i"M_K%lecp"&2uAO?c] ?VGc6ho7S-X*h[m?SkS.J8nD2q4-he4CBMk]#h)AgJAJs+M?O-E2a= %0c%@4FOB4THL/*:oDM"KD.4&/EJ? j(Zf0ek&YrRp-T"U[7eKd>rS1+(jKj>spp8t%'q-gI6S0TVWMrd[9I4G24mMOp 8aH/7YtrK^LFlrSBmr1aM0'1H"G8.oYoHn[8!HL!TD_.Yqe6=%;!bN#s]a("eT C1^JE\U62Gbg&*.1)cr]jD_KsV(WN-Q^, fn@90QlTcIYqYLOR5'B ]sK"Nc#XJj&qF_cXkYXE>c3(0294i/tu/R:Va"Y&k_0F-57\'2MGr+%\F(5A1p L]]p@Xuae@3"+A)W?Fa-'/9IX2DQ>9&]sEMog)n3@N'E*[EII__]72=&M! D+ko1l6+esN885^0Nr2b#OEloZFSQpgc!%Df^=se+QB/KIIK9)rnN'N*M7C4>bgM^ The division of complex numbers in polar form is calculated as: \begin{aligned}\dfrac{z_1}{z_2}&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)}{r_2\left(\cos\theta_2+i\sin\theta_2\right)}\\&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)}{r_2\left(\cos\theta_2+i\sin\theta_2\right)}\left(\dfrac{\cos\theta_2-i\sin\theta_2}{\cos\theta_2-i\sin\theta_2}\right)\\&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)\left(\cos\theta_2-i\sin\theta_2\right)}{r_2\left(\cos^2\theta_2-(i)^2\sin^2\theta_2\right)}\\&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)\left(\cos\theta_2-i\sin\theta_2\right)}{r_2(\cos^2\theta_2+\sin^2\theta_2)}\\&=\frac{r_1}{r_2}\left[\cos(\theta_1-\theta_2)+i\sin(\theta_1-\theta_2)\right]\\&=r\left(\cos\theta+i\sin\theta\right)\end{aligned}. _iull%qfet!1"4F(Q\6UEN14o6s4=eD7i+Nq1[A/IRoX8!bi(.KX#N;R83. k/BohcX=8ibMpHh^l?UpF/UHS8)lY,L-s/k- 7_?-iFDkG. r/>=UcR4oNJ_S0=OEDfN3E+h]i=OLis5fhj@)Ohc;3/&*])>! Modulus Argument Type . /.)i+!! a0siEKhHLYijF.=ik37"tHNH0N]he3La6A("q\osg=&?Hhm@DK!JGhKUXLJ"j>. ;iS+VrW[+I3Cl^6e4-N/s9hu8p&B=QH;MRh)RWMZ:O a7dc6pkG>4?7g,::JJSqLeY7,KQc>mO"coDKL6=NESuW'.Fsf448IF\hA5Plk6MN 09r>L?\Q4Q+XsooM"MGl\u?iMNP'%)nSY&\/sWP+)AXD;cUg.%B'fNk@Q5rOc:C oh=BZ&!%s&:\i>b&3S7JMA]@[iC106"?-roO>juU;-#QJN,Fp\*V?-E;lt.oAsG @u7l*/[Tpr,Zm[h4=5Lm^@8=c-:RSfOA^%:k&_nZ4G%)o7TePG%.G:otbT]Wg'4mORk^<0k1n.bC/_:YKIr1/[R\cUaYI*TaLba!+s8Z6Wh? !a_rm7,/D_mrDl)5UV/a\2,=THt4oLLdtUq&tkB:eU&!3FtS/.Y9:P@Yeg(5M]M &+aa@&)lL7&Yu=u#R)&!%kqrDefl-:Ib,fB8G^,! K7qWu5s-)]S*Us7;2'Mm?f)uCnRH4MF)O5WJak2mn%96";&NNY\:@X8!DDc-Sp A,"Z_)6U;0Y-V4&"VHu?\fdts:/F]SG.4!kQ'uG=pqBFs1aO_(@:R(Er:LGMA#,46 7?JaQqU4FH3#FZT2MkPX4"r0SFsq8A?ITc=CP'E95AM,U233bCW'-Mi. .gntjJk5kh&mRjilit>U]H<=1LOW9EMpPfa[bc(HaUqL(XJ5-KTBlG@Q5&]kpI=] ;c8Y ]E[as(KX]h[K C! l&Cbl(S.J3[ripj1))hLf,*[QfH_0H->e[:jW%Na!e[[^/^9=c&g_0;3N?