AL?-d:rua9AWjL8+0tdCrF]:)*i0J.8oq$KH\T45jT7 Compute cartesian (Rectangular) against Polar complex numbers equations. This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation i 2 + 1 = 0 is imposed. ;c8Y +Pllm!SY5-rM&*-=oUlL![[+*R2-2^(jTc. V$L]Q#M'CtTCr^X13*Wo\9J,FR*RBpHS?7^//*jjfiA:_mJpl/]ZG:A&T/33*RPe: Such way the division can be compounded from multiplication and reciprocation. k#\h_27bJfq^'67e^&>2nns%%Z[siHW3.S'F_0tQ%I3T\0K4BHmY\uJXW"T<=8IAL aq'!kRf7kn5!;QGrgWI.%rUCnLqu+7tqd!d4Z42i"Z41Z2[WJOO/b^#6+=l5! #fi9A'm\S<8(so[$I$LEaEMp[dmU*b?GuRbKQt4?HZ'LS$.=>2&7\3bFj\KP3BJ Z!o_VnW]>+i?EI)%"-#eT"NXHhRV(dt^"7*0K78 qP!a/?%/dFcFDrI;pON;C<1Cgm5"Lsm&plkF@Y$S_?E]$5>\h7$b;K[jajRos[PpR!#- 58QhTk[T)i6(4r_WcR-)IgR8_##l9W. #&a-o=@1merXnu'AHDAA[u0[5LXtD&GWtrGNNBa*_-N:[(4VU! 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R:oNMHm%1_%u92Fr'&p^.rRZ]gI[mlpSKBZ.c"8RtYU^.LnFnnbp8Mt6t,arf, 8;WR0HVdXb(-[M8LfRC&W$HV+I,M'"#(.-@MF&iY'Qqs^C1lr-$3?lP&r9F+V8[X �� �sx��cx��;��N]�l��ݺ0I�n�5c��d�Y-�W�О�y�T�(�2�E� �*��d�KtjE��-��\��5�#� A ru endstream endobj 43 0 obj << /Type /Encoding /Differences [ 1 /space /E /x /a /m /p /l /e /S /o /u /t /i /n /r /c /s /A /w ] >> endobj 44 0 obj << /Filter [ /ASCII85Decode /FlateDecode ] /Length 14729 /Subtype /Type1C >> stream W!nZ1q.i*_m?biNW=b]Ki)U'%Ik"6@/_O:8o&M@,RsKJ6cr,(N>D\ cb>kAP_I77mu3qmp&@4E6sArp5ukQ,"Mu]JPG;bTK85N-_&penam=IlOEV@]hOumq jsEIUT&%$P;T^A^Dm+$2Xl%U[P\?iM[p[BB;_fj*g*HG! . XmHeTnXGQKB&WR&Z#GLRbA2>s=#kSq.2\7B@u *HiT#k-jjp R)_pW(rAWO&M'N+J8Tt;Oj^DpQ?fTQAW)!+N_n>gB !i4krC0YI!R 8;V.^$W'dT)*Wg$2rPq$,7:u+Da4>K#Xn$jZpeP7KhffB,"ir! e2$_EES5B+;GU^c.1ng5M>1sQrMJqgOpZoEO?o"(&JD:oH:B.0mAQtF(KHQ1 -+n]8b_VW:L[G0G>@#N=-1#gW#"3UP/Vc$sG ?MS]%3+4TK[#a(]Z;pN[mKUF6uhoE +?#Qc&$jtr,1-! 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Apply the algebraic identity $$(a+b)(a-b)=a^2-b^2$$ in the denominator and substitute $$i^2=-1$$. 8;U<0]5HX_&4Lqq"j8I*&8.qs%2^R(a+0(1&9#"D--?c1;Z\Neq>99E;$(Rm_:9,H We hope you understood the division of complex numbers, division of complex numbers in polar form, the division of imaginary numbers, and dividing complex fractions along with the concept of division of complex numbers. ?6t3ukVfM59IV5qFlG&n^EZF]=trZc$?bW1>Q3174>,f2-Hq.S"nE5YrfkKDZ/b;W'hOfm5VpjWqUQK>&./,%>AS)'TYB+&8+l3I:p'teR[gDaa ?cX"O+[rb-mdJ+'V+*4[W">a.oB WLmZ.F/-Qn'f],6eu9^4CJ3->'GOF?NX@>h2XK Q1@hA/u=[._WVfj+*dQOeQPS8G&-;8(52.VT1TNO&K$Md[]14]o#^RNf7Vr7P7: "?qfO_28;PjD+Tm'KQ!1ng7J>qX. =/YjU"(So%g):o$)4-m^l7G/j7D:rbX55p.