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# Class Note for MATH 1310 at UH

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This 19 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 19 views.

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Date Created: 02/06/15

Seernn 24 De ni nn Page l 11le A enmplex numher ls a number that ean be wrmen m the form a bx where a and b are real numbers anal2 el The numbera ls calledthe real pan ofthe complex numbera 5 andthe numberb ls ealled the imaginary pan A complex number a bx where a 0 and b a 0 ls ealled a pure imaginary number For example 8 81 ls a pure lmaglnary number Every real number ls also a complex number wlth lmaglnary pan 0 For example the real number7 ean be Wntten as 7 01 Example Wnte eaeh numbenn the form abx andldenufy the real pan and the lmaglnary part 1 73 2 gr 3 bra5 4 35 s 3J4 5 3 Snlutinn m Example 1 1 7 73w The real part ls 73 andthe lmaglnary pan ls 0 2 lxUlx 2 2 The real part ls 0 andthe lmaglnary part ls 3617 7 61 The real part ls 7J2 and the lmaglnary part ls 6 Secl39nn 24 De ni nn P2g22uf19 45 3 5 4 gt6662 The real pan ls 7 and the lmagmary pan leg 3th 32 32 5 The real pan ls l and the lmagmary pan ls Example 2 Wnte the number 5 m the form n5 andldehtlfy the real pan and the lmagmary pan Snlntinn tn Example 2 505 The real pants 0 and the lmaglnary pan ls 5 Example 3 Wnte the number 74 m the form 15 andldenhfy the real pan andthe lmagmary pan Snlntinn tn Example 3 40 50 The real pants 74 andthe lmaglnary pants 0 Secl39nn 24 De ni nn P2geSuf9 Example 4 Wme the number 8 5 m the mm a 5 and ldenufy the real pan and me lmagmary pan Snlntinn u Example 4 7851 i 31 2 2 2 43 2 The real pam 74 andthelmaglnary pan Example 5 Wme the number 3394 m the formabx andldennfythe real pan and me lmagmary pan Snlntinn u Example 5 73x The real pam g andthelmaglnarypanls 7 Sesi39nn 24 Mainan Mimiiamnd Muiiipiiiai39nn Pageoom Addition Subtraction and Multiplication Additinn a aisai a saai Add the real parts a and s together and add the imaginary parts a and 1 together siihtrasu39nn aaiesai a bdl Subtraet the real parts a and s and subaaet the imaginary parts a and a Miiiu39piisatinn a ais ai as eaa aa Asi Multiply in the same manner as muitipiying binomiais and remember that i a a ais ai as aai Asi aaai2 as aai Asi eaa as eaa aa Asi Simplifying Powers nfi i 771711 To simplify ahigherpower of i sueh as 23 rewnte the expression as 3 5 123 7 1095 14 I Then use the powers of gven above to simpiity 12 5419 fo 5 7013 i Example 1 Evaluate eaeh expression 1 36123l 2 72ia13i 3 6 5x2 3x 5min 24 Mainnu Suhlnc nn m Mulu39plinl39nn 4 5 lt2 gtlt 3F9gt 6 lt J 7gtlt317gt Snlntinn m Example 1 1 3761281 P2ge uf9 32768x Addthzrealpms 22 Add hexmagnarypans 42 521 2 7217131 771273x Submmumpm H smmmgmpm 27 444 67 3 6512731 12718110171512 Mulnplyhkzbmnmm manmxau 52573 12718110115 hmmhenhahz 121571810x 27781 4 x Imus 2 xw I 1 71 5 2 73 E 2217331 27323x 71 Seelhu 24 Mainan Suhlnc nnrznd Mulu39plinl39nn P2ge uf9 6 ermW a r oaw 9J 12 xr3 xr4 xz 9m 12J 173 14 9 4 12J 73 1 13J 9J x Example 2 Evaluate the expresslon 87 3x 4 71 and express the results m the form aht Snlntinn tn Example 2 83147184 371 Addlhznalpnns 24 Addthzlnglmypnns 