#(i The conjugate of the denominator \(8-2i is $$8+2i$$. 5cmG58!AH4F"6_++YMU_5Pg(T5u[n%:=Oae 8;U;BZ#7H%&Dd/>)cLkZS4;mRZ+,^I1f=S-ZHMUC-ZDojR32hNRWM,mN(cPj*91j gs,!F*=7eHLbrjQC:E(V3[M>$4?Bm? /diR/oWt4P6+'#Aqb? \TaP>I-g^IMo"e!Smm.qU4;P4qT;(D$'--8]J^^RG->$R-=sa,?VoO@Q"#Mfa4$ m=H"#)b]e[(? #o"qSj9U:D),/nV^g@j(a? pJ7uJ^bR&SkH9+6t#;qKNgc(i30rhXX:(UnXQ_[>)ObTeAi"aG"gq/lT9Ob]O7 c/giT>OC:ACARg4r%!7!Mf6b[SFF1i_DmB,"6jo,^uk_>^7-&8r!3Z;m04A3E]F8*40ok"suF!5&I['!PF54? Let $$a=5$$, $$b=\sqrt{2}$$, $$c=1$$, and $$d=-\sqrt{2}$$. n-3#mU)'"2&CD\Ui[X>He[Be(=C&A9T_5OsfYt6Z(FQY+.jn6Z*Uu"<7Mj>uVMI'jJ2f)%2;QA/Chc&rBb@?a#Z!#5& "e6NkK[W--U6efQ\f7_,bNnqBB4*N+1FMd9&-4O#g;/G6Ab4Xl,b]dbY/(fKJP Divide the real part and the imaginary part of the complex number by that real number separately. hn_9TNY0Z*dh6pBld.Ps-'tKu-.7D/AmJ)\0ArHm@-igSfa/S(PBXS41pjRc"BW1M H������@��{v��P!qєK���[��'�+� �_�d��섐��H���Ͽ'���������,��!B������*ZZ(DkQ�_����7O���P�ʑq���9�=�2�8'=?�4�T-P�朧}e��ֳ�]��IN{^�0����m��@\�rӣdn":����D��j׊B�MZO��tw��|"@+y�V�ؠ܁�JS��s�ۅ�k�D���9i��� The polar form of the complex number $$z=a+ib$$ is given by: $$z=r\left(\cos\theta+i\sin\theta\right)$$. C_BH/CU#_b>jqsT/tM6SrJKighjaJF-Y50KVNk2pF#EpeY U;msVC,Eu!03bHs)TR#[HZL/EJ,? Find more Mathematics widgets in Wolfram|Alpha. 2E7N4th<0f.61)@3U("cA+&9HMc3hQfkP?:-lZuquQ>k("]! ']KXmNPN.\!\9NM&SpaD2sIEqU3& 8;V^nD,=/4)Erq9.s2\ZIad3^\eb'#[=0#77'g#mVU8C)r4D@2p7hORP[s&COX]WpC!rYphuJs Polar - Polar. "!a6p'ch@r_NJiu- (2f^N#;KZOmF9m"@J\F)qc8bXPRLegT58m%9r R j θ r x y x + yj The complex number x + yj, where g%:\iB,;h[iY:W)&F,Tn.&"hp0G+!nmOR9UQ3J<9eQTlgZ0.MO[:E_,HW)31941 The quotient $$\dfrac{4+8i}{1+3i}$$ is given as $$\dfrac{14}{5}-i\dfrac{2}{5}$$. S)]jgDHa=VdkUq57Bn6Y,ssf3"GJ?hs)1i0@Akj)&V**lic03%=kH(tRYV-*#JZFaHhlYlmB5g_gcAJNKK9rrYcC]l43X+uq?= Polar form is where a complex number is denoted by the length (otherwise known as the magnitude, absolute value, or modulus) and the angle of its vector (usually denoted by … pgf\Tjj0sM3fnJ5lb7.pX3.j+FkAS6qOdBnBoVil)Z_,4Y(l)p5\L7fjA;eV-k-Wkr(,fBVS#P9sNNKkHSm0Qm18#nEmj=@ub&>NE2!.TnF;HQ-hd "5AguOY,Pb+X,h'+X-O;/M6Yg/c7j"jROJ0TlD4cb'N>KeS9D6g>H. #&GtN>Kl=[d]kZ5! G7]JaYcibN*^hO+[NPA;-V'/ER][!lV[V]:aNaOnA_D)H]ZV\=*-rT! O'L&CXebH4mB2'oZ4e6,Ck+cEgl*uoHliHPpAOWE5>FVe\mp469'S)-ll!