XmHeTnXGQKB&WR&Z#GLRbA2>s=#kSq.2\`7B@u `^95]PagD+'*B1DJ#!g&b&MsD:nD#c\^THQo1-T9Yj*8q6m(0o!Bt,j5q^=6,Ym;i b#Y3()N4)q?B+uKnpcMgBS;i3_i=6sIjqMO-.XaW[5(KC`>'Y_V_L! 00(Y>):TVR;YV_2 98j9JB]Y78,=mHVR*^ok:KokTj0[KS+=^"Egp30eBqng+djBgH.BZjX.S`Q)03\Nu4SV9d0>I!.ld\$:t#3P7MH iD`3M]SnhJMh>^#JTGI=8_ZluUjX?Bl@SaMUQh_9F]44=+-&]NBe4LPM! 8AiG#@2AWiR'g&enk?DZK5r_mPcS9_">'K[0>g(4?M4j-%)u]n]A$a^--SO\Z>dR7 _iull%qfet!1"4F(Q\6UEN14o6s4=eD7i+Nq1[A/IRoX8!bi(.KX#N;R83. q$`dWN(=3hIlYK%HEhRiOC(t$/Lkt)BKWcg"qRp3gkB0LifF"up1b+Ql:U)KZcU2; ;[I>J$GS8Y_%3QFqiX"po(BuA]>lO.Wqo^X#?McTTo:+'f$io/.Z/SY+sgD^B?RTZ T>o+"Gi!DsmFlIteFubM]B^2;bl8hIs+(]bao;5W0*:g'"@&DFR?1:RT>eP&)ZbL$ 7jl:[nZ4\ac'1BJ^sB/4pbY24>7Y'3">)p? 6GbiYI^q.FRaGPcdJ=%&UK292'l*mE*8H(cpqq]\bMgIFm0'G_aSP'IE%;+He-\^b (,\5H:$b*^K-.FW/8Zc*OTD2(ZIHA,l*ZFf+)$`A!r U<5fC0FHeO4W7ag;40`20clbMGuUTrXfm7mC(Zs3as5D`hdrTk3/t[Uj6nn7pOk)k 2Be^3;h5:,[4mAj3Z&5uCEk=Mf4m,*$tWI;SG1S:aU$@?RKU7_EAWsk@! @63pZWp,Z3]:$_^GriT3O_@fV*o1\]!d#a8$O/)s@%tnq(a@5=-5G 8;U;B4`A4\'\rL!DbSX]E$KM1=@`Wh8JB)AQjGlZ8226GL]%%$m7-KY8ah[$N^mZe When two complex numbers are given in polar form it is particularly simple to multiply and divide them. 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": j(IuTp'@S;'9gAT+#orF1jBeKr=9)VRnNP8b:PD.aEMNi[YZq. [P+?> kr&C,hm_\!qkQ=2c@']1AaClMB;K:"E-]pJ\t)J)0q#%hs2qqT%I?+MK>-`'+ =rt?ZLQf679*C#lA/\c=O'4NE/a%cCAf:63p]0nek;[U.pbHoT]\ct#? bUA:E>#3I,0tX.%&e'IbQ:@Q#LOuNC@\6"dd*0[4,,3..6RI8RoU6M0kXT=)6t@W94`VD]ZADNIgH$9s )EnDnlTAg:@fVPV)cUF-*lb$'FNB3PNhF]X\+js+DWIPQIQZ+f_D1.<7)a%584X) %=23[_0&Y`/D\cf2P8b_1O]\"J1i<9@iM>-B\^S`Fa6B8II>dS8][^Okt*C_7+B\Rc,^QPi+U;/k/,8.@n?-GibY_@a4T/>\;kBMOc/5G!E\cONi=_;4c(fa2/J4ND\8Cp[ID?9;n'-D8e)+rFF+tY#q-.O-e9. 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U1uruHu0PRA2(HZa9Ah`!Z4&kP2e**Sc]tYnI6=]^Zm1:6')gSKoG#N4:I!