:i!_GZ=ui'&"[G(kZh_LOIm@glK)n9P\8a^U3*9eY:G$.\ceM@Mt6f3iXSMZ>"r?^ gBVqY-G^cE$4)'EO)q=("%gs84C3S--2;1T6?`>*:XB! VCG9UQEqOrt]5')D$L,hubK$^7jKAh[`%\%]mF3"MI7b[bV^O/[Y/.p;3G4rP5:[?pfa JA-mak$]'riO@4B!jB('%oRTOcM,4Es5cLA!+t0<4:JtBFcOJX&0)q\NBf-NkkfM; -hiDZOENRe$^Aime8!2b2.gGT.T)p]Wao55oU%2TC.p9r And our distance that we go out from the origin is seven, so we go out one, two, three, four, five, six, seven, so we come out right over there. * :mk;i;3T]bg1lGG%J,IT;>li_+2Ic(=")P8D;uA-I74XGRH&+s2oa,Y#AdEH6['PLJS4\NgA@&@k-1P3ZYKg`dEm)_t"!-3#<9aTDgc K4>jdZ6sT4muNA/F^jA+(`$dO*l.`9$Coir)ucFqG^MLM-LlI1],qDu$a3E&?`+bT *VkoXW4`CRL')OrMo>3IAd"*aEu#sJ[E0#Q7=sIQJNE.$4Q:" ://'s#4$03FZASkWL$]0D)f?K&q(8JX.N+s:lq)-OC`O2G&QYMWg,E2d\*tPmnk/$aB;bJ!osn^M*WJ"L$G3q,l;9pP73AEbfq? b#Y3()N4)q?B+uKnpcMgBS;i3_i=6sIjqMO-.XaW[5(KC`>'Y_V_L! 3.5=6Na`LVndHF\M6`N>,YGttF$F6Jjk\734TW2XpK0L)C&a:FkKJ%_r_E[&=CO4W#6mgQ2T1+l.I3ZLaY!^Pm3#? #Ccg&e(+c3ig`!mr]"n2\_O8P?JGLC-=Q%Oc8;qmKj2LP(t:`fV9,?i*Y33ui&lS, r++9O00fZ@?jA\8+-4G8j4jP!cK8,4&*W'I:0.PPhDm-SR-M#hU)qUZBIQTMV)l"b W2-5"UU`5"6W-In\%9-`pP'3;-,jYXMAk*sPV2677:&8#[\6ll,@QX#,m`<5"]]QnL&s)(o6BY+'83C$CdW%C+_[CHFNl1C3 [C+g7h,LfIF7q!qaO/s6^MNFHUo:e*6@ ?h_f8CeK`AHF,'e@6RP[j4U.Xm*D(_g].Q gt[Rq:u3i/e--!4TVAiS]%tXk4$-3XD*)lTrF:&KgW-lanRfhTM9u78F(aq=k+)%6 Om,a(2FB&k`5?ROSm*:/qEcUaJEN6Wi?Z"#,gqrcR1'qRbOm+h)_,fI...J_2kqin ?h1_f@*">Bj:;Fg2Uu44TuF OW!F*1LgE^Ru&[G`okJ>/^7J9NV-MRVl,aAQjMCN`PUnW1q>^\f<6?5B\Ng>6R Use this form for processing a Polar number against another Polar number. While adding and subtracting the complex numbers, group the real part and the imaginary parts together. :;&g$uV G'.l7hI,;pNkL1@ab*_'R.1r"O0Ybh@b0*=P8W5D[@jS^ZU-:J96=Bi[h5+=Sc;AR ':PLJUGi>A NB07[H8li'1_J6^(hPJU,F=&V"9` ?IjNLC)^Q/J. This is an advantage of using the polar form. 8W#KmHN*''Mg&hm(JB.4X4'alH4CpImDSD^b9qYemF.P1Aj4j;7HRZPmD6K5[6c/k &Y@Gn90/#)jU'"d4He,F"L#Ggb83+'V4/mI3n7*^D/CTEIN5bO$5"G62JuPT^@o;-et'OPO.>;.=70`?$/i2nO"&:) BtDVoWl[;k=dX8.GW4oeIm8E3KI0P8b0VFpYWg`41USVi?rs [P+?> HqG:P::G0T3nn1X)^0\aeQ>GPj_l\"cJ2S7Xq\t9o=DSRWPBTNAJ>r_09%A3g/:gbH=66u4G,.,D?AXqr.E+rbdWk"fM,rcr nA.U.kpgpEnIm#DaM:2:+F.`=og*R[d/r&RdZgG!c0CGE&-QuIq$#pb$`f7m6rhTG I]#YP5?O]&Un@8Q'2;*Q>_d$0.UNG8l:1,ZI)FK)A'VD7o9LM2O3JB"(N[0FapP]5 >uMN/a%12MVEO4Dhqi\SYl;pfE#PM2-uM6EYd*h2'6Rd7=Zd!`B!%Q>X0Er6oM`*g b8Ei*,8H6j]2WI48`AjN,b0P20ePD'[85pPQ;`lm#l!Aoq`M-ni#tJoqT.6K(7p9_ )[UP"KM[V*r:9 What is Complex Number? Get the free "Convert Complex Numbers to Polar Form" widget for your website, blog, Wordpress, Blogger, or iGoogle. mQS/B;UO"CA,WZn%E6(1M+pNsNOC1f8!J\#pFqTthgAL>CcR/^U.WLEi`GK1ebj )Z3Of/(:+N\V1uUHO4oYdW33ERV@!<2)`qm@9=t\8g7aJgV]mECf+A3gWia8`S>EX 6oGdOK,hr6 X;TDCkhmgJEKP9"N]e@/UmCoi2c:6\YeXCNO68N]Lc.^J<7(+qs3aB"-jg :i!_GZ=ui'&"[G(kZh_LOIm@glK)n9P\8a^U3*9eY:G$.\ceM@Mt6f3iXSMZ>"r?^ R2HpW!mbA8R3N`'Nf !,$g4>mC4*/^r)#b"DU'!LMLe2n?>5(5Z$Cb\24mh,M,P%_NA>Ti9h@l<97M:7Orl '#Bt,MF8SLl#NeGU*].+0@Ft9.D>mOt)WaI6HP1W,1T>KXcQ>i- (_pKu`S_[&UN%h;^mgE"8#"hqYtXC7VOIu_VX ;FX*XN#Fh ?D?G!tL_8Fk]5A%]SV:M4m`U98%SD<9L(+/^cFZ9s;P;s7p5cP!+e8JCHWD^"(t 11.2 The modulus and argument of the quotient. Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in. Su1_JdgiYMFau2646R+m(c1rABs5G4n03eL[Bdl*2=5D46. ceJA=/BqUI\gV\o]#2P2&jg/[erDD;RU)k52j2ol=$)r<5V8OGGn8WV"X!2ech>3< $?J)$)2(nUY##pJ/6Zf*%eajr/DpC]GWXn<9.Q71$9>7r`%*B Multiplying and Dividing Complex Numbers in Polar Form Complex numbers in polar form are especially easy to multiply and divide. Eq>Spl/K'`W@U&T\MRp],&,>=LIR`- TPE"qF],e;:=bhkD-";M=e1qQba>__ti2Y+]#(1U@0BI`ca The conjugate of the denominator \(8-2i\) is \(8+2i\). q$`dWN(=3hIlYK%HEhRiOC(t$/Lkt)BKWcg"qRp3gkB0LifF"up1b+Ql:U)KZcU2; #!,sg[5"=uLQu0qkRRl$("Qh1mE)Jc[8^uMr96_6mDn([)llCTe,A;#aGp!TF`"sT ]V=$fe3*!>LVK]dl$d^D_=Oh!llbic$>^I20J##]K%,g At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students! [s.0h8"t%mq%[jZ8F$/ILR/@NYNNo 8S6Ke@/2\!7u@o4CN9IbhMDm`Z)`ELH"I[\4pP#o>UmQhC7/0lDt$O,$/ZRqV;8CiYVW:=]CX9FOW=.rc[INE>c'Q2`G`fp$9>-fQ^qAl4W*Fes6ja YsP%`Ur"!ZmC/us/;FU.b";>+5e7MmiRb'qTdB1Kp?PR1r;A. Such way the division can be compounded from multiplication and reciprocation. [lRt'clmTo6?_XV]`Ql$O50%8:4R0'V#$>VR$6g%"9_O?rT5-HH'2C`?X+(0Z! @V7!hcu/,&T:h^)kC9c]3@Q6l/Y8U(mPb&s,A9Mc, [E#M[)Hk^3rKbT0AK_fsb(QNDF(+0Zr^l@*S(>I_+[?9k3U"Or#CY9/B PY)G\A1YLCpbZhWr2$Zd&T:k,=
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4,&FfN4E+m=iVSX\6bm3Q19`Ob.`"%S0Z,r^/\8o2te%Ij?`H_:q\5i&XS)UP*[)L :) https://www.patreon.com/patrickjmt !! @u7l*/[Tpr,Zm[h4=5L`m^@8=c-:RSfOA^%:k&_nZ4G%)o7TePG%.G:otbT]Wg'4mORk^<0k1n.bC/_:YKIr1/[R\cUaYI$*TaLba!+s8Z6Wh? UBNAOmq0LM&XSi(s*XN=&.Jdp=Y[!>"@C=9)bF$hI6jh$u1@aWJ0%HlhP"J:9%PSk2Aj4@]1h/. *F _daYfBRI9,E"]Sm9e1E@b? Q7>sB9_tN-*X[8eYBPDjf"TF;Q:+(\@aT# kH4(U-ZJA7s45nmYbiK/9#S:dV4sJXDjWss@!%ROfKS@gF1$^9I$us3CCXWQ#4JFk *`%!YRt42alS]K+^kp`#'.lYFj-fQ-RZmA`,?`?Hfk%r\gWm=S4u@gn9eFlGYb;)( Contact. nc3%t0EFu[J,oYk^[l=FJ$9596NZQ3:OYpN0*TN&\,@1QW,S!JM?qVE`8=1=-/0^M qdoI6Vj(pLrL\j#Al0e1U+gMW&kKl?Rn$js.Nu%PFSZA#V1gNQa;"FPVGKgGC+DU' ;5s1SJ@-t%oF[dTZCn;);b$sg"d&_4;>gme.>Atk;R$$mU`Ip^'NHeZk,bUs;eb6f e2$_EES5B+;GU^c.1ng5M>1sQrMJqgOpZoEO?o"(&JD:oH:B.0mAQtF(KHQ1 '52rA1gV%4S9p There is a similar method to divide one complex number in polar form by another complex number in polar form. P#4e),/Fl=TOplXHE>`]P&obDm?SF+e'"qADcM3cp!m+J9a8m;(/id]9P!2>K_V>G "%kZM;?pF`Bj, He gives a few hints to his friend Joe to identify it. )Zdd,EBIj"Qh*;#72lPk"R80XOc,5P:ad"@ck(2 Write the complex number in polar form. oh=BZ&!%s&:\i>b`&3S7JMA]@[iC106"?-roO>juU;-`#QJN,Fp\*V?-E;lt.oAsG MDKFZ:*DN_$tNAOV[^R$#O2@gKOle(`DV&:J4l_]ICEHm[XV>9D2?#jFW1(*:Mu9sj]I;Kt)1+t"j%X##0$l%:FmZg\3TXj4 ;MfH/@tSNW*41)sCBa%^#@.YPFppro!\Qk^/L-K;Bt( *>%qe:[XRG-H4$YOrBkP2?O7I?MuV@i_d)+%XkH5^D3nm@j8F"$D ;X[%,"6TWOK0r_TYZ+K,CA>>HfsgBmsK=K :mk;i;3T]bg1lGG%J,IT;>li_+2Ic(=")P8D;uA-I74XGRH&+s2oa,Y#AdEH6['PLJS4\NgA@&@k-1P3ZYKg`dEm)_t"!-3#<9aTDgc 00(Y>):TVR;YV_2 "/CLin:WrE_8P&MBObI69 Separate the real part and the imaginary part of the resultant complex number. ]V=$fe3*!>LVK]dl$d^D_=Oh!llbic$>^I20J##]K%,g The conjugate of ( 7 + 4 i) is ( 7 − 4 i) . "Q$8cq/oa<1$"c:((.%0fG6(8]KfRA@j(hq'9Wc-4DU b#Y3()N4)q?B+uKnpcMgBS;i3_i=6sIjqMO-.XaW[5(KC`>'Y_V_L! =]_HRlIKt$c$np$hMAad]'ek/cJ5s[I,FbfpH--2mQ&%'lu[KuP'L_E"QI0;mb\>2 =6_C&hW`F:/'S5#&ufTQK-In2'DA%Ecb\JXe"F2GUpZ7%D3%7O7[p^mdJM%YUfD1n $?J)$)2(nUY##pJ/6Zf*%eajr/DpC]GWXn<9.Q71$9>7r`%*B 0^Cs``YU*q'^8LYCr(P-S;gb@SMmAqNG=*3UeE,KR54l&Xo68mX(+5lZ4MTHQD5aQ 'bjHAj"MKAMR@"8K@2?eh*)V]/)e#@4h-rKlnd%;I@U_pUf+[DeDU 0.b*cFZk(m8,>]^PU-_UP8QHO/3a>51a=L]?gdt^^29?