47 124t Examples Evaluate the expresslon 7244045 41 and express the results m the form aht Snlntinn tn Example 3 2315412 5341 Submanhzxealpnns 7275 S nmc hzlngmaxypms 74 47 Example 4 Evaluate the expresslon 53t724t and express the results m the form aht Snlntinn tn Example 4 Secl39nn 24 Mainnu slumpmam Mulu39plinl39nn P2ge7uf9 53x 24 meme omen mum mm He Ha H H elo2otest1212 Mu uplyhkehnamlals 592 sxmztezzt4 alu2utashl2 10 1220 5z 522444 Example 5 Evaluate the expresslon lo 4xes and express the results m the form 15 Snlntinn tn Example 5 lo 4xes lo3xes4 wme menunbexslnthzfmmnbl 107634 Add39hexealpms IEI76 l r Addthzlmagmrypms 24 47 Example 5 Slmpllfy x 6 Snlntinn tn Example 5 24a 2 246mx4m 7617 2461mm 4 A I 604 12 m Apply pmpeny uf expehehts x e l A 2 Applypmpmyaapms M e ta l H 191 e 71 e P2geEuf9 The enmplex cnnillgzte ofthe eomplex number 15 ls the eomplex numberaehr 15 To slmpllfy the quotrent multrply both the numerator and olenomrnator by the em eomplex eomugate ofthe olenomrnator am 15 zed nehdoeeadr 54 em rid and The prooluet ofa eomplex number and rts eomplex eomugate ls a nonnegatlve real number whomehx mbwahxebir a2bz Example 1 Express each othe followmg m the form 15 23 D l 21 101 13 6 339 2 Snlutinn tn Example 1 23 172 The eomugate ofthe olenomrnator ls 12 and le 2112114 5 23 23 12 1 21 l 2 12 4x31612 14 724x31 o 5 47 1 Multrply numerator and olenomrnator of by the eomugate of the olenomrnator Secrnn 24 1mm P2ge uf9 2 Mu1trp1y nurnerator and denornrnator of 10 by the conjugate ofthe denornrnator 13 The conjugate ofthe denornrnator 15 173 and13l1311910 1o 10 131 13 13 131 7101 302 7 19 710130 7 10 730nm 7 10 30 10 l 3x 3 Mu1trp1y numerator and denornrnator of by the conjugate ofthe denornrnator 9 The conjugate ofthe denornrnator 15 39 and 3790mm 981 90 39 71m 718541 7 981 718541 7 90 18 54 77x 90 90 Eva1uate tne enpressron and express the resu1ts m the form 15 5 Secl39nn 24 Dirisim Page ln 1le Snlntinn tn Example 2 Muluply numerator and denommator by the eomugate ol the denommator and camugalz ri e 2 2 recallthatabxa been 5 17 l 375 35 35 3751 Example 3 x and express the results m the form 15 Evaluate the expresslon 4 Snlntinn tn Example 3 Muluply numerator and denommator by the eomugate of the denommator and camugalz ri e 2 2 recallthatabxa been 5 3917 39 2 247 24 2741 767121448173sz 7 416 76712118136 7 20 7636 7 20 1218 742 7 20 42 7 20 20 21 3 77x 10 10 Secl39nn 24 Dirisim Page ll 11le Example 4 Evaluate the expresslon 377 and expressthe result ln the folmabx Snlutinn tn Example 4 Frrst rewrrte 3770quot m fract on form Then muluply numerator and denomrnator can ate we 2 2 by the eomugate othe denomrnator and reeall that a hrnrhr a 5 3770quot Example 5 Evaluate the expresslon 3732 and express the result m the form 15 x Snlutinn tn Example Frrst rewrrte both numerator and denomrnator m the form a 5 Then multrply numerator and denomrnator by the eomugate othe denomrnator and reeall that can ate we 2 2 abx men 5 Secl39nn 24 Dirisim Page 12 11le Camp lex Runlsquuzdnli Equz nns 93s 7344715730 45 7 7 7306715x 773379 7 45 33 9 7373 7711 l 7quot Secl39nn 24 Camp lex Runls uf Quzdnl39l Equz nns Complex Roots of Quadratic Equations Ifthe alsemmmam D b 7 