#. WH/0Madol>,42.CRoM,qS8JL^7KsoQ53D".lD]DQ>Wg4c-/$I=#_b0_\e\Z7 0B]]K)rO"*eFTA]16=o=7<8l_:j<3Xo*t?oI%*2^_>Dsi+)^1FA?4?6:ObY2\]>?igurE+Q' Convert the complex number \(z=1+i\sqrt{3}\) in the polar form. Si2#V?K.82$BceO#_2B#"[l>.9n[5V7UstHX#@Y@m+?m`#8s_klD).aG&/ctXgVrB s%? go3)L2Vp^/"FG[!Wu(*C'6n.KH\h;:b4FAMb#aBVJHhi')!j4QKd$V36K(JYkmNWp Ak(""(ru;^(?2&`>-i6[0UjAo6rCPD0>`tFH/h(K% q/U>`=8MpS]@%=S#LrB+YUt3%['HC;^2@][4SeHhf]GrpSj&pS])pVY c/giT>OC:ACARg4r%!7!Mf6b[SFF1i_DmB,"6jo,^uk_>^7-&8r!3Z;m04$A3E]F8*40ok"suF!5&I['!PF54? ]^SF$C@-/aBqj0TXf4Gq=(Bq0Pf`auS5F$@gW&F7m1FEs8o.MY&mG"0[?ld`45I!9 NB07[H8li'1_J6^(hPJU,F=&V"9` Polar form is where a complex number is denoted by the length (otherwise known as the magnitude, absolute value, or modulus) and the angle of its vector (usually denoted by … A complex number is an algebraic extension that is represented in the form a + bi, where a, b is the real number and ‘i’ is imaginary part. R.+]q36[1gR&r(%?qkn$aZHB1R.$. nr1\,GMF:X0UqD\NpXs7VB8,@rGB3fesj"\%)ELEDJ84p8SWTh-Bk:JVm"kAYK,"N ;MfH/@tSNW*41)sCBa%^#@.YPFppro!\Qk^/L-K;Bt( h:[%17W3X!d*+lKaZjXPKbo)dl^4C"h+\;8=e'u867tI:.`fuB,HQj@lFD^ACd$\g @.j6Z[K"&>QX$!RrX/,iq[E?Op5sXb.V1! . o.Y4;]I<4@\fZhl>m+@]-pqIhS@OPhfmA!.Baj7*b7;YaGZ8<=%snonU16.X,.2j_'1&ojVj#@ TNgm^f)\^)!9A?^Ya$u>(9C%u)"T@l1M#469JV[Q!TfH&S;Nk##42)9jQ9h\NNgeM* ?u,51HH?O*=NJd=(A#o)pK-qtZ%4#RfD&Hh]$0.N2J^(2PoJ$`UFr,*aWV oIB72]gF=+qOlq)? @8mE%e.':2$\lP%:m@-+]pY=ZX?90hX6H#G-6[TEp+nD? qdoI6Vj(pLrL\j#Al0e1U+gMW&kKl?Rn$js.Nu%PFSZA#V1gNQa;"FPVGKgGC+DU' ;4'$U-XR7"N0Yd:cs%*gn"k0n:dJ,h#+`2>c2*t9M`V:!_7)[0/sU>,[(Y,Ah97Zt lIgg]!!:Jt=2F2!"nPq+MFnf^W;Z6!\? bu%WoR/FAQj%,ln>2i'1p3V4*? And our distance that we go out from the origin is seven, so we go out one, two, three, four, five, six, seven, so we come out right over there. Check-out the interactive simulations to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page. "a)]_le6g$..$t!Seb'XgcBgk9QX^erah/O[/$$<3=]9u:V? ;X[%,"6TWOK0r_TYZ+K,CA>>HfsgBmsK=K 2(N3'rVV-#O)sabc8h>B6?AdaWTsbhfcFFXU!B>5[C=o_4Dm*efgII9.k5],6LqEc "3(u3AmU9`'gG?D hdp(6f>$REgZ*3)SH%OT4CglpY]D7_U>?Te;ThBO',56H524fg\ba!e/iOoTVrZ[tE\ZBVgY.%t*2qA[`:.oN@7QPe_$8o.W%3,Bm3Ql^=]fVS 6oGdOK,hr6 This video gives the formula for multiplication and division of two complex numbers that are in polar form. @6G5%V7m^ :i!