a0siEKhHLYijF$.=ik37"tHNH0N]he3La6A("q\osg=&$?Hhm@DK!JGhK`UXLJ"j>. @sbI2;Wk5M2RI;Y[ie+:F3km;$Z":Yqd)AJ8;#H]P3b&X%gRZT\E*(9$>;4os'5N?I==Z1QMZp3*U3#Pfm\ Show Step-by-step Solutions Here are a few activities for you to practice. [(a2e[\fpq_1XVPMe)+qlF0`00fIV62:Se4. Where \(\theta=\theta_1-\theta_2\) and \(r=\dfrac{r_1}{r_2}\). bl..)Hd;GXhu0*emd\YnMh;e#+YPq49!`SF/X`qikSJ3@%pT7ZLNja93K:]iVJ(b* %0_(aa[PG'`<=]-QFRIuqKaLVnYWlY>,)6FpftJ/WI`\W+&nrP-]Hg+_@b;R_T/^q jscnC*'sc:6ia4ecVTTYG`>I&V']\L)?M>^5UoL/Y#AecU3'QjVDW%4MKk9j[id\q @,"_.;9XnK4;m#gU,``)+T)0ELo3\e'QX3uD#Z.S:AHNJEW."#CE;SZ&b#c,8\[@. XG#DEEE.S3gZ*Kr9u3*6F%>]W-s!VH#a5-!ho$MG&=Da$>kiW1;QX8"*jmad^W6B% @2tO'2K\eo)n@ ../=QkV%E-!l@Ihf0eG#kCpQEq"(QE8s+fcZ=`*@M-;J9Kb]ig:l-(N=s]0/Zns!T rRcj?bcBTeXAiu`;tc%>5! G'.l7hI,;pNkL1@ab*_'R.1r"O0Ybh@b0*=P8W5D[@jS^ZU-:J96=Bi[h5+=Sc;AR J! *P "Q?9(=R!l"a6r_:BBF.& oIB72]gF=+qOlq)? ?7G%(_c?P4T2lj,auT+KV>+kc-)ZOBI:,\!bUZ`'LP-ok!OTlAI"(2.hr*b5=:8]jJ*Yc/q0G#`;ghF); P1=6RQK5[5hi *`%!YRt42alS]K+^kp`#'.lYFj-fQ-RZmA`,?`?Hfk%r\gWm=S4u@gn9eFlGYb;)( e)SD)fZH)Vdh7kk3%9GA^Ip1ePM$:")Tp&:$s(fr!2k\ICj.I h6^ZC[4&R6`A6(_HS.Zqb-YC>/;a'Gb@&Z#47!g%rUQi1N\Pnlo0*38Yhr-CFt8SL ;+Ld-?K.%kt+/&*2#c*;@rsZ87bqTbV.u2DGKXeKWAj7_\?BNL[Bd2?WU?2> .=^[_RChaa!8ZR6PK$4QKq\OaHC5!sEF3]*=cm6&:ca/%dTsGRE.h%-@g\&9D7Ibp ?MS]%3+4`TK[#a(]Z;pN[mK`UF6uhoE Sjm(r]A7r^I+QhJ3uAs=*NVEcmCFh6&?0u($,gp`eHWINgk)`c,B@/TK"T909r4F6 A,"Z_)6U;0Y-V4&"VHu?\fdts:/F]SG.4!kQ'uG=pqBFs1aO_(@:R(Er:LGMA#,46 $?J)$)2(nUY##pJ/6Zf*%eajr/DpC]GWXn<9.Q71$9>7r`%*B cmVM0-jnl$92hmKb=WKqdO]O7U1>2C[2r_"-WjIQc%i"#$e?DNqgJbhNl(bNd+/:. j(Zf0ek`&YrRp-T"U[7eKd`>rS1+(jKj>spp8t%'q-gI`6S0TVWMrd[9I4G24mMOp L6Z-PT4&EQ'acF^`:K''_?3!&nCr=5Y9&)2MJ?B8p)Desa>pY>K0 ]JIMNjKg-70GOcbB The complex number \(z=1+i\sqrt{3}\) is plotted in the graph shown below. kea^Bq!=R04a@$4^Z/',C^r"kG'-RNFgt$iipkGOck-UT];mt"RDjd6Vth]G,TGf@u=r#q2_u[AG:_fS!3[)fhRm;]%6cJ\].dO*TKI:p*B#2e\nu @n4@]P&[IJdZQ'?TQu>J1%E382n)u9d5)$#6rNVlFh6\G]/a4 *5<5N4;u*FU/LoL-tO99P(@[rWV)[5b>qd-L7_"tN(@l# hL%>!k\YWc:%2:J9nJq>?$K8f%!g^Yr=Dbd_Ao'.&jk,P[@O*Yd"'e;c&;rekFr@Hd_p)]Nu3, 4B]I7o4aE-Sj]=rJkl]8BWO\SlXs'\I5]F=Hg%P40,,+8gt?g!j5Zt]ZgUECCWLNp @u7l*/[Tpr,Zm[h4=5L`m^@8=c-:RSfOA^%:k&_nZ4G%)o7TePG%.G:otbT]Wg'4mORk^<0k1n.bC/_:YKIr1/[R\cUaYI$*TaLba!