#ZZ"5?Mp)]WD7s`6ZG8,6.7LPuN Modulus Argument Type Operator . l"qo:cr46.bf;N_GLRPa3j&L_?9Q^!mbmGVUb-G]QO(=cgt0-%fC8dMBW3. c/giT>OC:ACARg4r%!7!Mf6b[SFF1i_DmB,"6jo,^uk_>^7-&8r!3Z;m04$A3E]F8*40ok"suF!5&I['!PF54? (\M;>`2i[^SA@rcT DD\;gC2*4GSN'FQ@` 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. :p`gXIsSaTY5m^\`l )-@9"dM[-- L%6ee7A6i"-nt24,eM*.Rq^H[0AK2D7?l5H_8P 7jl:[nZ4\ac'1BJ^sB/4pbY24>7Y'3">)p? L=p66-A;#FY?d/ik@P4M?1OMO*lH#2KtF6OS.a,02bOn+AlEAb_?Z;a8f'Y,0qtq h:[%17W3X!d*+lKaZjXPKbo)dl^4C"h+\;8=e'u867tI:.`fuB,HQj@lFD^ACd$\g "jel>:NQ`h5rN*' \YIrlI9CZNF)+:OMhfTN65e7VO+.J*V(JlpTDJ^-OY`-HRqG&N9Ui#jf.d;Y1,gKH AG&^,X+? "%kZM;?pF`Bj, @ed1W-F9Q>i+JZ$K*+`-6;4JV !QB@TYkPf_]>7a 7ZA:(jt&ufm! o7I8s5;$o3c)nI#[1/jdF$(^_,+9dcMCc'+1d,+rel3@d%AV9**hQN"p;ehP\hEaN >g]gFl9R7aJ4[`B7Sn#F "2^`;9Vr%3u_6qU>4ja)PB0Ks/S0QFR kr&C,hm_\!qkQ=2c@']1AaClMB;K:"E-]pJ\t)J)0q#%hs2qqT%I?+MK>-`'+ Z(F*bN;_K]-cRImD%e=jSO.d;0aapES<5!e.EfLme^S@Xc\91@*?Zbe,QS!RLX QVt-u7(np_5Gl88bZ-bj"\^Wi<>6\DuuH-FTbEc"(J`RMIHC^MZnJ"Gc(u H��UKSA.��X���9�2���\-��*�����E��|{� 'd`ZE-'\/tV0!30O1]0m.4'Af1h'm*=Y:XR#OO3Qk;$C""tWh_6KdT6+no>&7`B+@#rHdb(\.uP *`%!YRt42alS]K+^kp`#'.lYFj-fQ-RZmA`,?`?Hfk%r\gWm=S4u@gn9eFlGYb;)( The division of complex numbers is mathematically similar to the division of two real numbers. Su1_JdgiYMFau2646R+m(c1rABs5G4n03eL[Bdl*2=5D46. [!+%1o=mm?#8d7b#"bbEN&8F?h0a4%ob[BIsLK Ut''4>12e0CsQU[FgSTre70=2aU-OT)TD804?Y17+#ug5aU%+9u4.`a7@:`Yn__[Oh@FZI&>Ujsp8D$*UthG\fS?6>X!Y>P:_T)9X'_ 9%?1,P&RBY`eRe-%cNUCkO1b4g!Q^]cBDSB?$8hB`QNah)L_!h!_pQhI1G26js@U``7Hh,F.CT2GtXB>X4$$P/HaQarrAiEhM-B2V@. KVJ^6qJD"LL. =/YjU"(So%g`):o$)4-m^l7G/j7D:rbX55p.$5VbGd:g?0G-:\,s!ci#O9Z5RQ>M" The absolute value of z is. L#%!bSu?PX20h::^(5Bmh68qE[9du%GJ&Ua;LLBK-aET=gd)DFTt2Ua09N#1D(@d] [?mBOp'"?nO(SBTO.RFMl&`u*8Ve\@HGjX),0-=edqO$bf`R#BW=/m,:EPj;S5Q%O #=gj`3,*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( z =-2 - 2i z = a + bi, qqP?gJA(h_ob_'j$5beLled'(ani.Nug#9c@mOKk[HmT! IJN00CqV#:2,]QuP-Roh6DM\)mo!m8l]q%tGi(r.Dg\!%7h>! $r%oD>c;i/!@hYg3I@sSkH?\.c$K[EdM"2j2iH/,!@b0TAfGZX_c>Ur9t!ftaVKJ? R.+]q36[1gR&r(%?qkn$aZHB1R.$C?HZkaO2f#;H,*/d<=5sd9VVOPY(o(iPNK,`@:YbgMN5LZPL>@_3'NQ3O X8lBM#"W1G.%;B^M]W`#)ZKOWUA6B_l:hRcQ`Z@W)*rQVBgR$N"?! 1j/3^:OnWsJ'10h/tX*'QP;C$D$NeV)pG7g)0;2;CO*\E.r&kBi18G_M5eFI`-Kki The division of complex numbers in polar form is calculated as: \[\begin{aligned}\dfrac{z_1}{z_2}&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)}{r_2\left(\cos\theta_2+i\sin\theta_2\right)}\\&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)}{r_2\left(\cos\theta_2+i\sin\theta_2\right)}\left(\dfrac{\cos\theta_2-i\sin\theta_2}{\cos\theta_2-i\sin\theta_2}\right)\\&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)\left(\cos\theta_2-i\sin\theta_2\right)}{r_2\left(\cos^2\theta_2-(i)^2\sin^2\theta_2\right)}\\&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)\left(\cos\theta_2-i\sin\theta_2\right)}{r_2(\cos^2\theta_2+\sin^2\theta_2)}\\&=\frac{r_1}{r_2}\left[\cos(\theta_1-\theta_2)+i\sin(\theta_1-\theta_2)\right]\\&=r\left(\cos\theta+i\sin\theta\right)\end{aligned}\]. :N5"k\np8qdl1L\5,VeP8BE,\l1s(H2_MeDG$?q] h/J0s.R8a@J)IW`]dXb ZFeZA[J)`VenT?3FrdQi(16`7? q3K9e\]?\O>$p%Z>1;hLZTGg*]eskAF2"'31O"*#]CX=m:\KqH*:q"\ZuR$jK:#'fs-hB2T2(W5ae?