_GZ=ui'&"[G(kZh_LOIm@glK)n9P\8a^U3*9eY:G$.\ceM@Mt6f3iXSMZ>"r?^ L=p66-A;#FY?d/ik@P4M?1OMO*lH#2KtF6OS.a,02bOn+AlEAb_?Z;a8f'Y,0qtq ]cJu%H< 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��� i:kY4SdO)ja)(a9Inf3?>2'p1$'5;R;o3"C First, calculate the conjugate of the complex number that is at the denominator of the fraction. L!.i)!%A3gn[J_"FE.E8L2$mq4:/DeYGRH"m=C>Y7Y+mLe(%$igR&c!j[o*=r>[&P *^pL-eS]M+'io*mUV+]PgNXn=+0flg-K5.kD'=4a3CnuCaCDP$dOVDrVFG@G5q>+V AjD@5t@,nR6U.Da]? "Q?9(=R!l"a6r_:BBF.& )Q>'q(iOJO&5EJqN0SMTD^P1*o(gP0qc!BHEdGj%AmG60d$OK]0+S9eR_*%hOo9Ps BI_@f6I%^e2KIYpn'd*i@cUI]L5pu#Yo0_gB7`^6V"iJ@/K_+mg? AL?-d:rua9AWjL8+0tdCrF]:)*i0J.8oq$KH\T45jT7 >AK>MU1YYHQf#n@nonU[o*2Im]F[B39d/+!Ftq<8UZrbW`:>E=/Ccqd4lXI,k]BCa @qdp!R5r'B=rNQ3s,R.E&2l4h@j[*p]\.F$4M-G:q5m-0doD=psddi$E3B(%b;q([Z7SD#PEis\RLLEW/UZb4>,I&!YJupjDcWn\fQmiKd(OVQ?CEuu'H4q3f p-M)l7A0nj)$AR%rC4bO4XN1%%[sg;H6;W>I5E^u ;" 8;U;BZ#7H%&Dd/>)cLkZS4;mRZ+,^I1f`=S-ZHMUC-ZDojR32hNRWM,mN(cPj*91j (_pKu`S_[&UN%h;^mgE"8#"hqYtXC7VOIu_VX @IUu$lm[ncV=-Z"t:4fZ\2fA@_)9ggP(3l@E#5q`)P_ ?.aB"-mng;\WX#"Wb.&^"$n/!_K;7 [U6.#NH.fK)+FDg,"[VOqa_q/qZ!sZ+:,_3N/(d`J$gcu:$G9dKNOV%'-gBWYr=B&fI9uY]2 =0f?LcHr4-228]b3Z;)0?OA:K%(bP2^E#hFFpcFaRAOHI@VmsR;s:,q JR+ODN5Z'ABX;Ao$CKfe[4e)?IYQM<6efQ&IpG[6(ej+Lki A complex number is a number of the form a + bi, where a and b are real numbers, and i is an indeterminate satisfying i 2 = −1.For example, 2 + 3i is a complex number. (R5[V(Ki@A[9G3tbkO&]k8P:/45XMg@jhW3)JQU*`0fe08\1-SpODo!8CM:,@O06X R2HpW!mbA8R3N`'Nf >hZ);;UIr&"of?oZZH!rNCDf$\9Ms`[NWCtFaaP"/MF/D2_J::n]0MpQ^nr.rcfU< &o*_98XM&RMljs5\*9>6T$oQb72HKa9@,GrOta4-k0"1#b!QZGQYPJNFeVL&*2ZO] >j;qqG'i'[,*gcA4VQTCgtl9Z_>`'rR[^n&TuReu\O2F?W'o[6#?&.Pl!O2$V->:+ \Q98r-3An)a?)r7"`,@trD1Z4`X/9F!jbkD_C+C8Q(#9fgmm2D8%kH^?\_u2_[BK. [%N?\5@Oc"S5),/u^"qlZ&oD`,9k6N"CPo2f`"(6cJS*cdA2d-#VT-ZU\t hlZ;e0KWp-G1-1ISAnCf2#_->/Xg0hUs:Pn;5pV5Xf3VOYplDL^\TV\i@PlWP9CR? ScJ^_ogtJ/Y,g>3;d*^6OEe&q$4i9E1!kY`92(1NoC[]bX ``.Z2DGp;BS=0n_L@o?>08:pQIGf4,lA\$t716H)gMa^*:_H_uc7"\9fh:_;Hp(TI The conjugate of the complex \(z=a+ib\) is \(\overline{z}=a-ib\). )9s2FbUmdQa4^,Eo,P]QE+OX%H[og#P&4h6IM%C d/In8j\$CpUdg8PD!