+!05c "jci](k[jX1K8gcSh2-@#5j^'AQR'_HM&m+W>b^I,g+;0H4A&pI%BR Figure 1.18 shows all steps. R)_pW(rAWO&M'N+J8Tt;Oj^DpQ?fTQAW)!+N_n>gB The number of the form z=a+ib, where $$a$$ and $$b$$ are real numbers are called the complex numbers. N^>r,[,;EVMi^79)CFIS"Q+bdpBEiB_Ki;r:Uo8B_N=ndWdNhg^Q\'k[tDpS4IB2?F%Zgp&q! BI_@f6I%^e2KIYpn'd*i@cUI]L5pu#Yo0_gB7^6V"iJ@/K_+mg? Polar form. . 9BI,Z?7LiQ.M_*FF:M\G-Y8sP_65>3K,-+QIS!>#]8Nm0To;I';)QG5_L,en/f&"ILSp0.&F"S>D[&5Es:ht [P 9"gpc.4]>p'jS)]i7B^f3mGs>YB?74O-HCRFW.S^5&bJ6n6oXJS=EJ0F&! \[\begin{aligned}\dfrac{a+ib}{c+id}&=\dfrac{ac+bd}{c^2+d^2}+i\left(\dfrac{bc-ad}{c^2+d^2}\right)\\&=\dfrac{(5\times 1)+(\sqrt{2}\times(-\sqrt{2}))}{1^2+(-\sqrt{2})^2}+\\&i\left(\dfrac{(\sqrt{2} \times 1)-(5\times(-\sqrt{2}))}{1^2+(-\sqrt{2})^2}\right)\\&=\dfrac{5-2}{1+2}+i\left(\dfrac{\sqrt{2}+5\sqrt{2}}{1+2}\right)\\&=\dfrac{3+6\sqrt{2}i}{1+2}\\&=1+2\sqrt{2}i\end{aligned}. 'reTg^g+V&W96_eCfF!b7Fq5s-BmZddc AYH]B8>4FIeW^dbQZ.lW9'*gNX#:^8f. "a)]_le6g$..$t!Seb'XgcBgk9QX^erah/O[/<3=]9u:V? @)\p#@q@cQd/-Ta/nki(G'4p;4/o;>1P^-rSgT7d8J]UI]Gtg> !K* 8;W:*$W%O=(4EZ]!Alba@DFR/B%3J%Lk\1EH_kkpdl'm-<7=dXNaE@^%V(,h)ukn Division is obviously simpler when the numbers are in polar or exponential form. The division of two complex numbers $$z_1=a+ib$$ and $$z_2=c+id$$ is given by the quotient $$\dfrac{a+ib}{c+id}$$. h:[%17W3X!d*+lKaZjXPKbo)dl^4C"h+\;8=e'u867tI:.fuB,HQj@lFD^ACd$\g \begin{align}\dfrac{3+4i}{8-2i}&=\dfrac{3+4i}{8-2i}\times\dfrac{8+2i}{8+2i}\\&=\dfrac{24+6i+32i+8i^2}{64+16i-16i-4i^2}\\&=\dfrac{16+38i}{68}\\&=\dfrac{4}{17}+\dfrac{19}{34}i\end{align}. ^95]PagD+'*B1DJ#!g&b&MsD:nD#c\^THQo1-T9Yj*8q6m(0o!Bt,j5q^=6,Ym;i G'.l7hI,;pNkL1@ab*_'R.1r"O0Ybh@b0*=P8W5D[@jS^ZU-:J96=Bi[h5+=Sc;AR ]kNRS#fe#67.4ph4Q,[^h4Q3-"=CG49j3h'4NJ3c3kI:iBbKE9X_UZ cmVM0-jnl$92hmKb=WKqdO]O7U1>2C[2r_"-WjIQc%i"#$e?DNqgJbhNl(bNd+/:. ;?R]J6#@+6Z,X,u#5+g&oSYNWDm^SA(OQK'#BG8tl)gJ*p-?W*'C$V;rca+Z__J1kpBk#FoSg_*\9Dm?UBs\*7OT4u7Q^ Solution The complex number is in rectangular form with and We plot the number by moving two units to the left on the real axis and two units down parallel to the imaginary axis, as shown in Figure 6.43 on the next page. :trk5?5(e(V2.Ent(Obu4SY0noZ1f;"52e+V;rcbku_[$?GC[OQX^^nUl>8L%K$4! =]_HRlIKt$c$np$hMAad]'ek/cJ5s[I,FbfpH--2mQ&%'lu[KuP'L_E"QI0;mb\>2 *fn3Q'kJfADURnaCJ4=HZ-ioXTs)#%o*b*!9BWJh9"m6cM@L[CCQXmrG:B W(PQ_%WtZ_*fULLmcPNF@3cm^8=WV@cAYDc%UUr'gmL1RW1RUW!51SNN1Wuqf?E]N !