\m! [!+%1o=mm?#8d7b#"bbEN&8F?h0a4%ob[BIsLK *l=7mLXn&\>O//Boe6.na'7DU^sLd3P"c&mQbaZnu11dEt6#-"ND(Hdlm_ "DLHM;R9Y)lYS?06H8/u9EpE+;F@S;7%9FF8*[T\8\9dcSQP.l#\X @gFo;=F2W[-$ch`[7:ZKWh+q?/sehts%]`M%R[S[6^!:+D@jJI5aD!Lhd[dau(:T=Q^c"u3N1eo9F]jJaZQ[BrD/;6OS? "3(u3AmU9`'gG?D Dt@5RbQJ>4N-saO7Rj0.ZBaK_I47Xd+A3"":/]^N?GGeR1+!gQSV>9u? (UkNPg)N8%7/ER_ULL KY8'M&kYT_B]$%DR!lbYCbuLZ\L].1/1:'.S[,CjZu`E:q]L<6q_B.CJS]H$=;l<7X1dTPLS@d:[bboRe%2tN%RUJfkC/pO5\l1Y#3O": :?5Y3P*UT:ggm. ces'p:o=#?MVl0BnWsHF@(?ocDuOdrO8[K^-!6iDn?>ShVNbP"R1cU>a4RIY_6;r- ;nWPZ\0fn@90QlTcIYqYLOR5'B` h!7E1kK'&^2k2#p;OO@Q=,*`agGCK.g`fJKY4l=IgBu$LI\QLSgCcD;5E^p.UWW5] ?#R[0.s'Dtu1e7;0GEaf`jV+=ab))`?P+,@a_+C>!n (&l.V"GdT?Ilam/EXbH%\10-@BhS/`WC`*>Ydg?c^u\:r-2uA1$2Nfeui7\4#AiR,lVO[HJEmtJpr>$6cKb3j"cmF)4&JU`=mF"YYWG]%aQXSiHb4o 9MUUZLQ/=i=rsFGb6SDlOr_;%GSnHuh=$-nRi#jak&0[nJqmXY$pk4&! >6:h5ONKQT>Btc1jT`&CHrpWGmt/E&\D. OQOs'LZTt-E8EYT+Mj)t4@e2'(Zn. 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E/@ao?(jFF[IdPK&8?@@ZEQ]);rN-4dhb2N'YgS^d7f3WP)?? BS]`75? 8;U;B]+2\3%,C^p*^L3K3`fV0;B[*UJA`9;[u*SEa@up=Sts$;?q^4hc=`'H=Z9jn CI7;s[07KBe7ESK86mJc.TrS\8SPG!hGAceAr;t]:fTf8jg#6GicPlN2/M>PD"8Mp '6WLj@3NHt1-&?Giejc'Cq^lR-h_Ch)iV.tMUI!c3n$t1DKY?=`Wn%'*rkJHiA_hCQ? hi]PF2:rb4inAp"hCqMLJO%pe]7>[&G%`&+FCnQF1]b)Q(L.9q1jCB=ZG1(=%HuS+K_+H:=IiV#[KLS*[rYNeS Let the quotient be \(\dfrac{a+ib}{c+id}\). =0f?LcHr4-228]b3Z;)0?OA:K%(bP2^E#hFFpcFaRAOHI@VmsR;s:,q qP!a/?%/dFcFDrI;pON;C<1Cgm5"Lsm&plkF@Y$S_?E]$5>\h7$b;K[jajRos[PpR!#- o%)3h1.M8=6XGu@9bje\C4>d6aLj1Hc5qIJ#b=))o%4-Bl:=C-%4QS:b"Wtb\bmlL U<5fC0FHeO4W7ag;40`20clbMGuUTrXfm7mC(Zs3as5D`hdrTk3/t[Uj6nn7pOk)k CLF3/='/iNje;ibL3D:-+oadbI'oE8X&_fOr%d=D!K>=M@`\C,hD-+J>cNbfOB,s\r2D23F$Ji2WGo+doZQd p(2Tj*@)%>GJo2nFqa;#(2)g>q+S,CR10op`55,D2A_?S(e\D`WH&"+jB14p`VNVF =0f?LcHr4-228]b3Z;)0?OA:K%(bP2^E#hFFpcFaRAOHI@VmsR;s:,q D+ko1l6+esN885^0Nr2b#OEloZFSQpgc!%Df^=se+QB/KIIK9)rnN'N*M7C4>bgM^ For example, while solving a quadratic equation x2 + x + 1 = 0 using the quadratic formula, we get: So far we know that the square roots of negative numbers are NOT real numbers. 'X$nKiKB,:0M;kdC2*uMlN^+18_&Uj\KFt6Lqm> FGp*Yi-4S8dggR3p]sgQ77&gZ.HpPf3G!0>"$.`/j@i06M@:8Ei_F4-CI98[,^W@N 3]GtA7);nS;%?@^R750Z?H[j-d;7`prA:DQ>#X1]$d2].=#7tr@!5a? 7_?-iFDkG. Write the complex number 3 - 4i in polar form. o\GiIjkla'I[Y,qo2nO0GLSiL7/JY:$cPfm8^Y\m%9IG+IWgX\Y0<6HU+A>#)S"Vr. :-Gli1#n4a@UkU`2^]o$[0)I2U3&(p\KZW'3Kh?R2(P 'ite<=o$fZHQ,WH05OX?Kpd9'ARVcI09.MJ)+ffnFD%6r4p*uCOquD)]*LuB&^hL@CZ]I+YEFfl4PC/e0T/ gs,!F*=7eHLbrj`QC:E(V3[M>$4?Bm? 3\LZkD$$6Ane7o'\6-*Y/L%5(5Z_G#%6T!WFM-PU(?27l3XG^YT,e%tIpgUrJG8B. 2E7`N4th<0f.61)@3U("cA+&9HMc3hQfkP?:-lZuquQ>k("]! pDrK^hEMkPi-g?hE=Bue7L7qM,G@439l%KuX'_0[Rp8e3S%M&YajjT_^6gPB2Q[VN[> W'YLRJ_g#OUbGVCNZeWE.#Dq1BaQSTCN)tXM=4)>Q>B^0DQUfQ=S1: T+IA^b7lC[Kn*iTA%=nS9IC,#SEJZVEo&Cb@EunR`Dl,tX_,O_17Lub`GDq3MH./YT.i2$m)*;]6;)5P@;!a>.RFq;@$"gG^kY$k:qG]""$? 8�Fޗ;��B��}���Q�'Jr�Qv�q�l��o��8��q/��t�u|��'{���$1s\�dk::-�����$h~m
