 兲 r1r2关共cos 1 cos 2 sin 1 sin 2 兲 i 共sin 1 cos 2 cos 1 sin 2 兲兴. Using the sum and difference formulas for cosine and sine, you can rewrite this equation as z1z2 r1r2关cos共 1 2 兲 i sin共 1 2 兲兴. This establishes the first part of the rule shown below. The second part is left for you to verify (see Exercise 109). Product and Quotient of Two Complex Numbers Let z1 r1共cos 1 i sin 1 兲 and z2 r2共cos 2 i sin 2 兲 be complex numbers. z1z2 r1r2关cos共 1 2 兲 i sin共 1 2 兲兴 z1 r1 关cos共 1 2 兲 i sin共 1 2 兲兴, z2 0 z2 r2 Product Quotient

9781285057095 AppE.qxp 2/18/13 8:25 AM Page E7 Appendix E Complex Numbers E7 Note that this rule says that to multiply two complex numbers you multiply moduli and add arguments, whereas to divide two complex numbers you divide moduli and subtract arguments. Multiplying Complex Numbers Find the product z1z2 of the complex numbers. 冢 z1 2 cos 2 2 i sin , 3 3 冣 冢 z2 8 cos 11 11 i sin 6 6 冣 Solution 2 2 11 11 i sin 8 cos i sin 3 3 6 6 2 11 2 11 16 cos i sin 3 6 3 6 5 5 16 cos i sin 2 2 16 cos i sin 2 2 16关0 i共1兲兴 16i 冢 冣 冢 z1z2 2 cos 冤 冢 冤 冤 冣 冢 冣冥 冣 Multiply moduli and add arguments. 冥 冥 5 and are coterminal. 2 2 Check this result by first converting to the standard forms z1 1 冪3i and z2 4冪3 4i and then multiplying algebraically. Dividing Complex Numbers Find the quotient z1兾z2 of the complex numbers. z1 24共cos 300 i sin 300 兲, z2 8共cos 75 i sin 75 兲 Solution z1 24共cos 300 i sin 300 兲 z2 8共cos 75 i sin 75 兲 24 关cos共300 75 兲 i sin共300 75 兲兴 8 3关cos 225 i sin 225 兴 冤冢 冣 冢 冣冥 3 冪2 2 i 3冪2 3冪2 i 2 2 冪2 2 Divide moduli and subtract arguments.

9781285057095 AppE.qxp E8 2/18/13 Appendix E 8:25 AM Page E8 Complex Numbers Powers and Roots of Complex Numbers To raise a complex number to a power, consider repeated use of the multiplication rule. z r 共cos i sin 兲 z2 r 2 共cos 2 i sin 2 兲 z3 r 3共cos 3 i sin 3 兲 This pattern leads to the next theorem, which is named after the French mathematician Abraham DeMoivre (1667–1754). THEOREM E.1 DeMoivre’s Theorem If z r共cos i sin 兲 is a complex number and n is a positive integer, then zn 关r 共cos i sin 兲兴 n r n共cos n i sin n 兲. Finding Powers of a Complex Number Use DeMoivre’s Theorem to find 共 1 冪3i 兲 . 12 Solution First convert the complex number to trigonometric form. 冢 1 冪3i 2 cos 2 2 i sin 3 3 冣 Then, by DeMoivre’s Theorem, you have 共 1 冪3i兲12 冤 2冢cos 23 i sin 23 冣冥 12 2 i sin 12 3 4096 共cos 8 i sin 8 兲 4096. 冤 冢 212 cos 12 冣 冢 2 3 冣冥 Recall that a consequence of the Fundamental Theorem of Algebra is that a polynomial equation of degree n has n solutions in the complex number system. Each solution is an nth root of the equation. The nth root of a complex number is defined below. Definition of nth Root of a Complex Number The complex number u a bi is an nth root of the complex number z when z un 共a bi兲n.