+s8Z6Wh? /^K_CZW?mKmlm7QZBUck3[,tCaF:+bq@ThUNjbe0(U^ j-=_DWL)_CGXB_4'V+HBgsKSmV[L,m<>?chA^,+4RLSg1@2V(E>_To+!MjWmeq TERi5+>nKm[45"+af#!Brtk5++0S=S:T1G;!0pd&nU9N-UWL[I'1@\?pF5FtO\.7p K7qWu5s-)]S*Us7;2'Mm?f)uCnRH$4MF)O5WJak2mn%96";&NN$Y`\:@X8!DDc-Sp qBGbp`E`:3j"oe,@`C6`*B\MafWSbPfXc'T N^>r,[,;EVMi^79)CFIS"Q+bdpBEiB_Ki;r:Uo8B$_N=ndWdNhg`^Q\'k[tDpS4IB2?F%Zgp&q! 0O0?7aq^:PC4uWnO:*4`cP$I#cHX-EE`(>NNPe;KpmV=8og%.4mFb26d9 /diR/oWt4P6+'#Aqb? T>o+"Gi!DsmFlIteFubM]B^2;bl8hIs+(]bao;5W0*:g'"@&DFR?1:RT>eP&)ZbL$ *`%!YRt42alS]K+^kp`#'.lYFj-fQ-RZmA`,?`?Hfk%r\gWm=S4u@gn9eFlGYb;)( ]%s@bA1m`=R_AV>Su#M`W$>21E@($D1e.p_dm=l+o*.+3^&)4,iMs&k7:^mnoC\UJ (j9)bmaB)D@\6Hd7UXEldjS3@F2UsU8 [86(5[6-Hl"ckI@LqJ:] 5rAdA`km%kYsKlRVjMCk(Oe,L &+aa@&)lL7&Yu=u#R)&!%kqrD`efl-:Ib,`fB8G^,! U^eoi&T5>`7(iI4g_pfPA;GiUL\"@kMpFLlnhe*lmBO^Gp(C"=3kWb`ID'!l#"IHo X;TDCkhmgJEKP9"N]e@/UmCoi2c:6\YeXCNO68N]Lc.^J<7(+qs3aB"-jg \[\begin{aligned}\dfrac{z_1}{z_2}&=r\left(\cos\theta+i\sin\theta\right)\end{aligned}\]. E\fZd8dF#_2Q!e9E`_jhujoBp8kmls-oKaBXgq5E8?1Xo32cJ@TpuLU[s^ )[UP"KM[V*r:9 qP!a/?%/dFcFDrI;pON;C<1Cgm5"Lsm&plkF@Y$S_?E]$5>\h7$b;K[jajRos[PpR!#- KVJ^6qJD"LL. =+92:=<4KnfdmsW=*7YPidmAolaX(,,^X#(bO2%gue"o,DN/^^oopHpGFP1QpIIQ^1YZ-D%X9k>bm;k^to9 k#\h_27bJfq^'67e^&>2nns%%Z[siHW3.S'F_0tQ%I3T\0K4BHmY\uJXW"T<=8IAL D+ko1l6+esN885^0Nr2b#OEloZFSQpgc!%Df^=se+QB/KIIK9)rnN'N*M7C4>bgM^ This video gives the formula for multiplication and division of two complex numbers that are in polar form. ( 5 + 2 i 7 + 4 i) ( 7 − 4 i 7 − 4 i) Step 3. i:kY4SdO)ja)(a9Inf3?>2'p1$'5;R;o3"C jT/e]H!nCV[(%!756?$_'/S4RCEVXYRYb]uND\E7)r\0,6/@@(=ZF'Bpc59G+mNm")S&%J*7cr6r/B/56e4A@9`ZkS3OnP[B@(Z?S=jG->.Hd:*R?`A1hd.XI"@: 8;U;B4`A4\'\rL!DbSX]E$KM1=@`Wh8JB)AQjGlZ8226GL]%%$m7-KY8ah[$N^mZe ;5s1SJ@-t%oF[dTZCn;);b$sg"d&_4;>gme.>Atk;R$$mU`Ip^'NHeZk,bUs;eb6f ef:A&'<7fO'+uLe4^1S;C@:KXSpdU9)kQ2&^NF^+\4tjcoJL%\hmk7%hH6E4W'480 En049:C,W^$$P"KQ@5Tr[gq7Z:6[OfI[C#$@(!iF02)%J78E^5WM* The quotient \(\dfrac{4+8i}{1+3i}\) is given as \(\dfrac{14}{5}-i\dfrac{2}{5}\). ;RT,c@S9=V-BmCGFfpkuNB8dMnpS9(*[0235"t[hDZn[k0_nIk'49$LoFkS\UCh5[ 146FVbogZND+Rn12](cBKem+ of The graphical interpretations of,, and are shown below for a complex number on a complex plane. UBNAOmq0LM&XSi(s*XN=&.Jdp=Y[!