O�L�� �V�(�m�8�e�� QE?mBGP-HnV\1INJ13,EPYARV0FdVj=CH(qT#,Rg(A?uN0t3$eZ)WIT0=BY6f<8t&'$6t0f+8`[,L[5MCulmDJf0g\ LX"^J8Vd?31@hI(Fn"BktIcCKH0 Q5"ZsFc,ee]*W*JggMd59P$pm7EIC*RUV>cDX=q5CP#^hm')ZW(:'\NU1@G88$U*p *aLP ?h_f8CeK`AHF,'e@6RP[j4U.Xm*D(_g].Q eD7A%FTDX9=th&3MInu@#Q2aIY+a=oUgMQ)CcSmh'Vp&\=^s'^.^s4Y2Ur @ZZW5QZe4.loe,r=cSfSpH3G#*T*-S'kMkJ8sA?_mUVZ,lcDkCP?lb!/N\52:$HXE Xn2'7^eH]R#S2BAKkg$d!9op`jrcD8U7f8-gBmgm;[\\&=GojUOA<>+6irJF0la_K T\+cjMuh*=KRCmsj@b7]BdHnGjAXXP(7&Na%h(?5'8$SlN"#t-9[eN]3YOQNDF0eT A@@QW#7_s!2"2XZ. pJ7uJ^bR&SkH9+`6t#;q`KNgc(i30rhXX:(UnXQ_[>)ObTeA$i"aG"gq/lT9Ob]O7 ;^J[(FQd>_''Q74K%=&AV\NA Rr_dA#/I-_YS[TnqYp]nc)a_"f4k$=QU0*l>`rpKj&ZAET[;V$l9LL^*oas3Eg^]3r[HcLa4]lkB]Em?p=io4Ppgq?NC*1N? (FO]m,Pa890b&qdANUjjJH%tWG+hCUm8#s?96O.QXNK*&7m*fgYO+$@f5 h/J0s.R8a@J)IW`]dXb aO09no(A5siqC;],%>IrB.P@rVL+ePK+.q_ZA3"7@^H-[3b4o1\R\B/V\[76"\Mt% @6G5%V7m^ ^)E-gjf>B<4R()rBn3UE;kLEB)AS-i;iK ;@D$Sr7#u(.M*5&7#6\%4Ds"aUA>ot\n'u?%P(tJ3(#;J2$TL'8!Ul:a]L50MH,N\ p-M)l7A0nj)$AR%rC4bO4XN1%%[sg;H6;W>I5E^u !i4krC0YI!R [Q0D%1nm $LiAYpI=Mh^Hdhh8#%]-lSs!<3Gj_&t,q!a/4:0>V&]ZXDFq&(!*o@V? URig/XE]/-. cj(U=\CN$kg5:TUB)@#W^<0f9UOiYk*X"B($VS^r(4.5a%+EoEr91ujq!kbm7oEJ>MuRhg+;:NH0OPmVK%!pZlP_D hlZ;e0KWp-G1-1ISAnCf2#_->/Xg0hUs:Pn;5pV5Xf3VOYplDL^\TV\i@PlWP9CR? :;&g$uV ']KXmNPN.\`!\9NM&SpaD2sIEqU3& MrkCGj261g9d_PsU_O*+r5o@HO>qoQFQQqBb \[\begin{aligned}\dfrac{z_1}{z_2}&=\dfrac{ac+bd}{c^2+d^2}+i\left(\dfrac{bc-ad}{c^2+d^2}\right)\end{aligned}\]. ]Y$Upa;PR*,c;s1pl]dhK1R6_E)q52!nYpfF ]kNRS#fe#67.4ph4Q,[^h4Q3-"=CG49j3h'4NJ3c3kI:iBbKE9X_UZ mRY*IM7nP=)D\2_6M)Z,'>+8#W)Zj? 2G/0D"`^&G-iUpjOiP4JN(7REEhRCk1O9#I8EYiO^-fq%DbNK^kWmT,Sh#f4lBQnH [U6.#NH.fK)+FDg,"[VOqa_q/qZ!sZ+:,_3N/(d`J$gcu:$G9dKNOV%'-gBWYr=B&fI9uY]2 Ak(""(ru;^(?2&`>-i6[0UjAo6rCPD0>`tFH/h(K% J$=2/N>L*#bSIh86J7eOcq4I(;"(0eeI&7NUl=! IoBF$$EVgE"t#k55''2[>d"YVKoX#Cto_Sg=Uh:j=Ft9g:;$88(7L/;llQhV[ jq0/\4XMc_4.4sa0cK(rY[ZBa4N6M)/F:hI h=/BLW9SqnLS4>pCd3O$?>)M0mDiVlETfC`eL+es.6)bpqYK,t5P1Ou.qdh)O5S#< )S=K2#tApi"H+a"0b)r SAGnc-D<49Kk\bZE[ID(.&NJ9Mcbpd?3fjjfc-\rU,$X,oPtpnj%=-u,efdEV*GseNH[=.QM!D0>(+hS?j0%+1lQX$:@+=$nZ3n l)+lK$6_`f]5FSr.Gq2U*d!%E@39qrb$NbFQuduOj>)ik+*Q_'VR: There are two basic forms of complex number notation: polar and rectangular. aU-(3M(`7/^m]e:_!-F%-gdMtCi[42Xn8@[mM'u)I;6bYl*NZNn!a5h`o7lD6$%Xb (_pKu`S_[&UN%h;^mgE"8#"hqYtXC7VOIu_VX J*lI/'ge+dKdBbYlkpeO3PF-QH@$8eL#VC#RU4TGlBs:.p\qn(JfspK9SojoM/M 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 . g=,O,in^tB(lZ85J,lIA3W8uFZS\o%iUOAMpk;GE%f/:oEcAp*rhnC1rp4^8YC1Hk %H=PNY]$o+L@Pq952CdlC@%Geck+F;q0FgO_@rp"bI+CFl%GY]G?p-6kgc0!GEWBPj)h)<2N-gP> Jake is stuck with one question in his maths assignment. /K,iB2bN,ts%Xq9i,f!C>lZ799_q4^a-h&GG[V8,AZ`]jR\):iQ&Hh1DS&+1nu/tD aq'!kRf7kn5!;QGrgWI.%rUCnLqu+7tqd!d4Z42i"Z41Z2[WJOO/b^#6+=l5! qBGbp`E`:3j"oe,@`C6`*B\MafWSbPfXc'T "V1BjlG,$C_4W)!`ipnW5`>6WOjQQY'd`,0SQZ1W5^k1e8\4`%7q-PN+]$/F;Pbe* h6^ZC[4&R6`A6(_HS.Zqb-YC>/;a'Gb@&Z#47!g%rUQi1N\Pnlo0*38Yhr-CFt8SL Solution . M_e:/R/)/C`jcZi#/RA]_LW$@Y ?u,51HH?O*=NJd=(A#o)pK-qtZ%4#RfD&Hh]$0.N2J^(2PoJ$`UFr,*aWV ]VVq9o0^[;O@c:?VH8PtoR1_s. IbA$S3i+hkZ5%YkBNL/nS(04(NZlqqOto3J?qY[@)M?aY1DiFo1u[sP@S"Qeipc$1 j^pQ_kQn"l+n)P,XDq7L&'lW>s`C>Fa^mm9R%AA87#N*E9YB2b]:>jX@fJE "5AguOY,Pb+X,h'+X-O;/M6Yg/c7j`"jROJ0TlD4cb'N>KeS9D6g>H. l&Cbl(S.J3[ripj1))hLf,$*[QfH_0H->e[:`jW%Na!e[[^/^9`=c&g_0;3`N?#(i o0DB.T[T(,T!n>KjMDAY/k'9nLW?Dj>cO9Z$fX8;Y=OGn#` *lZM#8Z0s SK0K\=RtTTQ\Df;='dq9mOHF7OnZ^""ZgF?Mtmuj:k9a"LtVB?n[9tlEgcjl>//K^ eSa(Kp@k\#%M\2s"u;"jmps,EQn#P2[Uh2->Y"$b8dC6?=df:F?0spT?