8.3 POLAR FORM AND DEMOIVRE S THEOREM


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1 SECTION 8. POLAR FORM AND DEMOIVRE S THEOREM POLAR FORM AND DEMOIVRE S THEOREM Figure 8.6 (a, b) b r a 0 θ Complex Number: a + bi Rectagular Form: (a, b) Polar Form: (r, θ) At this poit you ca add, subtract, multiply, divide complex umbers. However, there is still oe basic procedure that is missig from the algebra of complex umbers. To see this, ider the problem of fidig the square root of a complex umber such as i. Whe you use the four basic operatios (additio, subtractio, multiplicatio, divisio), there seems to be o reaso to guess that i i. That is, To work effectively with powers roots of complex umbers, it is helpful to use a polar represetatio for complex umbers, as show i Figure 8.6. Specifically, if a bi is a ozero complex umber, the let be the agle from the positive x to the radial lie passig through the poit (a, b) let r be the modulus of a bi. So, a r, b r si, i i. r a b you have a bi r r si i from which the followig polar form of a complex umber is obtaied. Defiitio of Polar Form of a Complex Number The polar form of the ozero complex umber z a bi is give by z r i si where a r, b r si, r a b, ta ba. The umber r is the modulus of z is the argumet of z.
2 484 CHAPTER 8 COMPLEX VECTOR SPACES REMARK: The polar form of z 0 is give by z 0 i si where is ay agle. Because there are ifiitely may choices for the argumet, the polar form of a complex umber is ot uique. Normally, the values of that lie betwee are used, though o occasio it is coveiet to use other values. The value of that satisfies the iequality < Pricipal argumet is called the pricipal argumet is deoted by Arg(z). Two ozero complex umbers i polar form are equal if oly if they have the same modulus the same pricipal argumet. EXAMPLE Fidig the Polar Form of a Complex Number Fid the polar form of each of the complex umbers. (Use the pricipal argumet.) (a) i (b) i (c) i Solutio (a) Because a b, the r, which implies that r. From a r b r si, So, a r 4 z 4 i si 4. (b) Because a b, the r, which implies that r. So, a si b r r it follows that So, the polar form is z arcta i si arcta 0.98 i si0.98. (c) Because a 0 b, it follows that r so z i si arcta.. si b r., The polar forms derived i parts (a), (b), (c) are depicted graphically i Figure 8.7.
3 SECTION 8. POLAR FORM AND DEMOIVRE S THEOREM 485 Figure 8.7 (a) z = + i si 4 4 θ z = i [ ( ) ( )] 4 z = + i θ (b) z = [(0.98) + i si(0.98)] z = i θ ( ) (c) z = + i si EXAMPLE Covertig from Polar to Stard Form Express the complex umber i stard form. Solutio Because si, you ca obtai the stard form The polar form adapts icely to multiplicatio divisio of complex umbers. Suppose you are give two complex umbers i polar form z z r i si r i si. The the product of z is give by Usig the trigoometric idetities you have z 8 i si z 8 i si 8 z z z r r i si i si r r si si i si si. si si si si si z z r r i si. i 4 4i. This establishes the first part of the followig theorem. The proof of the secod part is left to you. (See Exercise 6.)
4 486 CHAPTER 8 COMPLEX VECTOR SPACES Theorem 8.4 Product Quotiet of Two Complex Numbers Give two complex umbers i polar form z z r i si r i si the product quotiet of the umbers are as follows. z z r r i si Product z z r r i si, z 0 Quotiet This theorem says that to multiply two complex umbers i polar form, multiply moduli add argumets, to divide two complex umbers, divide moduli subtract argumets. (See Figure 8.8.) Figure 8.8 z z z z z θ + θ r r r r z θ θ r r r r z z θ θ θ θ To multiply z z : Multiply moduli add argumets. To divide z by z : Divide moduli add argumets. EXAMPLE Solutio Multiplyig Dividig i Polar Form Determie z z z z for the complex umbers z 5 i si 4 Because you are give the polar forms of z z,you ca apply Theorem 8.4 as follows. z z 5 multiply divide z 5 z 4 4 subtract 4 add 6 i si 6 i si z i si subtract add i si 6. i si
5 SECTION 8. POLAR FORM AND DEMOIVRE S THEOREM 487 REMARK: Try performig the multiplicatio divisio i Example usig the stard forms z DeMoivre s Theorem The fial topic i this sectio ivolves procedures for fidig powers roots of complex umbers. Repeated use of multiplicatio i the polar form yields Similarly, 5 5 z 4 r 4 4 i si 4 z 5 r 5 5 i si 5. i z 6 6 i. z r i si z r i si r i si r i si z r i si r i si r i si. This patter leads to the followig importat theorem, amed after the Frech mathematicia Abraham DeMoivre ( ). You are asked to prove this theorem i Chapter Review Exercise 7. Theorem 8.5 DeMoivre s Theorem If z r i si is ay positive iteger, the z r i si. EXAMPLE 4 Solutio Raisig a Complex Number to a Iteger Power Fid i write the result i stard form. First covert to polar form. For i, r which implies that. By DeMoivre s Theorem, So, i i si. i i si ta () i si ()
6 488 CHAPTER 8 COMPLEX VECTOR SPACES i si i0) Recall that a equece of the Fudametal Theorem of Algebra is that a polyomial of degree has zeros i the complex umber system. So, a polyomial like px x 6 has six zeros, i this case you ca fid the six zeros by factorig usig the quadratic formula. x 6 x x x x x x x x Cosequetly, the zeros are x ±, x ± i, x ± i. Each of these umbers is called a sixth root of. I geeral, the th root of a complex umber is defied as follows. Defiitio of th Root of a Complex Number The complex umber w a bi is a th root of the complex umber z if z w a bi. DeMoivre s Theorem is useful i determiig roots of complex umbers. To see how this is doe, let w be a th root of z, where w s i si z r i. The, by DeMoivre s Theorem you have w s i si, because w z, it follows that s i si r i si. Now, because the right left sides of this equatio represet equal complex umbers, you ca equate moduli to obtai s r which implies that s r equate pricipal argumets to coclude that must differ by a multiple of. Note that r is a positive real umber so s r is also a positive real umber. Cosequetly, for some iteger k, which implies that k k,. Fially, substitutig this value for ito the polar form of w produces the result stated i the followig theorem.