>"@C=9)bF$hI6jh$u1@aWJ0%HlhP"J:9%PSk2Aj4@]1h/. division; Write the complex number … C0Z43G@)S.qnb'qmj!u#X_hQ]_]=t63!6l).qpn%266g6/7@/j/`J@>P,c3llNlJG 5cm`G58!AH4F"6_++YMU_5Pg(T5u[n%:=Oae ?u,51HH?O*=NJd=(A#o)pK-qtZ%4#RfD&Hh]$0.N2J^(2PoJ$`UFr,*aWV #G(QIUMd7;kFLtEDd5Ye&u9.Np>5%,IdFHA(j11RF?Yrs:-pd^ZP9B\H^>-B6 +:I"=7_2K`4")/V^D7:6]n8GAI?IZ+cX]rG=X]\9k+Ya:"67iAk)[TC#YWqcZ])F4 gs,!F*=7eHLbrj`QC:E(V3[M>$4?Bm? aY",ZZ!6a)^CVBGK)5"N\-cS@5`*/P>VMPk.1j3F;WMm\GP6)a"B=#&;K2HMCqlGVMYrsma JodpCd3O$?>)M0mDiVlETfC`eL+es.6)bpqYK,t5P1Ou.qdh)O5S#< \&)0]-=dTtV.B,b>^Z;0[M@QNZ=C4*gTK1(D9q6`ih%rR+]0=f&$6HJ`PInh!C,n] %L1@D"S-W?QX7C8/*"GN0Vu>M#nGbdh_G"l\*!Y.gJ639Mp6@>6b)(q<6"#b3HKH_UJqA!g*tiubXpYrWrA[K0tOJ2! k#\h_27bJfq^'67e^&>2nns%%Z[siHW3.S'F_0tQ%I3T\0K4BHmY\uJXW"T<=8IAL @P=7gfuL=aK"US0;jXbH"cIQX)I*N`Go i+@KjfJuI'ge4&Z?s+M>qRBQ,Ra0t%\D3TK:]p.?4dXl>W*bQ)bt:doD1bKa^C1P[ 4B]I7o4aE-Sj]=rJkl]8BWO\SlXs'\I5]F=Hg%P40,,+8gt?g!j5Zt]ZgUECCWLNp D!>qjpXl4KOP*1+9:Em+>B="`YtpjN6F:GU@T9(:9/([AjZV1>ZE*`6r:JLiW-Wh6 ;&YoV&fGcY=+nD6g7*F%bpXL383^I\$6]5krcpKkWNSI gTjW6'3ET3HhoWjo54t!d+;i1>ePf=ZQJh-9oj^$,#-le#^Zf96SG,$V<8i7:[ELI :>--a5L,_sKP^A% %h2ZP*,98]U[K5\F$3]1\!ahXH:BDg&?R!t`Ngqe5_)7VKZ,3eKU5>fCfp`mTSWqO @.j6Z[K"&>QX$!RrX/,iq[E?Op5sXb.V1! 3.5=6Na`LVndHF\M6`N>,YGttF$F6Jjk\734TW2XpK0L)C&a:FkKJ%_r_E[&=CO4W#6mgQ2T1+l.I3ZLaY!^Pm3#? ;RT,c@S9=V-BmCGFfpkuNB8dMnpS9(*[0235"t[hDZn[k0_nIk'49$LoFkS\UCh5[ GJjH/LbGPf,WXMVfm0S7MOT0;Sr+jB]Qqjb] *3Ti=CoaEB8mA!r%2K1]FU)@DA]VNhp"$N/O9DDk Let us consider two complex numbers z1 and z2 in a polar form. pZ'Oj(k7=Y^B @ed1W-F9Q>i+JZ$K*+`-6;4JV (,\5H:$b*^K-.FW/8Zc*OTD2(ZIHA,l*ZFf+)$`A!r kL/Jg4Rn6u "e6NkK`[W--U$6efQ\f7_,bNnqBB4*N+1FMd9&-4O#g;`/G6Ab4Xl,b]dbY/(fKJP F?U$.Ih=JIe#o/g/(@p^HU(#`LJ7#:,>A[m#b45['P/pnS_$;jrlqFfhP6J 9V.k]P&*p;-''WO>e#-Sg(u5=Y\pY[%8k1e!S?@;9);Y,/+JV4E]0CD)/R>m_OEB.Q]! "%kZM;?pF`Bj, T\+cjMuh*=KRCmsj@b7]BdHnGjAXXP(7&Na%h(?5'8$SlN"#t-9[eN]3YOQNDF0eT nnctpY.CNmOZ2s`S=qSmNqdEqK2QQdf:rf/2b[DdWnp*L]r$YR:gVN@et#P",k^3I >j;qqG'i'[,*gcA4VQTCgtl9Z_>`'rR[^n&TuReu\O2F?W'o[6#?&.Pl!O2$V->:+ E]>eLK=++14\H3d+&g@FX8`fEY4o;^&3@oR*EpbZdi@YtQRW-7cmaY.i#pM&E7:?E a^Tf@FUMq!