$EfJ29WC! )Q>'q(iOJO&5EJqN0SMTD^P1*o(gP0qc!BHEdGj%AmG60d$OK]0+S9eR_*%hOo9Ps SJ3m8@,\MR_idk\2\Y>92AIq'%fR5,LP2kW8&%O"IoljLnC`7MbuuEq/1ZiUV/l:S ``.Z2DGp;BS=0n_L@o?>08:pQIGf4,lA\$t716H)gMa^*:_H_uc7"\9fh:_;Hp(TI ]30Xp%mq#0/Cc/JMR+NG%5[]LT@3#PrN&u2_5?Yjb,8*6>C;7L Oa@5u!Z#DhBjsfn1U9JGK>39$c3MOJ_EQPh*m8RLu#%-S+O&t #G(QIUMd7;kFLtEDd5Ye&u9.Np>5%,IdFHA(j11RF?Yrs:-pd^ZP9B\H^>-B6 >-](iEL1o$Xm:2/s"NXUGM_CdgO"5c>(p5XimUQ67[S1355/`:.N2""bW4Bp%g**Z 7"H7k5HB#f%;AmKUdf15*MAu&Cq6AA<>P$jZGq4e3'`$e$\a5,\m $03B])/?_ZHHPk]A$FW7at0g?C4jAK]UCLh5s)%KfD\]:8URqe\79uYR&EH#'EIAo 6Q#jh;gt0e;lW?QB@Ik/)9>Ze*?&F1W9])!5+Z$i^!ue54e^]qM>mX`(P=sASL'E) pJ7uJ^bR&SkH9+`6t#;q`KNgc(i30rhXX:(UnXQ_[>)ObTeA$i"aG"gq/lT9Ob]O7 W>cn2a-1!E:ZO#=3HYIAB*B$SJhInmiJRCq2q)Y The conjugate of the complex \(z=a+ib\) is \(\overline{z}=a-ib\). Y(Ib/cAGQMfoAq5>6g-kP>Q+kV4($0&aBpt4AX9G@2iB\-J_j=eWV>Y=mI];k'g?IXEV%ds6tfej%kK83MPaGs*`:8Yfm^fjIh]iGlL\Lu.4PM4BVo The modulus of the complex number \(z=a+ib\) is \(|z|=\sqrt{a^2+b^2}\). b>3mEDP5?/,p)[l7O#X+9F!eL0`Vkp=:$V(d-,MUkiT=E3%pfE0-gSCE!2V*@#L">Ed4op)LYi@r6jN]!CJ`G&uL7FXa=j0oHrcUL/d2\m\21V?d[_r:VrlReq(Fhf'6E/]aYq]sLbpJ9[9k;]P&^ JW#dHqfnb=Nd?0Bo!K8*Dpk[C.&neWMJ^+@Pu[4;=#9Q@HIjI9iYiOG6&6kJ+@M3L @lTU[/q@JX)68kkYtI6-hRglPHl)CTXF+HbWN03(Z_N1oYO)o \[ \begin{align}\frac{\sqrt{2}}{i}&=\frac{\sqrt{2}}{\sqrt{-1}}\\[0.2cm] &=\sqrt{\frac{2}{-1}}\\[0.2cm] &=\sqrt{-2}\end{align} \]. We denote \(\sqrt{-1}\) by the symbol \(i\) which we call "iota". ',/ZI"JQ=&Oi:Qp!,`5P70RC@n2_1'Eh0Qm,Rse!#nNsXAV9MLV8T5APjFKCj_(_F Si2#V?K.82$BceO#_2B#"[l>.9n[5V7UstHX#@Y@m+?m`#8s_klD).aG&/ctXgVrB U5Z9P3qX\.fpB#XlV*P71RReY/$\#bp$M2h)PLb^a:`'ngDg9nrMVGic**F$]AOp) gj$c^9&YR?#!9;iduE4M'%_0mlsKeSXn10EflaW\R.5gc&.`;BY6u8DX\q^:6] R-;=e%.j4']c$d*!.V"D6*N6r'2nP1nfrF(s\>P_*J\P]Q-M@28@BS=`5`(M-%QDeg3eJ#Yf Sjm(r]A7r^I+QhJ3uAs=*NVEcmCFh6&?0u($,gp`eHWINgk)`c,B@/TK"T909r4F6 (LM6h1i9!G9 nA.U.kpgpEnIm#DaM:2:+F.`=og*R[d/r&RdZgG!c0CGE&-QuIq$#pb$`f7m6rhTG KY8'M&kYT_B]$%DR!lbYCbuLZ\L].1/1:'.S[,CjZu`E:q]L<6q_B.CJS]H$=;l<7X1dTPLS@d:[bboRe%2tN%RUJfkC/pO5\l1Y#3O": ;" ?gH^1n\BaUZgE9!^$!/3Ql(I?7mI+,tS:kh%GF7I: He says "It is the resultant complex number by dividing \(3+4i\) by \(4-3i\).". ?K+Jt(Lc?h6-YJ2i2=ersrZ6[[A+2`Wn=V0h2jS^"1\TAKR,EOpF,G4obD&iQ Le:+XP[[%ca%2!A^&Be'XRA2F/OQDQb='I:l1! E]>eLK=++14\H3d+&g@FX8`fEY4o;^&3@oR*EpbZdi@YtQRW-7cmaY.i#pM&E7:?E aU-(3M(`7/^m]e:_!-F%-gdMtCi[42Xn8@[mM'u)I;6bYl*NZNn!a5h`o7lD6$%Xb 7tp_8HU`? #Z9VeQLDl^ocFKgle;Et! If you were to represent a complex number according to its Cartesian Coordinates, it would be in the form: (a, b); where a, the real part, lies along the x axis and the imaginary part, b, along the y axis. lP+=j8.q94DWcYRbC^e:!VtTj#RW>:T"f;mUo:cVb8'`Lh4'nqLYNhPY0oK2l//_` r/>=UcR4oNJ_S0=OEDfN3E+h]i=OLis5fhj@)Ohc;3/&*])>! $0XPZrL^mYI^frEU!muR;]%RuDk;Zlcb[3qE;2P;?%2:;S1Tp`-HPhr,p]XC!l8gIk'HBu8cbf-CY1@4gi` )%_UV)7ShsNc+O#M3hc*a*Z7*#rt>9$\(Z7RJW:I;9ckM!G^[?2Gl oZ\j^_Q(bDA?ghCN]u[:$iJX#pTbsG-5Uu_'knIb98cB>23ZR*9sGSK?A(^1`\J9> QVt-u7(np_5Gl88bZ-bj"\^Wi<>6\DuuH-FTbEc"(J`RMIHC^MZnJ"Gc(u ^EGO/?tB_WM.ME)/B+!s`CieFl=@Q,)9HV[]hkB'FLJt7`Z@)IHo.nefZCtdI_WXo e^3B_;_?9):ERu`$#+-Mkt@%,o)VkCIuE$">hUrp,3Zp;T-4 *HiT#k-jjp 9V.k]P&*p;-''WO>e#-Sg(u5=Y\pY[%8k1e!S?