4a ofthe quadratll equatlon ax bxc 0 11 0 ls negatlve the equatlon has no real roots The roots ofthe equatlon are complex numbers and appear as a complex conjugate pair Example 1 Solve eael equaaon 1 x2360 2 x2x10 3 4x2x50 Snlutinn m Example 1 Secl39nn 24 amplex Runlsquuzdnli Equz nns Pig 1 11le The soluuons are x 61 andx 761 The dlscnmlnantD 52 74a 02 74mm 7144 and the mots ofthe equation es and 6 are complex eomugates 2 To solve x2 x10 subsmute a 11 1ahd 1 mto the quadrau formula ebbAZ 74a 2a 7 71L 12 7411 21 J 7 andx 2 The ducnmmant D b 7 4a 12 7411174 73 and the mots ofthe 1J5 1J5 equation 71 and e 71 are complex eomugates Secl39nn 2 Complex Runlsquuzdnli Equz nns Pig u my 3 To solve 4x2 x 5 0 substltute a 4 o 1and e 5 rmo the quadraue formula 75 rb2 7 4a 2a 71 6274900 24 The dlscnmlnant D 5 74a 12 74451780 779 and me roots ofthe e uatlon q 8 Example 2 Flnd all complex soluuorls ofthe quadraue equauor x2 3x12 0 lee results m the form 15 Snlutinn to Example 2 To solve the equauorr substltute a 1 o 3 and 12 rmo me quadraue formula Secl39nn 24 emplex Runlsquuzdnli Equz nns Page IS 11le 4a 73 3 74002 21 The soluuons are x Example 3 Fmd an complex solutions othe quadratic equauon xz ex 78 Gwe results m the form a 5 Snlntinn m Example 3 Begm by multiplying both sides by 2 to clear he equauon offracuons 2zeexj2eamp x272x716 2x1671616 x272x160 Subsmuce a A 72 and 16 mtothe quadraue formula Secl39nn 24 emplex Runlsquuzdnl39l Equz nns Pig m 11le rbtxbz 74a 2a 2 ix 22 4115 21 um 2 2 760 2 The solutions are x1 s1 51and x1J x Example 4 Fmd all complex solutions othe quadratic equauon 3x2 45 0 Gwe results m the form a 5 Snlntinn m Example 4 Subsmute a 3 b and e 5 mo the quadrath formula Secl39nn 24 emplex Runlsquuzdnl39l Equz nns Pig 17 11le The soluuons are Example 5 Fmd all complex solutions othe quadratic equauon 4x245 0 Gwe results m the form 15 Snlntinn m Example 5 4x2450 4x2457450745 The solutions are x 3 5 5 1 and xeix 2 Exercise Set 24 An Introduction to Complex Numbers State whether or not the following numbers can be classified as real complex or both 1 2 734752 79139 J3 State the real and the imaginary part of each of the following complex numbers 10 11 12 25i 3 76139 Simplify each of the following expressions and write the answer in the form a bi 13 14 15 16 17 5 2139 7 7 3139 6 7 8139 7 72 139 77 7 3i 7 4 6139 79139 7 75 4139 63 7 5139 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 752 7139 74139 3 5139 2139 9 7 139 5 374139 72 504 78 3139 472139 8139 7273i Simplify each of the following expressions and write the answer in the form a bi 35 36 J3 m Exercise Set 24 An Introduction to Complex Numbers 37 38 39 40 41 42 43 44 45 46 47 4 49 5 O 51 52 9 434 NJ J35 mm m zwan 3471243 JENIXJLE 343 4J79 27173 3472 Find all complex solutions of the following equations by first completing the square and then using the quadratic formula Express all answers in the form abi 53 x2 8x200 54 x2 76x 734 Find all complex solutions of the following equations Express all answers in the form a bi 55 56 57 58 59 60 61 62 63 64 65 66 67 68 x2 490 x2 160 25x290 100x2810 x2 76xl30 x2 710x74l x2 6x290 x2 78x280 3x25x74 5x273xl0

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