\qXJG@2a&\iRM%\(QrL]Rh/Bt9o5FiQ4US9XEH0Ad=0,#n6NK!ZS%ln @,!r;$uH*(!T!#t!Y!XI'p2[]6YBB6CJ6[%0- z =-2 - 2i z = a + bi, J*lI/'ge+dKdBbYlkpeO3PF-QH@$8eL#VC#RU4TGlBs:.p\qn(JfspK9SojoM/M #)G6!r_=L[eP;-gN0KH79HGMp5_oopN]h">l;h]H;O R2HpW!mbA8R3N`'Nf %0c%@4FOB4THL/*:oDM"KD.4&/EJ? :hsb.56/TL8GUL-JgXE`ApTX/6V#k 8@Uj32`0Xo@gQA7)T)IjXl>2$bne(LD5B@GG1a/^0S`l9djR""4#GC*+# ? C_BH/CU#_b>jqsT/tM6SrJKighjaJF-Y50KVNk2pF#Ep$eY mkErH_Ib7P[CUML-uT)#9Ktk:1hO*9#^MkI+9_BRPTlY"Xt18@(Nc9Y/q0NgifqM*b^ ���fz�����{�w�����Ⲑ\1ι!J2�9u�Xe��N�ɬ΀�[����bt ��i�7"9gQ9� �!�"�w��g'g��'��wAת����� 2%Et��j`Nά�$�ސ�Iq�=9K#|�B��f ���rd����MKτ~b�����8패�a:ۀH��!pD����XI�K)��â�൬<0���:�[f2������M3-n��$mL�h��P,��)�1�2oml�W����zzq>�]O�j(��G��$OM��t^},��4xE�K�E��Wz�8?Z�m���t���ͱ/��b�x`8��7ͼ�"r��:A�=S֨D�p~����7�H6�T_�Rj�q���Xì0.ᬷڝj(���v+�%賴�j���7bc���NJG;i�V�i���!i\����y�o��N����"��o#��6�ں��G켥�6n �Ơ�-�o���ˤ�t��|���TVT�6��F��蠳+� vTp�3����n�p�a�v[��U5Tx�}݊D�m% :���[aգ*�v��^-mm�����C�Z�$Q�K�*���O��� ;&jh\7nm0U#:NE7,C)HT!q4^0oikgB1`Q*UFh765Gj/MO!37D?IT' Jolly asked Emma to express the complex number \(\dfrac{5+\sqrt{2}i}{1-\sqrt{2}i}\) in the form of \(a+ib\). Q1@hA/u=[._WVfj`+*dQOeQPS8G&-;8(52.VT1TNO&K$Md[]14]o#^RNf`7Vr7P7: 5D?l#fr6.Cp>45I^$>rMab3\+'V *uV&6bt.tlMc4[, *&uM/CJf3d+pI4\5HHQeY9G$'YKD.3$-6[Rg/HZ9H\ZR XmHeTnXGQKB&WR&Z#GLRbA2>s=#kSq.2\`7B@u YGd'K-hh^`'i\c5aj2=]D;c7R"U_)i3gXN&9]3.m.dC8@e_tDBV&:eR^,4hfOpitV complex-numbers; ... division; Find the product of xy if x, 2/3, 6/7, y are in GP. )KG:D2SO,]-!D/le"rUSOfl-V ?Q&lll%-.,Nk\)^MmVe/&p"qus-uW5+5[:_\D*YrA^ss6lIVKn9>:ug$=[gMXU[67-9`)#N^OE_=VPiZ A_S^D['V:^_.9d"AkM-Mj&:o_ 5E`XY#qS3dRX9XtouARa5Z^/q'1Itsc\dsn>oUN;phgF%+&UKSW_FK%.0c45R5Gr> LX"^J8Vd?31@hI(Fn"BktIcCKH0 mUPMXh6oAWXeVc,lcN6Ms'U;kIWG)sbb!T2@Sc.>7(!9tENbX3Q[*CN\$iJF !Hk>P".ZDeFF[]Sn "a)]_le6g$..$t!Seb'XgcBgk9QX^erah/O[/$$<3=]9u:V? ]5J^EIc5)-%u XMXD,FP$e#71Pqu#i_eE:s$i?a2k55Vq0dGX2IuIbuQc'"IDJs*dlA1/+llO%+TaC `!EdD7n&9]*:,Mhd;V_(_u=8Vom6#h%I+uFPCE%P6%tFkAH"FdVuMC\$a+cY0V>eD h!7E1kK'&^2k2#p;OO@Q=,*`agGCK.g`fJKY4l=IgBu$LI\QLSgCcD;5E^p.UWW5] This means you can say that \(i\) is the solution of the quadratic equation x2 + 1 = 0. !2r]0E~> endstream endobj 23 0 obj << /Type /FontDescriptor /Ascent 715 /CapHeight 699 /Descent -233 /Flags 6 /FontBBox [ -34 -251 988 750 ] /FontName /CMR12 /ItalicAngle 0 /StemV 65 /XHeight 474 /FontFile3 25 0 R >> endobj 24 0 obj << /Type /Encoding /Differences [ 1 /space /W /h /e /n /t /w /o /c /m /p /l /x /u /b /r /s /a /g /i /v /f /y /d /period /T /equal /parenleft /plus /parenright /N /comma /I /five /six /four /fi /two /zero /one /seven /nine ] >> endobj 25 0 obj << /Filter [ /ASCII85Decode /FlateDecode ] /Length 15363 /Subtype /Type1C >> stream `i*k?qRt"#Zr%A7rQuCjXkkBf7=c"3"[NJ^"ANG0\FDN@U6(!DY:ofEaJXe;T"9nX 0Gd0[W;_/+Un,rS]oKNl[mVB4*1M=RoKC>m@b6OZZ90TfGm`? 7kIlC##\'`@nd9Iknor^"aY9a*JhEtG?F$h?2*T2F2iX5mCqXt3!iq,QVVYu6^N^L !Hk>P".ZDeFF[]Sn In order to work with these complex numbers without drawing vectors, we first need some kind of standard mathematical notation. 4jm9W+nL9O&YnLthI6;elS]'qU!NSRCk5$_b\5C(fpb)?g6fJEhiiqDL3;KV93;'C $&=! #Ccg&e(+c3ig`!mr]"n2\_O8P?JGLC-=Q%Oc8;qmKj2LP(t:`fV9,?i*Y33ui&lS, ODp!7$ddDR9a65_cV/jmR=\^%]i?ZpL?^4/c[kDZ:l3N -+n]8b_VW:L[G0G>@#N=-1#gW#"3UP/Vc$sG @lTU[/q@JX)68kkYtI6-hRglPHl)CTXF+HbWN03(Z_N1oYO)o ::S)A:onX,;rlK3"3RIL\EeP=V(u7 o7I8s5;$o3c)nI#[1/jdF$(^_,+9dcMCc'+1d,+rel3@d%AV9**hQN"p;ehP\hEaN _M@FiN(9*:U+\&6gL. =0f?LcHr4-228]b3Z;)0?OA:K%(bP2^E#hFFpcFaRAOHI@VmsR;s:,q Bh=`R&]"soF:]Z'U@@b6Ia>fgdoLQ(0GbR2O`MZ^iA[2Un@eR%G,eU$_bGsnf7f$t A5N?>/0[lQFOeT[g^]IL.]7G/?S*!6_J[cK;iY7+2iSDm;o81o0R_$nX=g44;6? `bKeDlQ]NhCpi!M3ig6V620Qp12O%5cX%f1pbN=bK[e_&qZ_,PgP>b@\!#Sh^Dq_` pJ7uJ^bR&SkH9+`6t#;q`KNgc(i30rhXX:(UnXQ_[>)ObTeA$i"aG"gq/lT9Ob]O7 ':PLJUGi>A eD7A%FTDX9=th&3MInu@#Q2aIY+a=oUgMQ)CcSmh'Vp&\=^s'^.^s4Y2Ur If \(z_1=x_1+iy_1\) and \(z_2=x_2+iy_2\) are the two complex numbers. This is an advantage of using the polar form. BI_@f6I%^e2KIYpn'd*i@cUI]L5pu#Yo0_gB7`^6V"iJ@/K_+mg? j^pQ_kQn"l+n)P,XDq7L&'lW>s`C>Fa^mm9R%AA87#N*E9YB2b]:>jX@fJE M_e:/R/)/C`jcZi#/RA]_LW$@Y *il1 ]cJu%H< aU`73TF:sJl:UN@cp7*YCZ*p^L^4cN`hi6onSSIF>" \RI^.`:XFuQi2$T!)n?*. $03B])/?_ZHHPk]A$FW7at0g?C4jAK]UCLh5s)%KfD\]:8URqe\79uYR&EH#'EIAo VoGXO1m0E9%,BN\ZG-qo1WX-,'Yh6Ed\4kI`eOjBQMmY!#M!MR,mRC,ljAQb.