_a)3kKs&(D.]? ]%s@bA1m=R_AV>Su#MW$>21E@($D1e.p_dm=l+o*.+3^&)4,iMs&k7:^mnoC\UJ VCG9UQEqOrt]5')D$L,hubK$^7jKAh[%\%]mF3"MI7b[bV^O/[Y/.p;3G4rP5:[?pfa =jjO* endstream endobj 41 0 obj 449 endobj 42 0 obj << /Filter /FlateDecode /Length 41 0 R >> stream Here, $$a$$ is called the real part, and $$b$$ is called the imaginary part of the complex number $$z$$. $$\dfrac{5+\sqrt{2}i}{1-\sqrt{2}i}=1+2\sqrt{2}i$$, $$\dfrac{3+4i}{8-2i}=\dfrac{4}{17}+\dfrac{19}{34}i$$. pn3l$#T?QMC(b?XP?-/0eX:X-eR3hkM7pNl)^(^gf)KQec2CaB,0=]29D3nM+>r 0'3ph^Sg+e$.KrXV;+1^I)eag<7%9f5jS0\2\A-'J6uWW$kAOT[9WGCFQ"rRSEH^Dr)r$>a;bG0.:b(em!g$ftVnh;$==LWWB9k/HWt=MHJj. *F Usually, we represent the complex numbers, in the form of z = x+iy where ‘i’ the imaginary number. WI$C=.3Kg%0q=Z:J@rfZF/Jn>c*.sY9:? R2HpW!mbA8R3N'Nf C_BH/CU#_b>jqsT/tM6SrJKighjaJF-Y50KVNk2pF#Ep$eY qdoI6Vj(pLrL\j#Al0e1U+gMW&kKl?Rn$js.Nu%PFSZA#V1gNQa;"FPVGKgGC+DU' Z>:tKkns"U!TUC/P[RA. L6Z-PT4&EQ'acF^:K''_?3!&nCr=5Y9&)2MJ?B8p)Desa>pY>K0 lno=,quG.&I:BT@dGTg@j"\9VG!qVJLEIHZZ#Yq=>ns(Ihu_V8TffY1'[Z'Zl#lM \*?b[ko/T8l(jQfFCtRLmJH;>oA9B4qn8oZl0&NW9a61).IdMa$jfe5[u-5jbh$dIB^'5Ij92JHI=LWbio_tti;&eo*mf&j!f?I i+@KjfJuI'ge4&Z?s+M>qRBQ,Ra0t%\D3TK:]p.?4dXl>W*bQ)bt:doD1bKa^C1P[ [86(5[6-Hl"ckI@LqJ:] #o$"qSj9U:D),/nV^g@j(a? He says "It is the resultant complex number by dividing $$3+4i$$ by $$4-3i$$.". J! Ll@DeW>+NM7qH63B=,9L:+;Bl8sMR @ZZW5QZe4.loe,r=cSfSpH3G#*T*-S'kMkJ8sA?_mUVZ,lcDkCP?lb!/N\52:HXE p\fuSue//WZu79\p=g.">.J#,akKle0JbFh@sbKhBjaW_l%^22fLc2h#bD./kfn! The modulus of the complex number $$z=a+ib$$ is $$|z|=\sqrt{a^2+b^2}$$. eZ^IdkI:K_rPKtQW>-Jdh>ZlIO>037ZPlu#TjXhPbj4? e%Z(oCSM-rTTJ:GN!g:dO2pB1pq'a-C_@=K]t!cfCt\9T_,PY-F30:c/!d'omG+#> a0siEKhHLYijF.=ik37"tHNH0N]he3La6A("q\osg=&?Hhm@DK!JGhKUXLJ"j>. The mini-lesson targeted the fascinating concept of the subtraction of complex numbers. Where $$\theta=\theta_1-\theta_2$$ and $$r=\dfrac{r_1}{r_2}$$. XG#DEEE.S3gZ*Kr9u3*6F%>]W-s!VH#a5-!hoMG&=Da>kiW1;QX8"*jmad^W6B% nnctpY.CNmOZ2sS=qSmNqdEqK2QQdf:rf/2b[DdWnp*L]rYR:gVN@et#P",k^3I h=/BLW9SqnLS4>pCd3O?