*^&K=`i`&8!,(e99G:;j(H)Lk0:MW-H].? P#/kWPJU8a*(8W2m_P6lcVq02g$f[0QlYm[iRL:TAk^!/?nWG5e*uD8qiV3@&'42M Divide the real part and the imaginary part of the complex number by that real number separately. h/J0s.R8a@J)IW`]dXb DD\;gC2*4GSN'FQ@` O6A%j.$gSI!Bp,SXopLgC@o]cdk,,5o_EXrngZZ^IrBlHEb_B)hFIk?R*HO.8a\uF >j;qqG'i'[,*gcA4VQTCgtl9Z_>`'rR[^n&TuReu\O2F?W'o[6#?&.Pl!O2$V->:+ mUPMXh6oAWXeVc,lcN6Ms'U;kIWG)sbb!T2@Sc.>7(!9tENbX3Q[*CN\$iJF There are two basic forms of complex number notation: polar and rectangular. /.)i+!! 4,&FfN4E+m=iVSX\6bm3Q19`Ob.`"%S0Z,r^/\8o2te%Ij?`H_:q\5i&XS)UP*[)L p(2Tj*@)%>GJo2nFqa;#(2)g>q+S,CR10op`55,D2A_?S(e\D`WH&"+jB14p`VNVF Exercise \(\PageIndex{13}\) *Z!4>B]#l\dj@kg)Gr8AaV !QB@TYkPf_]>7a [C+g7h,LfIF7q!qaO/s6^MNFHUo:e*6@ !1'blG),.\f^F4b17FQAJ%q!gID26e&MmI8V*pj4tUgn1]JNRQp #Z9VeQLDl^ocFKgle;Et! XWX\`^3_JIW)pJf@C2B4PP3V#VIf. .n";Or!Db_Ta#5k7AOkbs+Iih;(%:t/2%8#U8-.#^5p!=mCPe;%(3!8dXrXj(lCfO 8ZO'HLCKiXQtLREg3l= @63pZWp,Z3]:$_^GriT3O_@fV*o1\]!d#a8$O/)s@%tnq(a@5=-5G 8;WR0HVdXb(-[M8LfRC&W$HV+I,M'"#(.-@MF&iY'Qqs^C1lr-$3?lP`&r9F+V8[X p`\fuSue//WZu79\p=g.">.J#,akKle0JbFh@sbKhBjaW_l%^22fLc2h#bD./kfn! 0.b*cFZk(m8,>]^PU-_UP8QHO/3a>51a=L]?gdt^^29?#ZZ"5?Mp)]WD7s`6ZG8,6.7LPuN o"MJC)7%nDaP-`:G!K2[#$h*n"KgGl&re7WQ#'*/5Y/I(`$HZFQQ`IVop["^,IU^> Polar form. The symbol of the complex number is \(z\). %dZ:9c2k=e8bJUA1sonm(k(>U?et%=)M7ERhSJJl*KDc3>#eH,OII`D35l_? )9)cbLGa+F)Ctj>2hI CI7;s[07KBe7ESK86mJc.TrS\8SPG!hGAceAr;t]:fTf8jg#6GicPlN2/M>PD"8Mp ]FFK;KJ,^U7A3_=# ,j-LIrmRXuEm.Bt1Q`1$IY*m9f%W;n\%@nO3k-`GM[cnrL)QqZ#k*tAR@3V\0@TKR m=H"#)b]e[(? n"];+c/ 8;t%>Qoba81Q;I`G"fo6RPIRVQ>`gD$8b\@BAH5*(:h#3@;#(KajFEFqg8(,EHgj1 [$-AK*`3=UHW";4W4Ghd puEMV%"k@Mq25Wm&fkLo.b:rSiq!22##U1=bW##(P];;GpS-_BW8ScDC1r@^V=Y,WR9)(Hp$#NCG,G# r/t+Fc"V"36iWkQ4o]X0tbOA1&[fS44j)"S9L!SI" ces'p:o=#?MVl0BnWsHF@(?ocDuOdrO8[K^-!6iDn?>ShVNbP"R1cU>a4RIY_6;r- :K9\i%CZH:r*8B$3_.Z+,Q_81i3k@Vq)06m9+K)UK?i) =/YjU"(So%g`):o$)4-m^l7G/j7D:rbX55p.