9781285057095 AppE.qxp 2/18/13 8:25 AM Page E9 Appendix E Complex Numbers E9 To find a formula for an n th root of a complex number, let u be an n th root of z, where u s共cos i sin 兲 and z r共cos i sin 兲. By DeMoivre’s Theorem and the fact that un z, you have sn共cos n i sin n 兲 r共cos i sin 兲. Taking the absolute value of each side of this equation, it follows that s n r. Substituting r for s n in the previous equation and dividing by r, you get cos n i sin n cos i sin . So, it follows that cos n cos and sin n sin . Because both sine and cosine have a period of 2 , these last two equations have solutions if and only if the angles differ by a multiple of 2 . Consequently, there must exist an integer k such that n 2 k 2 k . n By substituting this value for in the next theorem. into the trigonometric form of u, you get the result stated THEOREM E.2 n th Roots of a Complex Number For a positive integer n, the complex number z r 共cos i sin 兲 has exactly n distinct n th roots given by 冢 n r cos 冪 2 k 2 k i sin n n 冣 where k 0, 1, 2, . . . , n 1. For k n 1, the roots begin to repeat. For instance, when k n, the angle 2 n 2 n n is coterminal with 兾n, which is also obtained when k 0. The formula for the n th roots of a complex number z has a nice geometric interpretation, as shown in Figure E.4. Note that because the n nth roots of z all have the same magnitude冪 r, n they all lie on a circle of radius冪r with center at the origin. Furthermore, because successive n r nth roots have arguments that differ by 2 兾n, the n roots are equally spaced along the circle. Figure E.4 Imaginary axis 2π n 2π n Real axis

9781285057095 AppE.qxp E10 2/18/13 Appendix E 8:25 AM Page E10 Complex Numbers Finding the n th Roots of a Complex Number Find the three cube roots of z 2 2i. Solution Because z lies in Quadrant II, the trigonometric form for z is z 2 2i 冪8 共cos 135 i sin 135 兲. By the formula for n th roots, the cube roots have the form 冢 6 8 cos 冪 135 360 k 135 360 k i sin . 3 3 冣 Finally, for k 0, 1, and 2, you obtain the roots 冪2 共cos 45 i sin 45 兲 1 i 兲 1.3660 0.3660i 冪2 共cos 285 i sin 285 兲 0.3660 1.3660i. 冪2 共cos 165 i sin 165 E Exercises Performing Operations In Exercises 1–24, perform the operation and write the result in standard form. 1. 共5 i兲 共6 2i兲 Writing a Complex Conjugate In Exercises 25–32, write the complex conjugate of the complex number. Then multiply the number by its complex conjugate. 25. 5 3i 26. 9 12i 3. 共8 i兲 共4 i兲 27. 2 冪5i 28. 4 冪2i 4. 共3 2i兲 共6 13i兲 29. 20i 30. 冪 15 31. 冪8 32. 1 冪8 6. 共8 冪 18 兲 共4 3冪2i兲 Writing in Standard Form In Exercises 33–42, write the 7. 13i 共14 7i兲 quotient in standard form. 2. 共13 2i兲 共 5 6i兲 5. 共 2 冪 8 兲 共5 冪 50 兲 8. 22 共 5 8i兲 10i 3 5 5 11 9. 共2 2i兲 共3 3 i兲 10. 共1.6 3.2i兲 共 5.8 4.3i兲 11. 冪 6 冪 2 12. 冪 5 冪 10 13. 共冪 10 兲2 14. 共冪 75 兲2 15. 共1 i兲共3 2i兲 16. 共6 2i兲共2 3i兲 10 2i 33. 6 i 34. 35. 4 4 5i 36. 3 1 i 37. 2 i 2 i 38. 8 7i 1 2i 39. 6 7i i 40. 8 20i 2i 41. 1 共4 5i兲2 42. 共2 3i兲共5i兲 2 3i 17. 6i共5 2i兲 Performing Operations In Exercises 43–46, perform the 18. 8i共9 4i兲 operation and write the result in standard form. 19. 共冪14 冪10i兲共冪14 冪10i兲 20. 共3 冪 5 兲共7 冪 10 兲 43. 2 3 1 i 1 i 44. 2i 5 2 i 2 i 21. 共4 5i兲2 45. i 2i 3 2i 3 8i 46. 3 1 i i 4 i 22. 共2 3i兲 2 23. 共2 3i兲2 共2 3i兲2 24. 共1 2i兲2 共1 2i兲2

