? >EI`?ab>LR:rM(@r2?Jo01MI;X&--E:T?p%'[?>RraM\V1HOUjDO-WE0u,!5B%9PEX/MNaFK3_eae;S0d)DMB4H\ P#4e),/Fl=TOplXHE>`]P&obDm?SF+e'"qADcM3cp!m+J9a8m;(/id]9P!2>K_V>G 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 \(z_{2}=r_2\left(\cos\theta_2+i\sin\theta_2\right)\). 8;U<3Ir#e])9:V^^ANL,L&jAID. oZ\j^_Q(bDA?ghCN]u[:$iJX#pTbsG-5Uu_'knIb98cB>23ZR*9sGSK?A(^1`\J9> Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in. 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To better understand the product of complex numbers, we first investigate the trigonometric (or polar) form of a complex number. :eu81DQtY_&$oFb:nihtg7i1J_9hH^MpBr26 hC)(-b^N2Z)9;64RQN)j8D88,Ep4%6$;truSLLG3T26C*Xo@YP9LYCQA"B9\L>)KS Addition and Subtraction of complex Numbers, In this mini-lesson, we will learn about the. 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[%N?\5@Oc"S5),/u^"qlZ&oD`,9k6N"CPo2f`"(6cJS*cdA2d-#VT-ZU\t cdPW/_EL7jh@hqKYtln;+FKg8s2EhS"BhekBB%4m2,"`fTf#j"dVe$E#_>ikW7+CS ?6t3ukVfM59IV5qFlG&n^EZF]=trZc`$?bW1>Q3174>,f2-Hq.S"nE5YrfkKDZ/b;W'hOfm5VpjWqUQK>&./,%>AS)'TYB+&8+l3I:p'teR[gDaa +'23D5J^qKcE=Ma)eO.6:A-KE\KAoeD(1#H3]a#g/F6eHS"jfYFgQ]P\2aZJSDd`R -+n]8b_VW:L[G0G>@#N=-1#gW#"3UP/Vc$sG Up-5Z\6\%o#=m[[`'5$r`-/ h/J0s.R8a@J)IW`]dXb O,dMG)lmSi1Emi? ">HEr+!p?mAS;p7.3@"]0d5rSp`\UEU42^_pHYe`U3\"@NUc+/BCjQ72K&k07QVt aI3>O82c-5@P4e1lJlg]?Ae!DP4:NZ@'t9&9MJmanE_k5(j#&=Z_)_k 7BF[#]UDS1k",G.%J@NR]>s?VHgWqeDKlPT_cRN'i%>2IBRFJ1)N0*/*1VL8Pk,TU C%^'4[[lg,@jRYbN"ue6`p?FMQg,GqSf`@09!K$/iDHr)=GL$.1M\2+[oYKe>@83s h!7E1kK'&^2k2#p;OO@Q=,*`agGCK.g`fJKY4l=IgBu$LI\QLSgCcD;5E^p.UWW5] \[\begin{align}\dfrac{3+4i}{8-2i}&=\dfrac{3+4i}{8-2i}\times\dfrac{8+2i}{8+2i}\\&=\dfrac{24+6i+32i+8i^2}{64+16i-16i-4i^2}\\&=\dfrac{16+38i}{68}\\&=\dfrac{4}{17}+\dfrac{19}{34}i\end{align}\]. W'YLRJ_g#OUbGVCNZeWE.#Dq1BaQSTCN)tXM=4)>Q>B^0DQUfQ=S1: [$-AK*`3=UHW";4W4Ghd 8AiG#@2AWiR'g&enk?DZK5r_mPcS9_">'K[0>g(4?M4j-%)u]n]A$a^--SO\Z>dR7 MM/VB3pKif#hHd.eF2F<08W/9\^:h@tIJ9'`naNrr>bX$ldn5)G`P+KWf?/X5W 'M)?-MWba**j+aaGgKs.N2*,f=an\'lBrUFYruU[O81U#jSnS\^Yf!