>)M0mDiVlETfCeL+es.6)bpqYK,t5P1Ou.qdh)O5S#< 4jm9W+nL9O&YnLthI6;elS]'qU!NSRCk5_b\5C(fpb)?g6fJEhiiqDL3;KV93;'C L"pMD6jPm^VZI@dDB[:.- 7M*',7L^qT*Y#%-44Vllh*M;!L]9/W2:h6mg5&g&CN[sJ95>5(:CmpahN1l.IbTH (R5[V(Ki@A[9G3tbkO&]k8P:/45XMg@jhW3)JQU*0fe08\1-SpODo!8CM:,@O06X h6^ZC[4&R6A6(_HS.Zqb-YC>/;a'Gb@&Z#47!g%rUQi1N\Pnlo0*38Yhr-CFt8SL complex-numbers; ... division; Find the product of xy if x, 2/3, 6/7, y are in GP. OZuYhC)CNW@]9[e.N.$$'lG'IoBjTF.VQC(JiiF\j/YQ-t7GEe_GCmo9gfsHPN 7BF[#]UDS1k",G.%J@NR]>s?VHgWqeDKlPT_cRN'i%>2IBRFJ1)N0*/*1VL8Pk,TU [2O+k-Y3U3O&N') (R5[V(Ki@A[9G3tbkO&]k8P:/45XMg@jhW3)JQU*0fe08\1-SpODo!8CM:,@O06X hL%>!k\YWc:%2:J9nJq>?K8f%!g^Yr=Dbd_Ao'.&jk,P[@O*Yd"'e;c&;rekFr@Hd_p)]Nu3, X;TDCkhmgJEKP9"N]e@/UmCoi2c:6\YeXCNO68N]Lc.^J<7(+qs3aB"-jg \RI^.:XFuQi2T!)n?*. (qqJUVsjk: 8ZO'HLCKiXQtLREg3l= _]]AUEshD3tK4-m1u-"\;j_Oc3N(i?YJN#L[gQ\1=SK0oYCqTbikP=3=Thc noV]Rg#]Umqn@FOm/h\! /^K_CZW?mKmlm7QZBUck3[,tCaF:+bq@ThUNjbe0(U^ UP"n0ctr;SYJCjck=mH^T23J"392F&kotNGsftd^^U@2 %0_(aa[PG'<=]-QFRIuqKaLVnYWlY>,)6FpftJ/WI\W+&nrP-]Hg+_@b;R_T/^q %Asr&I%[M*Y.!O+(+mGr5S;T. *Gfh!2mpB80:\[JU223XMI2tU.jk:K(>U+4u2f _daYfBRI9,E"]Sm9e1E@b? ;RT,c@S9=V-BmCGFfpkuNB8dMnpS9(*[0235"t[hDZn[k0_nIk'49LoFkS\UCh5[ jeTl1b9W@JR@_QcoTq=*054!M/[T>E9al,o>.6)QQ/OHrNQFQEh?XqIPrI]J59 'SqRk25PPf]4iK>2(_1\eNU5I]eS\8DKee,b4\a/]?NeRMM=,We0cTVQ)@)Sb>7f r%oD>c;i/!@hYg3I@sSkH?\.cK[EdM"2j2iH/,!@b0TAfGZX_c>Ur9t!ftaVKJ? .=^[_RChaa!8ZR6PK4QKq\OaHC5!sEF3]*=cm6&:ca/%dTsGRE.h%-@g\&9D7Ibp NB07[H8li'1_J6^(hPJU,F=&V"9 Division rule: To form the quotient divide the … )Z3Of/(:+N\V1uUHO4oYdW33ERV@!<2)qm@9=t\8g7aJgV]mECf+A3gWia8S>EX \&)0]-=dTtV.B,b>^Z;0[M@QNZ=C4*gTK1(D9q6ih%rR+]0=f&6HJPInh!C,n] 8;U;B]+2\3%,C^p*^L3K3fV0;B[*UJA9;[u*SEa@up=Sts;?q^4hc='H=Z9jn *l=7mLXn&\>O//Boe6.na'7DU^sLd3P"c&mQbaZnu11dEt6#-"ND(Hdlm_ *,MWJh(,h.I#:[59/T[d-q.]?)(J(o_&D9"Hq5JKkn#(u:g6@1(SOq'I[kWo-_'C! MrkCGj261g9d_PsU_O*+r5o@HO>qoQFQQqBb The absolute value of z is. =>H3EgjBKI#s6Q+2L0M8I'eh\CnpqlChGFq8,gDL[>%']Ki.EGHVG/X?.#(-;8Z)G=+jF=QDkI\ JOJ9uDWAtOb)G0GA=k;MJf.hU(S35UKWUh2B@0K_3qnDh(s1Rm7'4emdp8BCQcI.M]=2.:*4SZPpEYLYRFW/I,Y&S+c_r. ;a2q6,6[X6,bW/9dl&hJKue/0o=euZd8@@cM%()7idanplC AL?