7 SECTION 8. POLAR FORM AND DEMOIVRE S THEOREM 489 Theorem 8.6 th Roots of a Complex Number Figure 8.9 r th Roots of a Complex Number For ay positive iteger, the complex umber z r i si has exactly distict roots. These roots are give by r k i si where k 0,,,...,. REMARK: Note that whe k exceeds, the roots begi to repeat. For istace, if k, the agle is k which yields the same value for the sie ie as k 0. The formula for the th roots of a complex umber has a ice geometric iterpretatio, as show i Figure 8.9. Note that because the th roots all have the same modulus (legth) r, they will lie o a circle of radius r with ceter at the origi. Furthermore, the roots are equally spaced alog the circle, because successive th roots have argumets that differ by. You have already foud the sixth roots of by factorig the quadratic formula. Try solvig the same problem usig Theorem 8.6 to see if you get the roots show i Figure 8.0. Whe Theorem 8.6 is applied to the real umber, the th roots are give a special ame the th roots of uity. Figure i + i i i 6th Roots of Uity
8 490 CHAPTER 8 COMPLEX VECTOR SPACES EXAMPLE 5 Fidig the th Roots of a Complex Number Determie the fourth roots of i. Solutio I polar form, you ca write i as i i si so that r,. The, by applyig Theorem 8.6, you have i 4 4 k 4 4 i si k 4 4 k 8 Settig k 0,,, you obtai the four roots z i si z 5 5 i si z 9 9 i si z 4 i si as show i Figure 8.. i si 8 k. REMARK: I Figure 8. ote that whe each of the four agles, 8, 58, 98, 8 is multiplied by 4, the result is of the form k. Figure i si 5 + i si 9 + i si 9 + i si
9 SECTION 8. EXERCISES 49 SECTION 8. EXERCISES I Exercises 4, express the complex umber i polar form I Exercises 5 6, represet the complex umber graphically, give the polar form of the umber. (Use the priciple argumet.) 5. i 6. i 7. i 5 8. i 9. 6i i. 6i 4. i 5. i 6. 4 i I Exercises 7 6, represet the complex umber graphically, give the stard form of the umber. 7. i si i si 0. i 4 5 i si i i si. 8 i si i si i si i i si i si i si I Exercises 7 4, perform the idicated operatio leave the result i polar form I Exercises 5 44, use DeMoivre s Theorem to fid the idicated powers of the give complex umber. Express the result i stard form. 5. i 4 6. i 6 7. i 0 8. i7 9. i 40. i si i si i si [() i si()] 4[(9) i si(9)] (5) i si(5) i si [() i si()] [(6) i si(6)] 9[(4) i si(4)] 5[(4) i si(4)] 4. i si i si 0.5 i si 8 I Exercises 45 56, (a) use DeMoivre s Theorem to fid the idicated roots, (b) represet each of the roots graphically, (c) express each of the roots i stard form. 45. Square roots: 6 i si i si i si i si 6 4 i si 6 4 i si 9 9 i si i si 4
10 49 CHAPTER 8 COMPLEX VECTOR SPACES 46. Square roots: 47. Fourth roots: 48. Fifth roots: 9 i si i si 5 5 i si Square roots: 5i 50. Fourth roots: 65i 5. Cube roots: 5 i 5. Cube roots: 4 i 5. Cube roots: Fourth roots: i 55. Fourth roots: 56. Cube roots: 000 I Exercises 57 6, fid all the solutios to the equatio represet your solutios graphically. 57. x 4 i x x x x 64i 0 6. x 4 i 0 6. Give two complex umbers z r i si z r i si with z 0 prove that z z r r i si. 64. Show that the complex cojugate of z r i si is z r i si. 65. Use the polar form of z z i Exercise 64 to fid each of the followig. (a) zz (b) zz, z Show that the egative of z r i si is z r i si. 67. Writig (a) Let z r i si Sketch z, 6 i si 6. iz, zi i the complex plae. (b) What is the geometric effect of multiplyig a complex umber z by i? What is the geometric effect of dividig z by i? 68. Calculus Recall that the Maclauri series for e x, si x, x are e x x x si x x x!! x! x 4 4!... x5 5! x 7 7!... x x! x 4 x6 4! 6!.... (a) Substitute x i ito the series for e x show that e i i si. (b) Show that ay complex umber z a bi ca be expressed i polar form as z re i. (c) Prove that if z re i, the z re i. (d) Prove the amazig formula e i.
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