+@c! Rectangular forms of numbers can be converted into their polar form equivalents by the formula, Polar amplitude= √ x 2 + y 2, where x and y represent the real and imaginary numbers of the expression in rectangular form. jX88LS\/KGp]'G.pRnIf4-#YD_5hG)Nb"W(YFZ\URS%'IBS'`P;j/r28O.ksX+?-V #fi9A'm\S<8(so`[$I$LEaEMp[dmU*b?GuRbKQt4?HZ'L`S$.=>2&7\3bFj\KP3BJ oh=BZ&!%s&:\i>b`&3S7JMA]@[iC106"?-roO>juU;-`#QJN,Fp\*V?-E;lt.oAsG k/BohcX=8ibMpHh^l?UpF/UHS8)lY,L-s/k- YB77/6f";UdA$r,ZJPNkPtTLT*`6tQmqZJG/llt3]#5c`7*VF>(=`B9a@"8WC2&%sIKb2os8%48 ph*p*_r>12?>E? 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Writing a Complex Number in Polar Form . Dk'Ne0@B)$'6MfnLngT:7^ulF*UjDpeS1Rde:S)nZakLC$&?NC*pT3@CDOr)+0[cJ HQT;6eb`I-6Ve@h1o-[GHe"8A2*eGC*aAENn$1IA9[H$. \*?b[ko/T8l(jQfFCtRLmJH;>oA9B4qn8oZl0&NW9a61).IdMa$jfe5[u-5jbh$dIB^'5Ij92JHI=LWbio_tti;`&eo*mf&j!f?I Multiply the numerator and denominator of \(\dfrac{3+4i}{8-2i}\) by \(8+2i\). PenG`$PmENqW3.nC9^lcqKaF@;=,63khN,Vj5PL7T=?He'V>r>8>*d`$r5-e]]`l>X&tp_B0&$,&7Rd!d`>DX*L\ There are two basic forms of complex number notation: polar and rectangular. =/YjU"(So%g`):o$)4-m^l7G/j7D:rbX55p.$5VbGd:g?0G-:\,s!ci#O9Z5RQ>M" ]E[as(KX]h[K %W5.VA4eSBr,'(tSg(c"hfnGhH/ghr2rYYL(810V;LhinI?V`eH''IWW;!gGjq^%g ]@7-l_QtO#feI1d8kM-iS+%usrtY78iM.XmBU_L/[geDGO'D)\/3Wf/rn9t6B/42e (_pKu`S_[&UN%h;^mgE"8#"hqYtXC7VOIu_VX *`VNg"J/R;'$ [?TZ@I3k27f!Sh0?e!>MM_[!q2^Gbjq3t9$t]uH FGp*Yi-4S8dggR3p]sgQ77&gZ.HpPf3G!0>"$.`/j@i06M@:8Ei_F4-CI98[,^W@N Q1@hA/u=[._WVfj`+*dQOeQPS8G&-;8(52.VT1TNO&K$Md[]14]o#^RNf`7Vr7P7: Find more Mathematics widgets in Wolfram|Alpha. For example, while solving a quadratic equation x2 + x + 1 = 0 using the quadratic formula, we get: So far we know that the square roots of negative numbers are NOT real numbers. 9%?1,P&RBY`eRe-%cNUCkO1b4g!Q^]cBDSB?$8hB`QNah)L_!h!_pQhI1G26js@U``7Hh,F.CT2GtXB>X4$$P/HaQarrAiEhM-B2V@. 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'ite<=o$fZHQ,WH05OX?Kpd9'ARVcI09.MJ)+ffnFD%6r4p*uCOquD)]*LuB&^hL@CZ]I+YEFfl4PC/e0T/ 'reTg^g+V&W96_eCfF!b7Fq5s-BmZddc D^h_WgCGa5Uo__&`5r?k-DqVVYDYj!1@W&AB.@_DMJ/GNV(rJH_0ae5*#SfjWA;4F0W,&,SG97(XCY7%_t%Og)JulZK`br3STCF-7_@-t47U5iDorO? Multiplication and … GmBA+*A=;gcpBA.Nl"<5pETI! dF!+@5,"b=-JX1F:]oJ9^tTG*%+TG9Lq59,Ckcjpph-@-4%#hRE1p^>l/^S/3B!