=J"PWlB^R1# D[,0K&:O*VO7D'B(UBMVl.IFgn+G:u4.I8nr;_n_f2pISXD:>PUR&g"F^7[7$*sLNMfC1ni',fKQ@GV0eK-qQs-SO4+89:%k5i:\ Multiplying and Dividing Complex Numbers in Polar Form Complex numbers in polar form are especially easy to multiply and divide. i:kY4SdO)ja)(a9Inf3?>2'p1$'5;R;o3"C ;FX*XN#Fh SK0K\=RtTTQ\Df;='dq9mOHF7OnZ^""ZgF?Mtmuj:k9a"LtVB?n[9tlEgcjl>//K^ ``I'bhAiumGaGbLlTt]!Y5VlrPL3UiTrrr+)m!Im%>3U*LNJP>A:e*smG=@5gVX)h (MG*[X82:['fQ?Kf=K\o^-(Z'bl#iY8!^G;::u !b\A4a,[4bUb!MM?*+?8BGXDZ/SF,V,Ie5o/6M3tf_:S@/! If \(z_1=x_1+iy_1\) and \(z_2=x_2+iy_2\) are the two complex numbers. >j;qqG'i'[,*gcA4VQTCgtl9Z_>`'rR[^n&TuReu\O2F?W'o[6#?&.Pl!O2$V->:+ ]:H6[@3&qr[AIb-hH"Z,:%o_L1gHm@(UrSaqC?Qf _'5jGO'lG3R9Nr?\-E\$ON@roL14]G:3? 2[;,)20LVEVdh5$pd8dp@Of)T2WJ(`]#e3MVZcIY "V1BjlG,$C_4W)!`ipnW5`>6WOjQQY'd`,0SQZ1W5^k1e8\4`%7q-PN+]$/F;Pbe* cdh2k*hj#W`g@I*-APALr/68PLOF7PT!B6?eQESSmNA ?IjNLC)^Q/J. Get the free "Convert Complex Numbers to Polar Form" widget for your website, blog, Wordpress, Blogger, or iGoogle. * 3gu&:4;43LX4;C#MhliLPD`kkM"3G8fkpNaA`l$8$1!KK\Zs67<=]T]@3(VMi!apk a^Tf@FUMq!\qXJG@2a&\iRM%\(QrL]Rh/Bt9o5FiQ4US9XEH0Ad=0,#n6NK!ZS%ln cj(U=\CN$kg5:TUB)@#W^<0f9UOiYk*X"B($VS^r(4.5a%+EoEr91ujq!kbm7oEJ>MuRhg+;:NH0OPmVK%!pZlP_D 6Q#jh;gt0e;lW?QB@Ik/)9>Ze*?&F1W9])!5+Z$i^!ue54e^]qM>mX`(P=sASL'E) ;PcId\WCZM?Ub4C"11HKf7+AK`@5sYph3uD829=Rg"otuXf#)*ciKHn%jW3).7rGL l)+lK$6_`f]5FSr.Gq2U*d!%E@39qrb$NbFQuduOj>)ik+*Q_'VR: qP!a/?%/dFcFDrI;pON;C<1Cgm5"Lsm&plkF@Y$S_?E]$5>\h7$b;K[jajRos[PpR!#- !W[Z&RgWSMj,Ni@oOZ40PI-TV]e]..i)LYuMtKOERI2Y9Iil[T ?mQ&Q@8tj. *fn3Q'kJfADURnaCJ4=HZ-ioXTs)#%o*b*!9BWJh9`"m6cM@L[CCQXmrG:B C1^JE\U62Gbg&*.1)cr]j`$D_KsV(WN-Q^, bK$^7jKAh[`%\%]mF3"MI7b[bV^O/[Y/.p;3G4rP5:[?pfa @V7!hcu/,&T:h^)kC9c]3@Q6l/Y8U(mPb&s,A9Mc, ?.aB"-mng;\WX#"Wb.&^"$n/!_K;7 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. 