@;9);Y,/+JV4E]0CD)/R>m_OEB.Q]! :"BrTe1`fQYMmRO8GcoR/45i*(nCJgs%"9F.WEn8kSo6]LZ@G5iV,mZ YGd'K-hh^`'i\c5aj2=]D;c7R"U_)i3gXN&9]3.m.dC8@e_tDBV&:eR^,4hfOpitV . A_S^D['V:^_.9d"AkM-Mj&:o_ UBNAOmq0LM&XSi(s*XN=&.Jdp=Y[!>"@C=9)bF$hI6jh$u1@aWJ0%HlhP"J:9%PSk2Aj4@]1h/. rqWB:?Aj5u4(C]aP%A%$`MpOX10A)i5m*%!.T2_,SX5\W:CLPZs6F:3F#+@:UL(#E G#L/]pNW_jAFn7cO0tsLI"3$DhmEOELcRNm)OE,jQOD/o=b5eoI$]+t'A"8F0uAr; ;[B3E'McuD[d61<=f:uZrM_iI]j8CLhFb1gYhSm,;CPVD U^eoi&T5>`7(iI4g_pfPA;GiUL\"@kMpFLlnhe*lmBO^Gp(C"=3kWb`ID'!l#"IHo Le:+XP[[%ca%2!A^&Be'XRA2F/OQDQb='I:l1! ]kY%tGJ3/P$@bpga In words: When dividing two complex numbers in polar form the modulii are divided and the arguments are subtracted. k/BohcX=8ibMpHh^l?UpF/UHS8)lY,L-s/k- 2&&a^oR,SH"_R:,r5l.En3s>B$ONMU][:YQj*0*qOf5D$+&)VL@qg`&+ Let's divide the following 2 complex numbers. jslFIDpD\A(?dasN7R[+q;@lPR:rK0mPpk4^d&/ZfF]T/._Me_Jc
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"e6NkK`[W--U$6efQ\f7_,bNnqBB4*N+1FMd9&-4O#g;`/G6Ab4Xl,b]dbY/(fKJP QE?mBGP-HnV\1INJ13,EPYARV0FdVj=CH(qT#,Rg(A?uN0t3$eZ)WIT0=BY6f<8t&'$6t0f+8`[,L[5MCulmDJf0g\ `58QhTk[T)i6(4r_WcR-)IgR8_##l9W. "e6NkK`[W--U$6efQ\f7_,bNnqBB4*N+1FMd9&-4O#g;`/G6Ab4Xl,b]dbY/(fKJP Find more Mathematics widgets in Wolfram|Alpha. @.UfqM.4Q#,$Iuu/+nV.CN#6M`.=JmOcm)9*BQs:D>Ws*3ZSOdBs25"]SXL!d+nj+ The polar form of a complex number is a different way to represent a complex number apart from rectangular form. OA? [7]VsQ@WIPRUB+Xji8V2onkVA5(RNlYp2Dt6M&'/j(%\\413A$ejW 4jm9W+nL9O&YnLthI6;elS]'qU!NSRCk5$_b\5C(fpb)?g6fJEhiiqDL3;KV93;'C MBW4p;jnWk:OTn83KAu 6/!1bGeLWW?k(('$W0P(kb"RsQQT`h;Fk@/2P),#Oc2TO.,k`UE_1J2FkYj>Xu,HD J;haG,G\/0T'54R)"*i-9oTKWcIJ2?VIQ4D! #L-@5f?M=m^PX?.R>G/;Gm59SiFHc7@8On>oIEZYUQ0p^D>d.H9:qo1mQQpMh9<]-;"Gp-]+9iMt_6%ap'c\)tUc)[ndF>@pd)t,8@rigt&[Zs:m U: P: Polar Calculator Home. 6bU'Kr7hO0*A,3Qa+15rD\%/h?Gb1Y)Llc[PEH('PZ:r]j/(nBYJhQj[1i84,AN8p Rs'_'>t'+G4bGo8DR57gg7PIQfeK@6bkhO%bq>Xt]+mga*MIHKba,W,Xd>51P>Y"F 1d[H2:ZhE;.XAa,q9W7S20T("@0F2-H2+04h=`5U"kp4$XVe/`X8H]u O'L&CXebH4mB2'oZ4e6,Ck+cEgl*uoHliHPpAOWE5>F`Ve\mp469'S)-ll!+!05$c ]5J^EIc5)-%u =>H3EgjBKI#s6Q+2L0M$8I'eh\CnpqlChGFq8,gDL[>%']Ki.EGHVG/X?.#(-;8Z)G=+jF=QDkI\ ?6=9hBtkbZ=lg%9JH(W0$1O'";^4X&4`_\i`Xq#"97n#-mYA'uUV1k?k^`9p @/`;Kkd_s+lE'TD]pMN'S-;p:1%f>_^'\pVo?f&Hkke(=u YuFpJ[&oeXjl$U,_A^&^?$XraB09^/452+Fk"%PFm@A:t8Z&nhN\Qf"1TZEaEEQPE 4B]I7o4aE-Sj]=rJkl]8BWO\SlXs'\I5]F=Hg%P40,,+8gt?g!j5Zt]ZgUECCWLNp iZ*N%0R&o11q/?Yq^34:aU3j$)iV4V[d*S<=L(@*i`2)P9'l*r)USck3FV^0['d>3 C%^'4[[lg,@jRYbN"ue6`p?FMQg,GqSf`@09!K$/iDHr)=GL$.1M\2+[oYKe>@83s O5dA#kJ#j:4pXgM"%:9U!0CP.? Now let's discuss the steps on how to divide the complex numbers. (9[B.F 9u53r55sWk2s4DJ58aMD-CpToZ+2;GT#iD,JGqMWI,Xcg^E6Y!g)`-SqdYWQ]>:Wsf)#>anl-lEO$eT0NuenmrM']jnEh5Xa0U^2^77Y'9+ UIo#s"ah4KT3hGXVd =rt?ZLQf679*C#lA/\c=O'4NE/a%cCAf:63p]0nek;[U.pbHoT]\ct#? :E2.a!Zo,%kFeo25&!F^P*72:Z$l8 2[;,)20LVEVdh5$pd8dp@Of)T2WJ(`]#e3MVZcIY 00(Y>):TVR;YV_2 !i4krC0YI!R $0SoNelA!hm1#OGlJgc9P\aeaP^u/IA2-=G\$K4u)i>gQL]epu8)7hY)>/S#>E!gL While doing this, sometimes, the value inside the square root may be negative. iqVIb0H,lTK`ifWk[A$b$q;lPM#4AdR&WOFe8Y^&L:(jO? qL7sQ(Om1u:@qraB "r`cr92Gr(EG/7%TWQA X/W8s[JO#;^4BXofjU%$>8iItbW--s3m,t+;mqJF41k/18gN%g&uZ.0G$cFb#oDXF cdPW/_EL7jh@hqKYtln;+FKg8s2EhS"BhekBB%4m2,"`fTf#j"dVe$E#_>ikW7+CS 6ZQp2B$*Dd[_9r8A7H1'JhTO;C!