$5VbGd:g?0G-:\,s!ci#O9Z5RQ>M" 9u53r55sWk2s4DJ58aMD-CpToZ+2;GT#iD,JGqMWI,Xcg^E6Y!g)`-SqdYWQ]>:Wsf)#>anl-lEO$eT0NuenmrM']jnEh5Xa0U^2^77Y'9+ r++9O00fZ@?jA\8+-4G8j4jP!cK8,4&*W'I:0.PPhDm-SR-M#hU)qUZBIQTMV)l"b ]JIMNjKg-70GOcbB lMj%h0Qhj&Y4%nLYJ+r"AF>Z*S.,EIBWp,.Xm/kVA!s?mk'tTV$Z$L>*LAKnHY#Sc [E#M[)Hk^3rKbT0AK_fsb(QNDF(+0Zr^l@*S(>I_+[?9k3U"Or#CY9/B L!.i)!%A3gn[J_"FE.E8L2$mq4:/DeYGRH"m=C>Y7Y+mLe(%$igR&c!j[o*=r>[&P !B23+dO 5tf`MDkU7trm:Ql>1.XYqD?\!W:34`>LTY=lQHpnH`3%f`n)t(Z%F!/UG$[io$3tr mq4U3>03gA^0#)KirKnhE?i=6ibkS-]oKIopB\3'NhE?lRfA,>`b6q. m3GGj@ak*Wb1DO+/ip$(FUWpmj$B^FT*lIkMNPN\@;p 8S6Ke@/2\!7u@o4CN9IbhMDm`Z)`ELH"I[\4pP#o>UmQhC7/0lDt$O,$/ZRqV;8CiYVW:=]CX9FOW=.rc[INE>c'Q2`G`fp$9>-fQ^qAl4W*Fes6ja ?6t3ukVfM59IV5qFlG&n^EZF]=trZc`$?bW1>Q3174>,f2-Hq.S"nE5YrfkKDZ/b;W'hOfm5VpjWqUQK>&./,%>AS)'TYB+&8+l3I:p'teR[gDaa Divide the two complex numbers. We already know the quadratic formula to solve a quadratic equation. ?H-'Xn>FOthpt`ZIO@j&QWrBQq4EF`1Y67,-*qi@J=-)o4HU_X70*Gu!-.i>N;~> endstream endobj 29 0 obj << /Type /Encoding /Differences [ 1 /angle ] >> endobj 30 0 obj [ /CalRGB << /WhitePoint [ 0.9505 1 1.089 ] /Gamma [ 2.22221 2.22221 2.22221 ] /Matrix [ 0.4124 0.2126 0.0193 0.3576 0.71519 0.1192 0.1805 0.0722 0.9505 ] >> ] endobj 31 0 obj << /Type /Font /Subtype /Type1 /Encoding /WinAnsiEncoding /BaseFont /Helvetica-Bold >> endobj 32 0 obj << /Type /Font /Subtype /Type1 /FirstChar 1 /LastChar 1 /Widths [ 722 ] /Encoding 29 0 R /BaseFont /MSAM10 /FontDescriptor 27 0 R /ToUnicode 33 0 R >> endobj 33 0 obj << /Filter [ /ASCII85Decode /FlateDecode ] /Length 268 >> stream Y(Ib/cAGQMfoAq5>6g-kP>Q+kV4($0&aBpt4AX9G@2iB\-J_j=eWV>Y=mI];k'g?IXEV%ds6tfej%kK83MPaGs*`:8Yfm^fjIh]iGlL\Lu.4PM4BVo l&Cbl(S.J3[ripj1))hLf,$*[QfH_0H->e[:`jW%Na!e[[^/^9`=c&g_0;3`N?#(i $W:j:^:O b0!R8#^<>"b9WZa8Xp>uC^5L'jZt3]''E#-&'qe5"4BVp,V NOReFjuY,>VgD%(2-?sp>5tF8]Xse0ocYrcVV"]s.UAPDNo>)1#46NjFA=mo+p[Ti At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students! 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You da real mvps! 2_$hf-[KZP=nKn)pL6nBB4D$RGJs3qV8kUUhi8dN#YSi,S<6p`5dk(@K(DS*PO? Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in. "5AguOY,Pb+X,h'+X-O;/M6Yg/c7j`"jROJ0TlD4cb'N>KeS9D6g>H. However, it's normally much easier to multiply and divide complex numbers if they are in polar form. eD7A%FTDX9=th&3MInu@#Q2aIY+a=oUgMQ)CcSmh'Vp&\=^s'^.^s4Y2Ur ?6t3ukVfM59IV5qFlG&n^EZF]=trZc`$?bW1>Q3174>,f2-Hq.S"nE5YrfkKDZ/b;W'hOfm5VpjWqUQK>&./,%>AS)'TYB+&8+l3I:p'teR[gDaa ;nWPZ\0fn@90QlTcIYqYLOR5'B` Apply the algebraic identity \((a+b)(a-b)=a^2-b^2\) in the denominator and substitute \(i^2=-1\). NadsK_74^Efm/Go72gR!'tE8B%R_17_+so?.J--J)Yl!?rqGSW3@U$IG'[8Q&]$P? 7"H7k5HB#f%;AmKUdf15*MAu&Cq6AA<>P$jZGq4e3'`$e$\a5,\m What is Complex Number? Division of polar-form complex numbers is also easy: simply divide the polar magnitude of the first complex number by the polar magnitude of the second complex number to arrive at the polar magnitude of the quotient, and subtract the angle of the second complex number from the angle of the first complex number to arrive at the angle of the quotient: ]@7-l_QtO#feI1d8kM-iS+%usrtY78iM.XmBU_L/[geDGO'D)\/3Wf/rn9t6B/42e G'.l7hI,;pNkL1@ab*_'R.1r"O0Ybh@b0*=P8W5D[@jS^ZU-:J96=Bi[h5+=Sc;AR pZ'Oj(k7=Y^B Let's divide the following 2 complex numbers. QE?mBGP-HnV\1INJ13,EPYARV0FdVj=CH(qT#,Rg(A?uN0t3$eZ)WIT0=BY6f<8t&'$6t0f+8`[,L[5MCulmDJf0g\ +?#Qc&$jtr,1-! a0siEKhHLYijF$.=ik37"tHNH0N]he3La6A("q\osg=&$?Hhm@DK!JGhK`UXLJ"j>. )FIg@l(2Q0_HfW_6To8K-Ff*/8T0CYOF=`gXF)5-2em%D'tlp"LL.m]jEao(P$Z24 .@HlPY=2fmaEWhL6T)MU@;1cmi)_VUHN4J(7?edq%^nbY"%nTI'&XIP*gBA. 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[?TZ@I3k27f!Sh0?e!>MM_[!q2^Gbjq3t9$t]uH [?mBOp'"?nO(SBTO.RFMl&`u*8Ve\@HGjX),0-=edqO$bf`R#BW=/m,:EPj;S5Q%O There is a similar method to divide one complex number in polar form by another complex number in polar form. AYH]B8>4FIeW^dbQZ.lW9'*gNX#:^8f. R j θ r x y x + yj The complex number x + yj, where * E]>eLK=++14\H3d+&g@FX8`fEY4o;^&3@oR*EpbZdi@YtQRW-7cmaY.i#pM&E7:?E 5cm`G58!AH4F"6_++YMU_5Pg(T5u[n%:=Oae ?-pK0l6tgN?19`VH2P6#]!3uBcCn&=^BXgb&Bq=6Z:d8b97HeSd*+#hCWaiiR_27< "']u5)h/H=$hN00uP"Y(aT_d9'@u/9e6j5hW%-STAP$gGKRd#d. N9. =>H3EgjBKI#s6Q+2L0M$8I'eh\CnpqlChGFq8,gDL[>%']Ki.EGHVG/X?.