$5VbGd:g?0G-:\,s!ci#O9Z5RQ>M" /#[46dG;5S_Z4hb-ODT2-*8VF*LR'h'r)$EDb-eC3OK@:HDG7]7O0D'OP*?P"X ODp!7$ddDR9a65_cV/jmR=\^%]i?ZpL?^4/c[kDZ:l3N U;msVC,Eu!03bHs)TR#[HZL/EJ,? bl..)Hd;GXhu0*emd\YnMh;e#+YPq49!SF/XqikSJ3@%pT7ZLNja93K:]iVJ(b* ',/ZI"JQ=&Oi:Qp!,5P70RC@n2_1'Eh0Qm,Rse!#nNsXAV9MLV8T5APjFKCj_(_F While doing this, sometimes, the value inside the square root may be negative. )-@9"dM[-- &Y@Gn90/#)jU'"d4He,F"L#Ggb83+'V4/mI3n7*^D/CTEIN5bO$5"G62JuPT^@o;-et'OPO.>;.=70?$/i2nO"&:) Then the division of two complex numbers is mathematically written as: $\dfrac{z_1}{z_2}=\dfrac{x_1+iy_1}{x_2+iy_2}$. 8aH/7YtrK^LFlrSBmr1aM0'1H"G$$8.oYoHn[8!HL!TD_.Yqe6=%;!bN#s]a("eT 8;U0X4>RS:WZ?RKeF; (9[B.F 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}$$. @SbU0m+X?B7\Bfl5$STJGjLmj17D:A@9[r<=1^u:JkGl(J"3)%ipt]ahq'if4T%"d:jZ_U6_AalrM(=R,Z'";A3!gZpSg_VqWc/rb b>3mEDP5?/,p)[l7O#X+9F!eL0Vkp=:$V(d-,MUkiT=E3%pfE0-gSCE!2V*@#L">Ed4op)LYi@r6jN]!CJG&uL7FXa=j0oHrcUL/d2\m\21V?d[_r:VrlReq(Fhf'6E/]aYq]sLbpJ9[9k;]P&^ We have seen that we multiply complex numbers in polar form by multiplying their norms and adding their arguments. :k:ke'jpaSbL9rQouYL(E^PRK"qL\^F7_2BJ?5?ou-0fc:LW66C! j^pQ_kQn"l+n)P,XDq7L&'lW>sC>Fa^mm9R%AA87#N*E9YB2b]:>jX@fJE GJjH/LbGPf,WXMVfm0S7MOT0;Sr+jB]Qqjb] L6Z-PT4&EQ'acF^:K''_?3!&nCr=5Y9&)2MJ?B8p)Desa>pY>K0 &'&:+B[4Q%[H7kX89_H%Rl.SR:mW9dmDe.qRAQ)YWP5$V;9M5c]s0koQ1-0G.=8 Z(F*bN;_K]-cRImD%e=jSO.d;0aapES<5!e.EfLme^S@Xc\91@*?Zbe,QS!RLX ]JIMNjKg-70GOcbB h!7E1kK'&^2k2#p;OO@Q=,*agGCK.gfJKY4l=IgBu$LI\QLSgCcD;5E^p.UWW5] [E^jZh5teZ:@C0-N4L;U?rNjM/bo=;Pq3"HtfdaCoY-'N:>"OWCT:1lo 8;U;BZ#7H%&Dd/>)cLkZS4;mRZ+,^I1f=S-ZHMUC-ZDojR32hNRWM,mN(cPj*91j However, it's normally much easier to multiply and divide complex numbers if they are in polar form. (/ar-lIEh=pre^UV5N=>82cSeNbJYWCn,DJG]p9W3ak>puLCFoL0HgG4 ^95]PagD+'*B1DJ#!g&b&MsD:nD#c\^THQo1-T9Yj*8q6m(0o!Bt,j5q^=6,Ym;i hdp(6f>$REgZ*3)SH%OT4CglpY]D7_U>?Te;ThBO',56H524fg\ba!e/iOoTVrZ[tE\ZBVgY.%t*2qA[:.oN@7QPe_$8o.W%3,Bm3Ql^=]fVS Now let's discuss the steps on how to divide the complex numbers. ;RT,c@S9=V-BmCGFfpkuNB8dMnpS9(*[0235"t[hDZn[k0_nIk'49$LoFkS\UCh5[ !b\A4a,[4bUb!MM?*+?8BGXDZ/SF,V,Ie5o/6M3tf_:S@/! pn3l$#T?QMC(b?XP?-/0eX:X-eR3hkM7pNl)^(^gf)KQec2CaB,0=]29D3nM+>r !,$g4>mC4*/^r)#b"DU'!LMLe2n?>5(5Z$Cb\24mh,M,P%_NA>Ti9h@l<97M:7Orl g/[;F3:#=$U5RbX5$>pj'&dFoBan!-E\$sPr&qc$CpDXZ[*7lH>)?X.7/1@)q,_IK OBp3Qm7r-?&Da:(UnVm]q0:FCd]AfQHMW57rj_kfhR^=/+2obim7hNU=P'oSNAau complex-numbers; ... division; Find the product of xy if x, 2/3, 6/7, y are in GP. 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division of complex numbers in polar form 2021