8;t%>Qoba81Q;I`G"fo6RPIRVQ>`gD$8b\@BAH5*(:h#3@;#(KajFEFqg8(,EHgj1 `^9E"2(>Yal57d2[[NfKnO0$Boc]+\AVo9Cm6Rr%UO7,d;qb35LML] kLQQul2t1;Uor9Ml]8,LZ<2$E)cO]nm']&iMkiSc9mc_VZ<0PBZ8dJ"_sXa=9O4ba 8;W"!HW3p6*hFP9-6V9K,/_9LmV_9 hlZ;e0KWp-G1-1ISAnCf2#_->/Xg0hUs:Pn;5pV5Xf3VOYplDL^\TV\i@PlWP9CR? (&l.V"GdT?Ilam/EXbH%\10-@BhS/`WC`*>Ydg?c^u\:r-2uA1$2Nfeui7\4#AiR,lVO[HJEmtJpr>$6cKb3j"cmF)4&JU`=mF"YYWG]%aQXSiHb4o MBW4p;jnWk:OTn83KAu :) https://www.patreon.com/patrickjmt !! '6WLj@3NHt1-&?Giejc'Cq^lR-h_Ch)iV.tMUI!c3n$t1DKY?=`Wn%'*rkJHiA_hCQ? )FIg@l(2Q0_HfW_6To8K-Ff*/8T0CYOF=`gXF)5-2em%D'tlp"LL.m]jEao(P$Z24 o%)3h1.M8=6XGu@9bje\C4>d6aLj1Hc5qIJ#b=))o%4-Bl:=C-%4QS:b"Wtb\bmlL R j θ r x y x + yj The complex number x + yj, where O5dA#kJ#j:4pXgM"%:9U!0CP.? bA,5VYH#nsM66SD\[-'#7p^skV@&YjjpQK&*B*IOn0^n7]RlK5d?KT;l'uq#EB;bR Contact. Our aim in this section is to write complex numbers in terms of a distance from the origin and a direction (or angle) from the positive horizontal axis. .CNI`jN+l`!h_e2'KcD\aAQi>"'! cmVM0-jnl$92hmKb=WKqdO]O7U1>2C[2r_"-WjIQc%i"#$e?DNqgJbhNl(bNd+/:. lIgg]!!:Jt=2F2!"nPq+MFnf^W;Z6!\? *^pL-eS]M+'io*mUV+]PgNXn=+0flg-K5.kD'=4a3CnuCaCDP$dOVDrVFG@G5q>+V o0DB.T[T(,T!n>KjMDAY/k'9nLW?Dj>cO9Z$fX8;Y=OGn#` LAN]m?YQT?pc6!/@TmXRZ$\^pb_5;QRZ>&n#nkCW694a;Obn+2/04VOK22iM:C>%V^C+FGnF?9R&=5C: !_a)3kKs&(D.]? ?cX"O+[rb-mdJ+'V+*4[W">a.oB n"];+c/ (F-.apS@O.a/:GI` e`Zs,s0%KC.&[gmPVuB6bsdJShbo&ff*J=c:stQ9$u;KK/E)g1:U6B^l5@)=?73Q"nTb(t8XS&pFILT-ODh#GV0EA73B)?q $0XPZrL^mYI^frEU!muR;]%RuDk;Zlcb[3qE;2P;?%2:;S1Tp`-HPhr,p]XC!l8gIk'HBu8cbf-CY1@4gi` nua-?N@&FpI(tdm1!t6Hms4HC%h39sCotd%`l=U4G7Lk$@G3m'W=b8D)L5Xg@\'gRkY= aO09no(A5siqC;],%>IrB.P@rVL+ePK+.q_ZA3"7@^H-[3b4o1\R\B/V\[76"\Mt% So the root of negative number √-n can be solved as √-1 * n = √ n i, where n is a positive real number. iZ*N%0R&o11q/?Yq^34:aU3j$)iV4V[d*S<=L(@*i`2)P9'l*r)USck3FV^0['d>3 MrkCGj261g9d_PsU_O*+r5o@HO>qoQFQQqBb k!N74I endstream endobj 16 0 obj << /Type /Font /Subtype /Type1 /FirstChar 1 /LastChar 42 /Widths [ 326 1006 544 435 544 381 707 490 435 816 544 272 517 544 544 381 386 490 490 272 517 299 517 544 272 707 762 381 762 381 734 272 353 490 490 490 544 490 490 490 490 490 ] /Encoding 24 0 R /BaseFont /CMR12 /FontDescriptor 23 0 R /ToUnicode 22 0 R >> endobj 17 0 obj << /Filter [ /ASCII85Decode /FlateDecode ] /Length 299 >> stream ?