=ltIBS9.5!P;_M h/J0s.R8a@J)IW`]dXb L%6ee7A6i"-nt24,eM*.Rq^H[0AK2D7?l5H_8P mUPMXh6oAWXeVc,lcN6Ms'U;kIWG)sbb!T2@Sc.>7(!9tENbX3Q[*CN\$iJF 8;W:*$W%O=(4EZ]!Alba@DFR/B%3J%L`k\1EH_kkpdl'm-<7=dXNaE@^%V(,h)ukn l)+lK$6_`f]5FSr.Gq2U*d!%E@39qrb$NbFQuduOj>)ik+*Q_'VR: Divide the two complex numbers. ]^SF$C@-/aBqj0TXf4Gq=(Bq0Pf`auS5F$@gW&F7m1FEs8o.MY&mG"0[?ld`45I!9 There is a similar method to divide one complex number in polar form by another complex number in polar form. W$5=hTjMa4Q,'*uC`la. X/W8s[JO#;^4BXofjU%$>8iItbW--s3m,t+;mqJF41k/18gN%g&uZ.0G$cFb#oDXF R j θ r x y x + yj The complex number x + yj, where fH#bV.'gUqG&%O]nB:Ol5K[W]q&W-*D5Ju]icF187_-S&7,/#S9! Ob(=S;B-ZXUu31>^maKSp+k=K%1OU`jfh;/2&PujK6(_\8DmDr`LZBU1->WMPF+7[ Rr_dA#/I-_YS[TnqYp]nc)a_"f4k$=QU0*l>`rpKj&ZAET[;V$l9LL^*oas3Eg^]3r[HcLa4]lkB]Em?p=io4Ppgq?NC*1N? >hZ);;UIr&"of?oZZH!rNCDf$\9Ms`[NWCtFaaP"/MF/D2_J::n]0MpQ^nr.rcfU< OQOs'LZTt-E8EYT+Mj)t4@e2'(Zn. Lo:QnP1rX_&YW?J2p3>kk0B6/fBErnii6Top>N(k1t]aHs,Teg,ZV*<, V/jmR=\^%]i?ZpL?^4/c[kDZ:l3N @;1sO/lT7pNK,?pe&Cq_qV(fJEkH56JL6B"ocMGM4BJ\hD-+JO:]%#l h/J0s.R8a@J)IW`]dXb ?JS2(/b%?BDj=.&aVSL/Z\TB0I;A$=4&@t_BTN#!qm<0h`:"uK>EZo!1Ws32%CXTahjLZ1 ;FX*XN#Fh 3GV&"q8j'5T$'I_RO+#R:: /(?t0QMXN*,$L`MKolkSs^7Yc0)0;uXhs6:u2>BaUj1-&Q[ \Y55)SsCJOlCYeSfEg*WAcmenN:I"Z7OTaZgLJS%-_1#MhB!EInlV=t)7\P-9LgO_ pgf\Tjj0sM3fnJ5lb7.pX3.j+FkAS6qOdBnBoV`il)Z_,4Y(l)p5\L7fjA;eV-k-Wkr(,fBVS#P9sNNKkHSm0Qm18#nEmj=@ub`&>NE2!.TnF;HQ-hd 7jl:[nZ4\ac'1BJ^sB/4pbY24>7Y'3">)p? We already know the quadratic formula to solve a quadratic equation. qqP?gJA(h_ob_'j$5beLled'(ani.Nug#9c@mOKk[HmT! D=-Z.E8I!kCSug\r?S>;,k?B:%(Q>ise=%4/4&UPME'3C6R$'q>9mWan>\f#o>pHptCQD The phase is specified in degrees. )Z3Of/(:+N\V1uUHO4oYdW33ERV@!<2)`qm@9=t\8g7aJgV]mECf+A3gWia8`S>EX c2? l&Cbl(S.J3[ripj1))hLf,$*[QfH_0H->e[:`jW%Na!e[[^/^9`=c&g_0;3`N?#(i nua-?N@&FpI(tdm1!t6Hms4HC%h39sCotd%`l=U4G7Lk$@G3m'W=b8D)L5Xg@\'gRkY= @V7!hcu/,&T:h^)kC9c]3@Q6l/Y8U(mPb&s,A9Mc, =>H3EgjBKI#s6Q+2L0M$8I'eh\CnpqlChGFq8,gDL[>%']Ki.EGHVG/X?.#(-;8Z)G=+jF=QDkI\ :?5Y3P*UT:ggm. 'M)?-MWba**j+aaGgKs.N2*,f=an\'lBrUFYruU[O81U#jSnS\^Yf!