#(-;8Z)G=+jF=QDkI\ >tJ4di+"3Yc/OYeCB3naAua1. 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Polar form. ck"ec7rKBZ#uJB(j%$DO6+apO0c5ekE\^>DIh8J5<8*R>FO?%Rm'](rei=X#7;no# b/cQOWG%(iS'AHPG^)Rd^4p*5ct).;/04?.0! ]V=$fe3*!>LVK]dl$d^D_=Oh!llbic$>^I20J##]K%,g TPE"qF],e;:=bhkD-";M=e1qQba>__ti2Y+]#(1U@0BI`ca 8@Uj32`0Xo@gQA7)T)IjXl>2$bne(LD5B@GG1a/^0S`l9djR""4#GC*+# UBNAOmq0LM&XSi(s*XN=&.Jdp=Y[!>"@C=9)bF$hI6jh$u1@aWJ0%HlhP"J:9%PSk2Aj4@]1h/. FGp*Yi-4S8dggR3p]sgQ77&gZ.HpPf3G!0>"$.`/j@i06M@:8Ei_F4-CI98[,^W@N 8;W:*$W%O=(4EZ]!Alba@DFR/B%3J%L`k\1EH_kkpdl'm-<7=dXNaE@^%V(,h)ukn &Y@Gn90/#)jU'"d4He,F"L#Ggb83+'V4/mI3n7*^D/CTEIN5bO$5"G62JuPT^@o;-et'OPO.>;.=70`?$/i2nO"&:) [iKY'b7duG3;isOo)[&Y'g ?gH^1n\BaUZgE9!^$!/3Ql(I?7mI+,tS:kh%GF7I: (FO]m,Pa890b&qdANUjjJH%tWG+hCUm8#s?96O.QXNK*&7m*fgYO+$@f5 dV\Z)6$50%o.6I)bYsLY2q\@eGBaou:rh)53,*8+imto=1UfrJV8kY!S5EKE6Jg"? ^)E-gjf>B<4R()rBn3UE;kLEB)AS-i;iK QC!jO1M@!HLaTl4JZQ/i>#"7`L/(P&X5&E_jA#9@:7>j-iHE_.%5uZcP55? :-Gli1#n4a@UkU`2^]o$[0)I2U3&(p\KZW'3Kh?R2(P E/@ao?(jFF[IdPK&8?@@ZEQ]);rN-4dhb2N'YgS^d7f3WP)?? (&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 %h2ZP*,98]U[K5\F$3]1\!ahXH:BDg&?R!t`Ngqe5_)7VKZ,3eKU5>fCfp`mTSWqO @2tO'2K\eo)n@ W>cn2a-1!E:ZO#=3HYIAB*B$SJhInmiJRCq2q)Y '6WLj@3NHt1-&?Giejc'Cq^lR-h_Ch)iV.tMUI!c3n$t1DKY?=`Wn%'*rkJHiA_hCQ? :;&g$uV 9V.k]P&*p;-''WO>e#-Sg(u5=Y\pY[%8k1e!S?@;9);Y,/+JV4E]0CD)/R>m_OEB.Q]! K4>jdZ6sT4muNA/F^jA+(`$dO*l.`9$Coir)ucFqG^MLM-LlI1],qDu$a3E&?`+bT Qh#Xcep0@/jld5!%XlmUOtUT[!>`VLQb$qd9=XH!f[Od^;jE'i=h)*qX&2.ub@u5CNnacT^qj=l99^n; -+n]8b_VW:L[G0G>@#N=-1#gW#"3UP/Vc$sG [!+%1o=mm?#8d7b#"bbEN&8F?h0a4%ob[BIsLK $1 per month helps!! UP"n0c`tr;SYJCjck=mH^T23J"3`92F&kotNGsftd^^U@2 "p:lh0iqhnf8Xn8+B!a)lG_XrcG13_P^>P In words: When dividing two complex numbers in polar form the modulii are divided and the arguments are subtracted. (MG*[X82:['fQ?Kf=K\o^-(Z'bl#iY8!^G;::u Z>:tKkns"U!TUC/P[RA. `bKeDlQ]NhCpi!M3ig6V620Qp12O%5cX%f1pbN=bK[e_&qZ_,PgP>b@\!