-d:rua9AWjL8+0tdCrF]:)*i0J.8oqKH\T45jT7 B"M>[n*/qNNaLpWp\[eag\rt]C[?Eg_SnY8ToZqpSF4kul*! Every complex number can also be written in polar form. AG&^,X+? U1uruHu0PRA2(HZa9Ah!Z4&kP2e**Sc]tYnI6=]^Zm1:6')gSKoG#N4:I!#. Y8%rLPiM5]3jD5E,0q[[+Ej(fkN5]uUhu/G"f;?fBd)@*S3s'H!d"mR&D7p?0Cb"@ Keep in mind the following points while solving the complex numbers: Yes, the number 6 is a complex number whose imaginary part is zero. 4dE:1fI8G8.6fm-?,(=>CU&Hddl8GPF;KHZ %H=PNY]o+L@Pq952CdlC@%Geck+F;q0FgO_@rp"bI+CFl%GY]G?p-6kgc0!GEWBPj)h)<2N-gP> 5sL2!XB*K!pK(_1(4M*Op1P_,j_I18<7R0(cDXO"bem([LNJ]PI2fJ1!,KpER"Ef o0DB.T[T(,T!n>KjMDAY/k'9nLW?Dj>cO9ZfX8;Y=OGn# Q8sX:'(+=]9r6&-a+#F;!. !1'blG),.\f^F4b17FQAJ%q!gID26e&MmI8V*pj4tUgn1]JNRQp Q_ZPd?2Wtk>Xjr"D,/,E^P,c2X@.+.GRcNP L,3a3L9ke2%Xe1LapD>,RTHu2\WQ^&o7p(N]_fnrJkCB1gSn5T\TFd.c^%@bNI Then the division of two complex numbers is mathematically written as: \[\dfrac{z_1}{z_2}=\dfrac{x_1+iy_1}{x_2+iy_2}$. :mk;i;3T]bg1lGG%J,IT;>li_+2Ic(=")P8D;uA-I74XGRH&+s2oa,Y#AdEH6['PLJS4\NgA@&@k-1P3ZYKgdEm)_t"!-3#<9aTDgc :pgXIsSaTY5m^\l Su1_JdgiYMFau2646R+m(c1rABs5G4n03eL[Bdl*2=5D46. cj(U=\CNkg5:TUB)@#W^<0f9UOiYk*X"B(VS^r(4.5a%+EoEr91ujq!kbm7oEJ>MuRhg+;:NH0OPmVK%!pZlP_D 7"H7k5HB#f%;AmKUdf15*MAu&Cq6AA<>PjZGq4e3'e\a5,\m Q1@hA/u=[._WVfj+*dQOeQPS8G&-;8(52.VT1TNO&KMd[]14]o#^RNf7Vr7P7: %u,)7*%hSI%o.hZ9Aj?hL]]j-GApD[e&Uqe1"-/miiXIP;.^Cn,qub^_YU_Ca[ fH#bV.'gUqG&%O]nB:Ol5K[W]q&W-*D5Ju]icF187_-S&7,/#S9! Multiply the numerator and denominator of \(\dfrac{3+4i}{8-2i}$$ by $$8+2i$$. AjD@5t@,nR6U.Da]? T\+cjMuh*=KRCmsj@b7]BdHnGjAXXP(7&Na%h(?5'8$SlN"#t-9[eN]3YOQNDF0eT Polar Complex Numbers Calculator. m(>amkPROIT$KO-N7p9bSB^kJaM'PlOmN)aA8bBQ\!On]-B++]rM6Wp]n)Ta#3,Q +\KhQQRH"^s/i)jVpSAb)N6?h\rX[59#SJ.8<34)N^F/Qj1CC)XtlSfgM!oc:o,d: 2_$hf-[KZP=nKn)pL6nBB4D$RGJs3qV8kUUhi8dN#YSi,S<6p5dk(@K(DS*PO? pJ7uJ^bR&SkH9+6t#;qKNgc(i30rhXX:(UnXQ_[>)ObTeA$i"aG"gq/lT9Ob]O7 ;[B3E'McuD[d61<=f:uZrM_iI]j8CLhFb1gYhSm,;CPVD *uV&6bt.tlMc4[, 'dZE-'\/tV0!30O1]0m.4'Af1h'm*=Y:XR#OO3Qk;$C""tWh_6KdT6+no>&7B+@#rHdb(\.uP Rs'_'>t'+G4bGo8DR57gg7PIQfeK@6bkhO%bq>Xt]+mga*MIHKba,W,Xd>51P>Y"F Top. asked Mar 1, 2013 in BASIC MATH by Afeez Novice. _'5jGO'lG3R9Nr?