Section 8.3 Polar Form of Complex Numbers


 Damian Charles
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1 80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the sum of a real number and an magnary number Whle these are useful for expressng the solutons to quadratc equatons, they have much rcher applcatons n electrcal engneerng, sgnal analyss, and other felds Most of these more advanced applcatons rely on propertes that arse from lookng at complex numbers from the perspectve of polar coordnates We wll begn wth a revew of the defnton of complex numbers Imagnary Number The most basc complex number s, defned to be 1, commonly called an magnary number Any real multple of s also an magnary number Example 1 Smplfy 9 We can separate 9 as 9 1 We can take the square root of 9, and wrte the square root of 1 as A complex number s the sum of a real number and an magnary number Complex Number A complex number s a number z a + b, where a and b are real numbers a s the real part of the complex number b s the magnary part of the complex number 1 Plottng a complex number We can plot real numbers on a number lne For example, f we wanted to show the number, we plot a pont:
2 Secton 8 Polar Form of Complex Numbers 81 To plot a complex number lke, we need more than just a number lne snce there are two components to the number To plot ths number, we need two number lnes, crossed to form a complex plane Complex Plane In the complex plane, the horzontal axs s the real axs and the vertcal axs s the magnary axs magnary real Example Plot the number on the complex plane The real part of ths number s, and the magnary part s  To plot ths, we draw a pont unts to the rght of the orgn n the horzontal drecton and unts down n the vertcal drecton magnary real Because ths s analogous to the Cartesan coordnate system for plottng ponts, we can thnk about plottng our complex number z a + b as f we were plottng the pont (a, b) n Cartesan coordnates Sometmes people wrte complex numbers as z x + y to hghlght ths relaton Arthmetc on Complex Numbers Before we dve nto the more complcated uses of complex numbers, let s make sure we remember the basc arthmetc nvolved To add or subtract complex numbers, we smply add the lke terms, combnng the real parts and combnng the magnary parts Example Add and + 5 Addng ( ) + ( + 5), we add the real parts and the magnary parts Try t Now 1 Subtract + 5 from We can also multply and dvde complex numbers
3 8 Chapter 8 Example Multply: ( + 5) To multply the complex number by a real number, we smply dstrbute as we would when multplyng polynomals ( + 5) Example 5 Dvde ( + 5 ) ( ) To dvde two complex numbers, we have to devse a way to wrte ths as a complex number wth a real part and an magnary part We start ths process by elmnatng the complex number n the denomnator To do ths, we multply the numerator and denomnator by a specal complex number so that the result n the denomnator s a real number The number we need to multply by s called the complex conjugate, n whch the sgn of the magnary part s changed Here, + s the complex conjugate of Of course, obeyng our algebrac rules, we must multply by + on both the top and bottom ( + 5 ) ( + ) ( ) ( + ) To multply two complex numbers, we expand the product as we would wth polynomals (the process commonly called FOIL frst outer nner last ) In the numerator: ( + 5 )( + ) Expand Snce 1, ( 1) Smplfy + Followng the same process to multply the denomnator ( )( + ) Expand (1 + ) Snce 1, 1 (1 ( 1)) 17 Combnng ths we get
4 Secton 8 Polar Form of Complex Numbers 8 Try t Now Multply and + Wth the nterpretaton of complex numbers as ponts n a plane, whch can be related to the Cartesan coordnate system, you mght be startng to guess our next step to refer to ths pont not by ts horzontal and vertcal components, but usng ts polar locaton, gven by the dstance from the orgn and an angle Polar Form of Complex Numbers Remember, because the complex plane s analogous to the Cartesan plane that we can thnk of a complex number z x + y as analogous to the Cartesan pont (x, y) and recall how we converted from (x, y) to polar (r, θ) coordnates n the last secton Brngng n all of our old rules we remember the followng: x cos(θ ) r x r cos(θ ) y sn(θ ) r y r sn(θ ) y tan(θ ) x + y r x magnary r θ x x + y y real Wth ths n mnd, we can wrte z x + y r cos( θ) + r sn( θ) Example Express the complex number usng polar coordnates On the complex plane, the number s a dstance of from the orgn at an angle of, so cos + sn Note that the real part of ths complex number s 0 In the 18 th century, Leonhard Euler demonstrated a relatonshp between exponental and trgonometrc functons that allows the use of complex numbers to greatly smplfy some trgonometrc calculatons Whle the proof s beyond the scope of ths class, you wll lkely see t n a later calculus class
5 8 Chapter 8 Polar Form of a Complex Number and Euler s Formula θ The polar form of a complex number s z re, where Euler s Formula holds: θ re r cos( θ ) + r sn( θ ) Smlar to plottng a pont n the polar coordnate system we need r and θ to fnd the polar form of a complex number Example 7 Fnd the polar form of the complex number 7 Treatng ths s a complex number, we can consder the unsmplfed verson 7+0 Plotted n the complex plane, the number 7 s on the negatve horzontal axs, a dstance of 7 from the orgn at an angle of from the postve horzontal axs The polar form of the number 7 s 7 e Pluggng r 7 and θ back nto Euler s formula, we have: 7e 7cos( ) + 7 sn( ) as desred Example 8 Fnd the polar form of + On the complex plane, ths complex number would correspond to the pont (, ) on a Cartesan plane We can fnd the dstance r and angle θ as we dd n the last secton r x + r y ( ) + r To fnd θ, we can use cos(θ ) cos( θ ) Ths s one of known cosne values, and snce the pont s n the second quadrant, we can conclude that θ The polar form of ths complex number s x r e +