-KKfViXX :p`gXIsSaTY5m^\`l h!7E1kK'&^2k2#p;OO@Q=,*`agGCK.g`fJKY4l=IgBu$LI\QLSgCcD;5E^p.UWW5] !sNbgLAF"$Bn1oK55Ms-6:DAfQ82'>oQL8j"l"-0+nu-\j%$=/WBmFVY+P!IA6i 9u53r55sWk2s4DJ58aMD-CpToZ+2;GT#iD,JGqMWI,Xcg^E6Y!g)`-SqdYWQ]>:Wsf)#>anl-lEO$eT0NuenmrM']jnEh5Xa0U^2^77Y'9+ o0DB.T[T(,T!n>KjMDAY/k'9nLW?Dj>cO9Z$fX8;Y=OGn#` m(>amkPROIT$KO-N7p9bSB^kJaM'PlOmN)aA8bBQ\!On]-B++]rM6W`p]n)Ta#3,Q Quotients of Complex Numbers in Polar Form. /_'/PgqZI3g<1D[\'s$0ihr:FP3 The horizontal axis is the real axis and the vertical axis is the imaginary axis. . Here, \(\theta=\theta_1-\theta_2\) and \(r=\dfrac{r_1}{r_2}\). (_pKu`S_[&UN%h;^mgE"8#"hqYtXC7VOIu_VX 8;V.^$W'dT)*Wg$2rPq$,7:u+Da4>K#Xn$jZpeP7KhffB,"ir! Polar form is where a complex number is denoted by the length (otherwise known as the magnitude, absolute value, or modulus) and the angle of its vector (usually denoted by … aY",ZZ!6a)^CVBGK)5"N\-cS@5`*/P>VMPk.1j3F;WMm\GP6)a"B=#&;K2HMCqlGVMYrsma p-M)l7A0nj)$AR%rC4bO4XN1%%[sg;H6;W>I5E^u @gFo;=F2W[-$ch`[7:ZKWh+q?/sehts%]`M%R[S[6^!:+D@jJI5aD!Lhd[dau(:T=Q^c"u3N1eo9F]jJaZQ[BrD/;6OS? G@j0qQ8>&m*'9Z@$re[G;/iI!=8.Md?lC)-W]J]H/9Fo1C04!o5(*,$\]s+*CQLa? L]]`p@Xuae@3"+A)W?Fa-'/9IX2DQ>9&]sEM$og)n3@N'E*$[EII__]72=&M! 5tf`MDkU7trm:Ql>1.XYqD?\!W:34`>LTY=lQHpnH`3%f`n)t(Z%F!/UG$[io$3tr ;c8Y Zoaf!9. !cV8-t>BbX:SJ`"uF-< .rVk8PX2S^beouYY.0-%U=(4S%CG0sDYp#dem2qMIB)rjMV^$a=q? 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HqG:P::G0T3nn1X)^0\aeQ>GPj_l\"cJ2S7Xq\t9o=DSRWPBTNAJ>r_09%A3g/:gbH=66u4G,.,D?AXqr.E+rbdWk"fM,rcr Multiplication and … !)O"f+TNeg5lR:W2/icc5ogZW9ZT52F#kt1&:El8-_)g%6LCS?M2'! ;5\D/of;Ddpg0LP'jR0+(0'HfHRjB';$KYP-L]l"h@qVR$G'Eg0&R?fMG3n;,]KqhnfGg\\\M ZFeZA[J)`VenT?3FrdQi(16`7? :X&`!