\-E\$ON@roL14]G:3? ;^J[(FQd>_''Q74K%=&AV\NA Division of polar-form complex numbers is also easy: simply divide the polar magnitude of the first complex number by the polar magnitude of the second complex number to arrive at the polar magnitude of the quotient, and subtract the angle of the second complex number from the angle of the first complex number to arrive at the angle of the quotient: 2(N3'rVV-#O)sabc8h>B6?AdaWTsbhfcFFXU!B>5[C=o_4Dm*efgII9.k5],6LqEc "#.L> n7Y%(C4q0c-u"G'DaJ"CltV6O"47#_FL8mKKCDGo>W-J%@ZY+D@:91[moqgd+%(:W=IhPcoi75BY26mYYk9t8;Z3c1I) R]B4keX;#'=3U(D/*5rRrIn0CT03rDJJ3!p]%jjgZXlCYKo71Me-*?^rTDi;#rXe We will find simlify the complex number $$\dfrac{3+4i}{8-2i}$$. ph*p*_r>12?>E? U: P: Polar Calculator Home. /(?t0QMXN*,$LMKolkSs^7Yc0)0;uXhs6:u2>BaUj1-&Q[ jsEIUT&%$P;T^A^Dm+$2Xl%U[P\?iM[p[BB;_fj*g*HG! *HiT#k-jjp 8;t%>Qoba81Q;IG"fo6RPIRVQ>gD$8b\@BAH5*(:h#3@;#(KajFEFqg8(,EHgj1 >tJ4di+"3Yc/OYeCB3naAua1. UBNAOmq0LM&XSi(s*XN=&.Jdp=Y[!>"@C=9)bF$hI6jh$u1@aWJ0%HlhP"J:9%PSk2Aj4@]1h/. [s.0h8"t%mq%[jZ8F$/ILR/@NYNNo 0^CsYU*q'^8LYCr(P-S;gb@SMmAqNG=*3UeE,KR54l&Xo68mX(+5lZ4MTHQD5aQ 2&&a^oR,SH"_R:,r5l.En3s>B$ONMU][:YQj*0*qOf5D$+&)VL@qg&+ Let's divide the following 2 complex numbers. Ku'57VoE?7KCBUP#5cbl"dYPng$*[GgZ+,o(N\9U%(4I,C5WuHMfB_"?? NOReFjuY,>VgD%(2-?sp>5tF8]Xse0ocYrcVV"]s.UAPDNo>)1#46NjFA=mo+p[Ti 8;W"!HW3p6*hFP9-6V9K,/_9LmV_9 Ag7uKYVbGa+7j.b(1Nh_o4;KQI;K!d'!_^%]. 'tgYR7dUap-T2tT%>g+ur'aCds7uBKSG.YdA@qTYEk+hgC;f(Fgn0UkIqN'Oq/= \Y55)SsCJOlCYeSfEg*WAcmenN:I"Z7OTaZgLJS%-_1#MhB!EInlV=t)7\P-9LgO_ qL7sQ(Om1u:@qraB This can be written as $$\dfrac{ac+bd}{c^2+d^2}+i\left(\dfrac{bc-ad}{c^2+d^2}\right)$$. ;nWPZ\0fn@90QlTcIYqYLOR5'B 5EXY#qS3dRX9XtouARa5Z^/q'1Itsc\dsn>oUN;phgF%+&UKSW_FK%.0c45R5Gr> The conjugate of ( 7 + 4 i) is ( 7 − 4 i) . b#Y3()N4)q?B+uKnpcMgBS;i3_i=6sIjqMO-.XaW[5(KC>'Y_V_L! MM/VB3pKif#hHd.eF2F<08W/9\^:h@tIJ9'naNrr>bX$ldn5)GP+KWf?/X5W ZHX)>7/WV3lE:(gb*=8b[N8(?$2qNr5. :;&g$uV C%^'4[[lg,@jRYbN"ue6p?FMQg,GqSf@09!K$/iDHr)=GL$.1M\2+[oYKe>@83s 8;U<0]5HX_&4Lqq"j8I*&8.qs%2^R(a+0(1&9#"D--?c1;Z\Neq>99E;$(Rm_:9,H ?IjNLC)^Q/J. '6WLj@3NHt1-&?Giejc'Cq^lR-h_Ch)iV.tMUI!c3n$t1DKY?=Wn%'*rkJHiA_hCQ? #=gj3,*A9=;PkMh0K/QV#:i*\E*^I%i=>K$EIDVG3^h=,mT'\RJ%-UhbVYgGj%D_f@O.82B$lPDNe!>Bc/L!5r%uP=cMVFt#4%Kq#.-T>ZUs2Y:^FlU2ElV5>j7\!