9781285057095 AppE.qxp 2/18/13 8:25 AM Page E11 Appendix E Complex Numbers E11 Using the Quadratic Formula In Exercises 47–54, use the Performing Operations In Exercises 83–86, perform the Quadratic Formula to solve the quadratic equation. operation and leave the result in trigonometric form. 47. x2 2x 2 0 48. x2 6x 10 0 49. 4x2 16x 17 0 50. 9x2 6x 37 0 51. 4x2 16x 15 0 52. 9x2 6x 35 0 53. 16t2 4t 3 0 83. 冤3冢cos 3 i sin 3 冣冥冤4冢cos 6 i sin 6 冣冥 84. 冤 2 冢cos 2 i sin 2 冣冥冤6冢cos 4 i sin 4 冣冥 3 85. 关53共cos 140 86. cos共5 兾3兲 i sin共5 兾3兲 cos i sin Writing in Standard Form In Exercises 55–62, simplify i sin 140 兲兴关23共cos 60 i sin 60 兲兴 DeMoivre’s Theorem to find the indicated power of the complex number. Write the result in standard form. the complex number and write it in standard form. 87. 共1 i兲5 55. 6i3 i 2 88. 共2 2i兲6 56. 4i 2 2i 3 89. 共 1 i兲10 57. 5i5 90. 共1 i兲12 58. 共 i兲3 91. 2共冪3 i兲7 60. 共冪 2 兲6 93. 92. 4共1 冪3i兲3 59. 共冪 75 兲3 1 i3 62. 1 共2i兲3 Using DeMoivre’s Theorem In Exercises 87–94, use 54. 5s2 6s 3 0 61. 冢cos 54 i sin 54 冣 94. 冤 2冢cos i sin 冣冥 2 2 10 8 Absolute Value of a Complex Number In Exercises 63–68, plot the complex number and find its absolute value. 63. 5i 64. 5 65. 4 4i 66. 5 12i 67. 6 7i 68. 8 3i Finding n th Roots In Exercises 95–100, (a) use Theorem E.2 to find the indicated roots of the complex number, (b) represent each of the roots graphically, and (c) write each of the roots in standard form. 95. Square roots of 5共cos 120 i sin 120 兲 96. Square roots of 16共cos 60 i sin 60 兲 冢 97. Fourth roots of 16 cos Writing in Trigonometric Form In Exercises 69–76, 冢 represent the complex number graphically, and find the trigonometric form of the number. 98. Fifth roots of 32 cos 69. 3 3i 70. 2 2i 99. Cube roots of 71. 冪3 i 72. 1 冪3i 73. 2共1 冪3i兲 74. 75. 6i 76. 4 5 2 共冪3 i兲 4 4 i sin 3 3 5 5 i sin 6 6 冣 冣 125 共1 冪3i兲 2 100. Cube roots of 4冪2共1 i兲 Writing in Standard Form In Exercises 77–82, represent Solving an Equation In Exercises 101–108, use Theorem E.2 to find all the solutions of the equation and represent the solutions graphically. the complex number graphically, and find the standard form of the number. 101. x 4 i 0 102. x3 1 0 103. x5 243 0 104. x 4 81 0 105. x3 64i 0 106. x6 64i 0 77. 2共cos 150 i sin 150 兲 78. 5共cos 135 i sin 135 兲 3 79. 2共cos 300 i sin 300 兲 3 80. 4共cos 315 i sin 315 兲 冢 81. 3.75 cos 冢 82. 8 cos 3 3 i sin 4 4 i sin 12 12 冣 107. x3 共1 i兲 0 109. Proof 冣 108. x 4 共1 i兲 0 Given two complex numbers z1 r1共cos 1 i sin 1兲 and z2 r2共cos 2 i sin 2兲 show that z1 r1 关cos共 1 2兲 i sin共 1 2兲兴, z2 r2 z2 0.

To add (or subtract) two complex numbers, you add (or subtract) the real and imaginary parts of the numbers separately. a bi, bi b a bi. a i2 1. i 1 1. x x2 1 0 Appendix E Complex Numbers E1 E Complex Numbers Definition of a Complex Number For real numbers and the number is a complex number.If then is called an imaginary number.

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