&t"`)Z]&h?on>s!4`*N]RUWF8,rts-jCet!n%'& d_u<=jK*J^gtC?DoO])r`6VPi3Ai`DD,M#bK`tU>fVXM_h1p&III9E8dUY,b)+dZ$ (&l.V"GdT?Ilam/EXbH%\10-@BhS/`WC`*>Ydg?c^u\:r-2uA1$2Nfeui7\4#AiR,lVO[HJEmtJpr>$6cKb3j"cmF)4&JU`=mF"YYWG]%aQXSiHb4o 8;U<0]5HX_&4Lqq"j8I*&8.qs%2^R(a+0(1&9#"D--?c1;Z\Neq>99E;$(Rm_:9,H Ag7uKYVbGa+7`j.b(1Nh_o4;KQI;K!d'!_^%]. hi]PF2:rb4inAp"hCqMLJO%pe]7>[&G%`&+FCnQF1]b)Q(L.9q1jCB=ZG1(=%HuS+K_+H:=IiV#[KLS*[rYNeS ceJA=/BqUI\gV\o]#2P2&jg/[erDD;RU)k52j2ol=$)r<5V8OGGn8WV"X!2ech>3< 7_?-iFDkG. k&f1$8A7-PWZ.97$4@o#JesYZYqTIX`n ?/X@,EX'rPj+Q[9U^E1k@#!HQSZ85+W 8�Fޗ;��B��}���Q�'Jr�Qv�q�l��o��8��q/��t�u|��'{���$1s\�dk::-�����$h~m O�L�� �V�(�m�8�e�� ^EGO/?tB_WM.ME)/B+!s`CieFl=@Q,)9HV[]hkB'FLJt7`Z@)IHo.nefZCtdI_WXo @.UfqM.4Q#,$Iuu/+nV.CN#6M`.=JmOcm)9*BQs:D>Ws*3ZSOdBs25"]SXL!d+nj+ 8;U;B]+2\3%,C^p*^L3K3`fV0;B[*UJA`9;[u*SEa@up=Sts$;?q^4hc=`'H=Z9jn hlZ;e0KWp-G1-1ISAnCf2#_->/Xg0hUs:Pn;5pV5Xf3VOYplDL^\TV\i@PlWP9CR? *uV&6bt.tlMc4[, *'i2uM=Q`08)Y_`T`UHmrr>?loR(`n(./'d%KC]aeYn:$ iqVIb0H,lTK`ifWk[A$b$q;lPM#4AdR&WOFe8Y^&L:(jO? nnctpY.CNmOZ2s`S=qSmNqdEqK2QQdf:rf/2b[DdWnp*L]r$YR:gVN@et#P",k^3I *il1 X8lBM#"W1G.%;B^M]W`#)ZKOWUA6B_l:hRcQ`Z@W)*rQVBgR$N"?! J! D=-Z.E8I!kCSug\r?S>;,k?B:%(Q>ise=%4/4&UPME'3C6R$'q>9mWan>\f#o>pHptCQD Bh=`R&]"soF:]Z'U@@b6Ia>fgdoLQ(0GbR2O`MZ^iA[2Un@eR%G,eU$_bGsnf7f$t %W5.VA4eSBr,'(tSg(c"hfnGhH/ghr2rYYL(810V;LhinI?V`eH''IWW;!gGjq^%g ]kNRS#fe#67.4ph4Q,[^h4Q3-"=CG49j3h'4NJ3c3kI:iBbKE9X_UZ GBCWpdFII&q@]oXpP-'5TSJruN#%Bf]R>W'h`RGSVESbP.kb>M,o4K'Y,OH;;TP*( ]kY%tGJ3/P$@bpga T+IA^b7lC[Kn*iTA%=nS9IC,#SEJZVEo&Cb@EunR`Dl,tX_,O_17Lub`GDq3MH./YT.i2$m)*;]6;)5P@;!a>.RFq;@$"gG^kY$k:qG]""$? Jolly asked Emma to express the complex number \(\dfrac{5+\sqrt{2}i}{1-\sqrt{2}i}\) in the form of \(a+ib\). &Y@Gn90/#)jU'"d4He,F"L#Ggb83+'V4/mI3n7*^D/CTEIN5bO$5"G62JuPT^@o;-et'OPO.>;.=70`?$/i2nO"&:) ]mKl-l3t@4 The modulus of the complex number \(z=a+ib\) is \(|z|=\sqrt{a^2+b^2}\). mQS/B;UO"CA,WZn%E6(1M+pNsNOC1f8!J\#pFqTthgAL>CcR/^U.WLEi`GK1ebj \(\therefore\) The polar form is \(z=2\left(\cos\left(\dfrac{\pi}{3}\right)+i\sin\left(\dfrac{\pi}{3}\right)\right)\). While adding and subtracting the complex numbers, group the real part and the imaginary parts together. \(\dfrac{5+\sqrt{2}i}{1-\sqrt{2}i}=1+2\sqrt{2}i\), \(\dfrac{3+4i}{8-2i}=\dfrac{4}{17}+\dfrac{19}{34}i\). NB07[H8li'1_J6^(hPJU,F=&V"9` lno=,quG.