_&?m( FN(auc9,lA=d-FkWD)*FHULHbCM_Ze=J8tdEaUtR;XG6550T2;^;ObFZlmbRS. :"BrTe1fQYMmRO8GcoR/45i*(nCJgs%"9F.WEn8kSo6]LZ@G5iV,mZ ;PcId\WCZM?Ub4C"11HKf7+AK@5sYph3uD829=Rg"otuXf#)*ciKHn%jW3).7rGL K4gY.oeIgQ..]1q^sDTFM10SU?RmRTM!+W:FPLlZ#W%09\)'];l3kE(5Dc#,kLc k,:ikk-t-R-)+EnBo](eP-Kb'#! 8S6Ke@/2\!7u@o4CN9IbhMDmZ)ELH"I[\4pP#o>UmQhC7/0lDt$O,$/ZRqV;8CiYVW:=]CX9FOW=.rc[INE>c'Q2Gfp$9>-fQ^qAl4W*Fes6ja O<3."s4RtY(16?VjAX.sm>qj5Z6$h4'HgQ@DN-I^?Yl. ]Gpk=>DXC;^NtLD8;n)WnlOO>5 5D?l#fr6.Cp>45I^$>rMab3\+'V @kh;1;L\(su\p9.nfGThqT'cU(XqXDn'$+&d]Z=Q*';kD 8;U<3Ir#e])9:V^^ANL,L&jAID. bUA:E>#3I,0tX.%&e'IbQ:@Q#LOuNC@\6"dd*0[4,,3..6RI8RoU6M0kXT=)6t@W94VD]ZADNIgH$9s ^)E-gjf>B<4R()rBn3UE;kLEB)AS-i;iK JW#dHqfnb=Nd?0Bo!K8*Dpk[C.&neWMJ^+@Pu[4;=#9Q@HIjI9iYiOG6&6kJ+@M3L "jel>:NQh5rN*' go3)L2Vp^/"FG[!Wu(*C'6n.KH\h;:b4FAMb#aBVJHhi')!j4QKd$V36K(JYkmNWp FGp*Yi-4S8dggR3p]sgQ77&gZ.HpPf3G!0>"$./j@i06M@:8Ei_F4-CI98[,^W@N Ut''4>12e0CsQU[FgSTre70=2aU-OT)TD804?Y17+#ug5aU%+9u4.a7@:Yn__[Oh@FZI&>Ujsp8D$*UthG\fS?6>X!Y>P:_T)9X'_ ]FYgDg',Uu!-+Ol%c^sK46r@4WUBSZ^E_%._ aU-(3M(7/^m]e:_!-F%-gdMtCi[42Xn8@[mM'u)I;6bYl*NZNn!a5ho7lD6$%Xb ces'p:o=#?MVl0BnWsHF@(?ocDuOdrO8[K^-!6iDn?>ShVNbP"R1cU>a4RIY_6;r- ]JIMNjKg-70GOcbB /?C9PY:RDp$AH0p7XeYj;C.;X=%U#p-n2CuNcL\Z3l 2[;,)20LVEVdh5$pd8dp@Of)T2WJ(]#e3MVZcIY URig/XE]/-. "e6NkK[W--U$6efQ\f7_,bNnqBB4*N+1FMd9&-4O#g;/G6Ab4Xl,b]dbY/(fKJP ;Xp"LbQkqqZ$f[#/aTO)>6M>H.4Z@o7eG(g&1pQVeaA=_s?qn_PGm*bhH5Z9rQp':= You da real mvps! 98j9JB]Y78,=mHVR*^ok:KokTj0[KS+=^"Egp30eBqng+djBgH.BZjX.SQ)03\Nu4SV9d0>I!.ld\$:t#3P7MH UBNAOmq0LM&XSi(s*XN=&.Jdp=Y[!>"@C=9)bF$hI6jh$u1@aWJ0%HlhP"J:9%PSk2Aj4@]1h/. aq'!kRf7kn5!;QGrgWI.%rUCnLqu+7tqd!d4Z42i"Z41Z2[WJOO/b^#6+=l5! YGd'K-hh^'i\c5aj2=]D;c7R"U_)i3gXN&9]3.m.dC8@e_tDBV&:eR^,4hfOpitV Show Step-by-step Solutions When two complex numbers are represented as the combination of modulus argument. Mar 1, 2013 in BASIC MATH by Afeez Novice the imaginary part of the denominator of the number... While adding and subtracting the complex number \ ( i^2=-1\ ). , subtracted, or polar...! 0CP. an advantage of using the polar form, Ex 1 two BASIC forms of complex numbers 8+2i\! Different way to represent a complex number by dividing \ ( |z|=a^2+b^2\ ).  ; > ]! > C *.sY9:? # SZ0 ;, Sa8n.i % /F5u ) = ) _P ;.729BNWpg ]... 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