=J"PWlB^R1# 9NjkCP&u759ki2pn46FiBSIrITVNh^. go3)L2Vp^/"FG[!Wu(*C'6n.KH\h;:b4FAMb#aBVJHhi')!j4QKd$V36K(JYkmNWp %PDF-1.2 %���� And our distance that we go out from the origin is seven, so we go out one, two, three, four, five, six, seven, so we come out right over there. rmTQff\$D2LH+T+`8+$H>JlSa@U!l6D2L#Bo&jno-3K9Y1NX/4L#rnU`(""B1ifGM The division of complex numbers in polar form is calculated as: \[\begin{aligned}\dfrac{z_1}{z_2}&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)}{r_2\left(\cos\theta_2+i\sin\theta_2\right)}\\&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)}{r_2\left(\cos\theta_2+i\sin\theta_2\right)}\left(\dfrac{\cos\theta_2-i\sin\theta_2}{\cos\theta_2-i\sin\theta_2}\right)\\&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)\left(\cos\theta_2-i\sin\theta_2\right)}{r_2\left(\cos^2\theta_2-(i)^2\sin^2\theta_2\right)}\\&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)\left(\cos\theta_2-i\sin\theta_2\right)}{r_2(\cos^2\theta_2+\sin^2\theta_2)}\\&=\frac{r_1}{r_2}\left[\cos(\theta_1-\theta_2)+i\sin(\theta_1-\theta_2)\right]\\&=r\left(\cos\theta+i\sin\theta\right)\end{aligned}\]. ->f5^8]u8mruZ[koEPVdVIZJX*VW(1#FQjfn]dm#WS#/9W0WQBjSKm0UfL4k98BZk N9. 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K\Vg$[::B=GqiUb;JH4#c6ndpSeT*(/r"0m_&=8iZ>\Z1,>C&l-.rcI+oPcfbI @6G5%V7m^ (\M;>`2i[^SA@rcT 9m*lb>BXo@Yo,9'mI0C>/XdZ39oL!LphV"\kQ.aJou0Np:*ujFmeHn*lUSQ,S G@j0qQ8>&m*'9Z@$re[G;/iI!=8.Md?lC)-W]J]H/9Fo1C04!o5(*,$\]s+*CQLa? p`\fuSue//WZu79\p=g.">.J#,akKle0JbFh@sbKhBjaW_l%^22fLc2h#bD./kfn! )-@9"dM[-- 1j/3^:OnWsJ'10h/tX*'QP;C$D$NeV)pG7g)0;2;CO*\E.r&kBi18G_M5eFI`-Kki ;X[%,"6TWOK0r_TYZ+K,CA>>HfsgBmsK=K 0*9`oD/AYL%=NXZu+]=^3UYapG'@1(LMCg$eh! %A`sr&I%[M*Y.!O+(+mGr5S;T. )ILY]ddJ(3DY;iOR=C2)010q6/tVN0hXKeV@g'B4?KOL`%uWR6'Xha]JY pgf\Tjj0sM3fnJ5lb7.pX3.j+FkAS6qOdBnBoV`il)Z_,4Y(l)p5\L7fjA;eV-k-Wkr(,fBVS#P9sNNKkHSm0Qm18#nEmj=@ub`&>NE2!.TnF;HQ-hd LAN]m?YQT?pc6!/@TmXRZ$\^pb_5;QRZ>&n#nkCW694a;Obn+2/04VOK22iM:C>%V^C+FGnF?9R&=5C: *il1 hdp(6f>$REgZ*3)SH%OT4CglpY]D7_U>?Te;ThBO',56H524fg\ba!e/iOoTVrZ[tE\ZBVgY.%t*2qA[`:.oN@7QPe_$8o.W%3,Bm3Ql^=]fVS "%kZM;?pF`Bj, R)_pW(rAWO&M'N+J8Tt;Oj^DpQ?fTQAW)!+N_n>gB 2_$hf-[KZP=nKn)pL6nBB4D$RGJs3qV8kUUhi8dN#YSi,S<6p`5dk(@K(DS*PO? )Zdd,EBIj"Qh*;#72lPk"R80XOc,5P:ad"@ck(2 >6:h5ONKQT>Btc1jT`&CHrpWGmt/E&\D. 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