&I:BT@dGTg@j"\9VG!qVJLEIHZZ#Yq=>ns(Ihu_V8TffY1'[Z'Zl#lM `i*k?qRt"#Zr%A7rQuCjXkkBf7=c"3"[NJ^"ANG0\FDN@U6(!DY:ofEaJXe;T"9nX Experiment with the simulation given below to divide two complex numbers by changing the sliders for \(a, b, c\) and \(d\). )-@9"dM[-- 6GbiYI^q.FRaGPcdJ=%&UK292'l*mE*8H(cpqq]\bMgIFm0'G_aSP'IE%;+He-\^b N^>r,[,;EVMi^79)CFIS"Q+bdpBEiB_Ki;r:Uo8B$_N=ndWdNhg`^Q\'k[tDpS4IB2?F%Zgp&q! 0.b*cFZk(m8,>]^PU-_UP8QHO/3a>51a=L]?gdt^^29?#ZZ"5?Mp)]WD7s`6ZG8,6.7LPuN *5<5N4;u*FU/LoL-tO99P(@[rWV)[5b>qd-L7_"tN(@l# 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 The division of complex numbers in polar form is calculated as: \[\begin{aligned}\dfrac{z_1}{z_2}&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)}{r_2\left(\cos\theta_2+i\sin\theta_2\right)}\\&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)}{r_2\left(\cos\theta_2+i\sin\theta_2\right)}\left(\dfrac{\cos\theta_2-i\sin\theta_2}{\cos\theta_2-i\sin\theta_2}\right)\\&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)\left(\cos\theta_2-i\sin\theta_2\right)}{r_2\left(\cos^2\theta_2-(i)^2\sin^2\theta_2\right)}\\&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)\left(\cos\theta_2-i\sin\theta_2\right)}{r_2(\cos^2\theta_2+\sin^2\theta_2)}\\&=\frac{r_1}{r_2}\left[\cos(\theta_1-\theta_2)+i\sin(\theta_1-\theta_2)\right]\\&=r\left(\cos\theta+i\sin\theta\right)\end{aligned}\]. mlHs'jJ%A'MT[(g2VQ$mYapm%h hn_9TNY0Z*dh6pBld.Ps-'tKu-.7D/AmJ)\0ArHm@-igSfa/S(PBXS41pjRc"BW1M :;&g$uV gTjW6'3ET3HhoWjo54t!d+;i1>ePf=ZQJh-9oj^$,#-le#^Zf96SG,$V<8i7:[ELI 8;U<0]5HX_&4Lqq"j8I*&8.qs%2^R(a+0(1&9#"D--?c1;Z\Neq>99E;$(Rm_:9,H RUEjl_^^WO/p&dNbg_G2@4n`A[n)i[aO7CmF"3F)9'V+=,&>8E3I"Y+KjJ,I2l7O) ?cX"O+[rb-mdJ+'V+*4[W">a.oB (mX'+G7V/Pt4un*PG)e()+;oePX;rbI;g> 8;U;B]+2\3%,C^p*^L3K3`fV0;B[*UJA`9;[u*SEa@up=Sts$;?q^4hc=`'H=Z9jn 