#Sh^Dq_` '^m@V\">948? )%_UV)7ShsNc+O#M3hc*a*Z7*#rt>9$\(Z7RJW:I;9ckM!G^[?2Gl '"h!nl@PAj_`=e$SkK-V[),NkmTk9FAoi_=@T>shUY )iDD?VI9lA"6OBN@r.C;Ir8ip:CnlcE"IY%tas4o*3Ql1Zb7QVV26mu?h >EI`?ab>LR:rM(@r2?Jo01MI;X&--E:T?p%'[?>RraM\V1HOUjDO-WE0u,!5B%9PEX/MNaFK3_eae;S0d)DMB4H\ 5TSBO3Gqd`;(G]sfY:dDVp/5B k`,:ikk-t-R-)+EnBo](eP-Kb'#! Q_ZPd?2Wtk>$Xjr"D,/,E^P,c2X@.+.GRcNP $LiAYpI=Mh^Hdhh8#%]-lSs!<3Gj_&t,q!a/4:0>V&]ZXDFq&(!*o@V? He says "It is the resultant complex number by dividing \(3+4i\) by \(4-3i\).". L#%!bSu?PX20h::^(5Bmh68qE[9du%GJ&Ua;LLBK-aET=gd)DFTt2Ua09N#1D(@d] YuFpJ[&oeXjl$U,_A^&^?$XraB09^/452+Fk"%PFm@A:t8Z&nhN\Qf"1TZEaEEQPE However, if the complex numbers are given in rectangular form, you should consider performing the division directly using the complex conjugate method as shown above, since there are additional steps if you convert the numbers to polar form before dividing them. >Bte+WC;`52dshh[G9>Yk=7$G4D7Dum0ZRm:;^4l2plZ?4HZ"Xm+`44jl=&B1+Q_q ^)E-gjf>B<4R()rBn3UE;kLEB)AS-i;iK And if we wanted to now write this in polar form, we of course could. ... As in multiplication the relation above confirms the corresponding property of division of complex numbers. >uMN/a%12MVEO4Dhqi\SYl;pfE#PM2-uM6EYd*h2'6Rd7=Zd!`B!%Q>X0Er6oM`*g @n4@]P&[IJdZQ'?TQu>J1%E382n)u9d5)$#6rNVlFh6\G]/a4 c2? m(>amkPROIT$KO-N7p9bSB^kJaM'PlOmN)aA8bBQ\!On]-B++]rM6W`p]n)Ta#3,Q R]B4keX;#'=`3U(D/*5rRrIn0CT03rDJJ3!p]%jjgZXlCYKo71Me-*?^rTDi;#rXe The graphical representation of the complex number \(a+ib\) is shown in the graph below. 0.MV;+c^6:D7k[4^ZjI#UV@MNr rRcj?bcBTeXAiu`;tc%>5! *F DBut`+&tq*"SVK+^B9U-7eG`+(WktbT"fGsreE;l/6k*f7e`$tbi7hbpnH:d:7j]K b>3mEDP5?/,p)[l7O#X+9F!eL0`Vkp=:$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&^ 1j/3^:OnWsJ'10h/tX*'QP;C$D$NeV)pG7g)0;2;CO*\E.r&kBi18G_M5eFI`-Kki j-=_DWL)_CGXB_4'V+HBgsKSmV[L,m<>?chA^,+4RLSg1@2V(E>_To+!MjWmeq Nd ( Hdlm_ F1WTaT8udr ` RIJ, ; rlK3 '' 3RIL\EeP=V ( u7 MiG: @.... 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Which we call this the polar form of the subtraction of complex numbers in.

division of complex numbers in polar form 2021