6 Secton 8 Polar Form of Complex Numbers 85 Note we could have used check the quadrant y tan(θ ) nstead to fnd the angle, so long as we remember to x Try t Now Wrte + n polar form Example 9 Wrte e n complex a + b form e cos + sn Evaluate the trg functons 1 + Smplfy + The polar form of a complex number provdes a powerful way to compute powers and roots of complex numbers by usng exponent rules you learned n algebra To compute a power of a complex number, we: 1) Convert to polar form ) Rase to the power, usng exponent rules to smplfy ) Convert back to a + b form, f needed Example 10 Evaluate ( + ) Whle we could multply ths number by tself fve tmes, that would be very tedous To compute ths more effcently, we can utlze the polar form of the complex number In an earler example, we found that ( + ) + e Usng ths, Wrte the complex number n polar form e ( ) e m Utlze the exponent rule ( ab ) a m n On the second factor, use the rule ( a ) a m b m mn
7 8 Chapter 8 ( ) 78 e 9 e Smplfy At ths pont, we have found the power as a complex number n polar form If we want the answer n standard a + b form, we can utlze Euler s formula 9 78e 9 78cos + 78sn 9 9 Snce s cotermnal wth, we can use our specal angle knowledge to evaluate the sne and cosne cos + 78sn 78 (0) + 78(1) 78 We have found that ( + ) 78 The result of the process can be summarzed by DeMovre s Theorem DeMovre s Theorem z r cos θ sn ( ) ( ) n n If ( ) + ( θ), then for any nteger n, z r cos( nθ) + sn ( nθ) We omt the proof, but note we can easly verfy t holds n one case usng Example 10: 9 9 ( + ) cos + sn 78 cos + sn 78 ( ) Example 11 Evaluate 9 To evaluate the square root of a complex number, we can frst note that the square root 1 s the same as havng an exponent of : 1/ 9 (9) To evaluate the power, we frst wrte the complex number n polar form Snce 9 has no real part, we know that ths value would be plotted along the vertcal axs, a dstance of 9 from the orgn at an angle of Ths gves the polar form: 9 9e
8 Secton 8 Polar Form of Complex Numbers e 1/ (9 ) Use the polar form 1/ 9 9 1/ e e 1/ 1 1/ Use exponent rules to smplfy Smplfy e Rewrte usng Euler s formula f desred cos + sn Evaluate the sne and cosne + Usng the polar form, we were able to fnd a square root of a complex number 9 + Alternatvely, usng DeMovre s Theorem we can wrte 9e 1/ cos + sn and smplfy Try t Now Wrte ( + ) n polar form You may remember that equatons lke x have two solutons, and  n ths case, though the square root only gves one of those solutons Lkewse, the square root we found n Example 11 s only one of two complex numbers whose square s 9 Smlarly, the equaton z 8 would have three solutons where only one s gven by the cube root In ths case, however, only one of those solutons, z, s a real value To fnd the others, we can use the fact that complex numbers have multple representatons n polar form Example 1 Fnd all complex solutons to z 8
9 88 Chapter 8 1/ Snce we are tryng to solve z 8, we can solve for x as z 8 Certanly one of these solutons s the basc cube root, gvng z To fnd others, we can turn to the polar representaton of 8 Snce 8 s a real number, s would st n the complex plane on the horzontal axs at an angle of 0, gvng the polar form 8 e 0 Takng the 1/ power of ths gves the real soluton: 0 ( 8 ) 1/ 8 1/ 0 0 e ( e ) 1/ e cos(0) + sn(0) However, snce the angle s cotermnal wth the angle of 0, we could also represent the number 8 as 8 e Takng the 1/ power of ths gves a frst complex soluton: 1/ 1/ 1 8e 8 1/ e e cos + sn + 1+ ( ) ( ) To fnd the thrd root, we use the angle of, whch s also cotermnal wth an angle of 0 1/ 1/ 1 8e 8 1/ e e cos + sn + 1 Altogether, we found all three complex solutons to z 8, 1+ z, 1+, 1 ( ) ( ) Graphed, these three numbers would be equally spaced on a crcle about the orgn at a radus of 1 Important Topcs of Ths Secton Complex numbers Imagnary numbers Plottng ponts n the complex coordnate system Basc operatons wth complex numbers Euler s Formula DeMovre s Theorem Fndng complex solutons to equatons Try t Now Answers 1 ( ) ( + 5 ) 1 9 ( )( + ) n polar form s e
10 Secton 8 Polar Form of Complex Numbers 89 Secton 8 Exercses Smplfy each expresson to a sngle complex number Smplfy each expresson to a sngle complex number (5 ) ( + ) + 8 ( ) ( ) 9 ( 5 + ) ( ) 10 ( ) 11 ( + ) ( ) 1 ( ) ( + ) 5 ( ) 1 ( ) (5) 1 ( + )( 8) 15 ( + ) ( ) 1 ( ) 1+ ( + ) 17 ( ) ( + ) 18 ( + )( ) Rewrte each complex number from polar form nto a + b form 9 e 0 e 1 e 8e e 5 5e 7 Rewrte each complex number nto polar 5 8 re θ form
11 90 Chapter 8 Compute each of the followng, leavng the result n polar 51 e e 5 5 e e re θ 5 form e e 5 e 55 e e 10 5 e 57 1 e 58 9e Compute each of the followng, smplfyng the result nto a + b form 59 ( + ) 8 0 ( + ) Solve each of the followng equatons for all complex solutons z z 7 z 1 8 z 1
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