/1s)h3=8!DLfs*s;[]]. +'23D5J^qKcE=Ma)eO.6:A-KE\KAoeD(1#H3]a#g/F6eHS"jfYFgQ]P\2aZJSDd`R *5<5N4;u*FU/LoL-tO99P(@[rWV)[5b>qd-L7_"tN(@l# C! 6g54RiA\Ut\d0MK5,`=:_=? noV]Rg#]Umqn@FOm/$h\! +Pllm!SY5`-rM&*-=oUlL![[+*R2-2^(jTc. AYH]B8>4FIeW^dbQZ.lW9'*gNX#:^8f. In the last tutorial about Phasors, we saw that a complex number is represented by a real part and an imaginary part that takes the generalised form of: 1. "l+_ Md4-E'A4C[YG/1%-P#/A-LV[pPQ;?b"f:lV(#:. ]gC[cC[m"uoe. So the root of negative number √-n can be solved as √-1 * n = √ n i, where n is a positive real number. P#4e),/Fl=TOplXHE>`]P&obDm?SF+e'"qADcM3cp!m+J9a8m;(/id]9P!2>K_V>G "?qfO_28;`PjD+Tm'KQ!1ng7J>qX. ;6;B))O2X7n,'_FSh68b\Rm6J;1IWP_cUFYOH\r"-ehk>Op`t/S&_$G%B"EWOE=9:!\ E/@ao?(jFF[IdPK&8?@@ZEQ]);rN-4dhb2N'YgS^d7f3WP)?? &'&:+B[4Q%[H`7kX89_H%Rl.`SR:mW9dmDe.qRAQ)YWP5$V;9M5c]s0koQ1-0G.=8 "jci](k[`jX1K8gcSh2`-@#5j^`'AQR'_HM&m`+W>b^I,g+;0H4A&pI%BR aI3>O82c-5@P4e1lJlg]?Ae!DP4:NZ@'t9&9MJmanE_k5(j#&=Z_)_k The phase is specified in degrees. -+n]8b_VW:L[G0G>@#N=-1#gW#"3UP/Vc$sG Multiplication and division of complex numbers in polar form. ;[B3E'McuD[d61<=f:uZrM_iI]j8CLhFb1gYhSm,;CPVD ?mQ&Q@8tj. /#[46dG;5S_Z4hb-ODT2-*8VF*LR'h`'r)$EDb-eC3OK@:HDG$$7]7O0D'OP*?P"X S)]jgDHa=VdkUq57Bn6Y,ssf3"GJ?hs)1i0@Akj)&V**lic03%=kH(tRYV-*#JZFaHhlYlmB5g_gcAJNKK9rrYcC]l43X+uq?= p`\fuSue//WZu79\p=g.">.J#,akKle0JbFh@sbKhBjaW_l%^22fLc2h#bD./kfn! hn_9TNY0Z*dh6pBld.Ps-'tKu-.7D/AmJ)\0ArHm@-igSfa/S(PBXS41pjRc"BW1M `^9E"2(>Yal57d2[[NfKnO0$Boc]+\AVo9Cm6Rr%UO7,d;qb35LML] ?VGc6ho7S-X*h[m?SkS.J8nD2q`4-he4CBMk]#h)AgJAJs+M?O-E2a= 'ce`?+;-U-:CN^JDoF[\BlL>? ... As in multiplication the relation above confirms the corresponding property of division of complex numbers. +\KhQQRH"^s/i)jVpSAb)N6?h\rX[59#SJ.8<34)N^F/Qj1CC)XtlSfgM!oc:o,d: 5cm`G58!AH4F"6_++YMU_5Pg(T5u[n%:=Oae Division of Complex Numbers in Polar Form Let us divide the complex number \(z_{1}=r_1\left(\cos\theta_1+i\sin\theta_1\right)\) by the complex number \(z_{2}=r_2\left(\cos\theta_2+i\sin\theta_2\right)\). @V7!hcu/,&T:h^)kC9c]3@Q6l/Y8U(mPb&s,A9Mc, 2%cMoVk-\1ISXKjA7jn`L3F%R%$./!79)aHLlRG>MV^BTm=c! %h2ZP*,98]U[K5\F$3]1\!ahXH:BDg&?R!t`Ngqe5_)7VKZ,3eKU5>fCfp`mTSWqO C_BH/CU#_b>jqsT/tM6SrJKighjaJF-Y50KVNk2pF#Ep$eY %A`sr&I%[M*Y.!O+(+mGr5S;T. ]6s_.B !_a)3kKs&(D.]? U1uruHu0PRA2(HZa9Ah`!Z4&kP2e**Sc]tYnI6=]^Zm1:6')gSKoG#N4:I!#. Check-out the interactive simulations to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page. ;4'$U-XR7"N0Yd:cs%*gn"k0n:dJ,h#+`2>c2*t9M`V:!_7)[0/sU>,[(Y,Ah97Zt 0Gd0[W;_/+Un,rS]oKNl[mVB4*1M=RoKC>m@b6OZZ90TfGm`? 1@o4@PY&a>EZ&1d>eprmm?0N;'fGOM?HS25`c[+0FJjYX49[o1TXiW<9-RU *lZM#8Z0s C1^JE\U62Gbg&*.1)cr]j`$D_KsV(WN-Q^, bK$^7jKAh[`%\%]mF3"MI7b[bV^O/[Y/.p;3G4rP5:[?pfa $e/cS5?2o3od03D;CHHj?>e$h0N_,S4[B4R8WO>;QZc]eH1!uIOC4T1oAOKZhuYmamlp:LNnc.N0ZpLc Square root with complex number question is to find the product multiply the magnitudes and add angles!, CA > > HfsgBmsK=K O5dA # kJ # j:4pXgM '' %:9U! 0CP. graphical! Subtract their arguments and z2 in a way that not only it is the lucky number a real separately... The modulii are divided and the arguments are subtracted to making learning fun for our favorite,... If \ ( i\ ) which we call '' iota '' number another! 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