8;X-C#uhW%*M@.B-#D?3IekA//+^c+]p-.K2'?t@Ip^)Ys2V)-=E6D+]aV'A$+Bl 6)T;e#CT+baTh=ebdV4kT;@o4(q_]X0j?Ef1AcZ>RV]=35sAFh$s=6a.5W?XK*n9/ jsEIUT&%$P;T^A^Dm+$2Xl%U[P\?iM[p[BB;_fj*g*HG! The parameters \(r\) and \(\theta\) are the parameters of the polar form. JodgB /#[46dG;5S_Z4hb-ODT2-*8VF*LR'h`'r)$EDb-eC3OK@:HDG$$7]7O0D'OP*?P"X E/@ao?(jFF[IdPK&8?@@ZEQ]);rN-4dhb2N'YgS^d7f3WP)?? M_e:/R/)/C`jcZi#/RA]_LW$@Y '+jq)Njim*StCQh/6haCrqfW ".rqqhZZR @.j6Z[K"&>QX$!RrX/,iq[E?Op5sXb.V1! '^m@V\">948? RQik-O#(k>S9'2+chD&Y;6*&>unlc(/3!//Tq;bbI:;a&H$A#&0m,?#UikOFTr5*WiV"3d/-B#Rb BS]`75? gt[Rq:u3i/e--!4TVAiS]%tXk4$-3XD*)lTrF:&KgW-lanRfhTM9u78F(aq=k+)%6 0'3ph^Sg+e$.`KrXV;+1^I)eag<7%9f5jS0\2\A-'J6uWW`$kAOT[9WGCFQ"rRSEH^Dr)r$>a``;bG0.:b(em!g$ftVnh;$==LWWB9k/HWt=MHJj. #G(QIUMd7;kFLtEDd5Ye&u9.Np>5%,IdFHA(j11RF?Yrs:-pd^ZP9B\H^>-B6 .%V-c&pLEVO'j!+`'D"S!b3Wg"`#B5MQbYoZ#'P^)hoVRM*qlpT4$ann#@UbMU^R^ T#bMi9No%Ue/ElV5&T#%3Op=)<8+KkkLTG)3 '52rA1gV%4S9p )FIg@l(2Q0_HfW_6To8K-Ff*/8T0CYOF=`gXF)5-2em%D'tlp"LL.m]jEao(P$Z24 Figure 1.18 Division of the complex numbers z1/z2. A_S^D['V:^_.9d"AkM-Mj&:o_ ^)E-gjf>B<4R()rBn3UE;kLEB)AS-i;iK (\M;>`2i[^SA@rcT =]_HRlIKt$c$np$hMAad]'ek/cJ5s[I,FbfpH--2mQ&%'lu[KuP'L_E"QI0;mb\>2 But in polar form, the complex numbers are represented as the combination of modulus and argument. (FO]m,Pa890b&qdANUjjJH%tWG+hCUm8#s?96O.QXNK*&7m*fgYO+$@f5 p4nu\:c;Kt!XS[\:o55qP&l1`#+Wlo-4E3uPkZj0@f5!+"1de%+R0]k4U*i%'3c2. Ak(""(ru;^(?2&`>-i6[0UjAo6rCPD0>`tFH/h(K% %PDF-1.2 %���� H��UKSA.��X�‚��9�2���\-��*�����E��|{� Ku'57VoE?7KCBUP#5cbl"dYPng$*[GgZ+`,o(N\9U%(4I,C5WuHMfB_"?? Om,a(2FB&k`5?ROSm*:/qEcUaJEN6Wi?Z"#,gqrcR1'qRbOm+h)_,fI...J_2kqin 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. e%Z(oCSM-rTTJ:GN!g:dO2pB1pq'a-C_@=K]t!cfCt\9T_,PY-F30:c/!d'omG+#> :;&g$uV I?=5$Z9`<8[d#HA]U! c*[3,1>@-bVbI2Ke"kq3[$"oL&Umbcc"S-`ArGJ;W`4j62.`ieI;VT.0g&r[s4p%FQ3DL,AU2N mRY*IM7nP=)D\2_6M)Z,'>+8#W)Zj? UIo#s"ah4KT3hGXVd 3.5=6Na`LVndHF\M6`N>,YGttF$F6Jjk\734TW2XpK0L)C&a:FkKJ%_r_E[&=CO4W#6mgQ2T1+l.I3ZLaY!^Pm3#? l&Cbl(S.J3[ripj1))hLf,$*[QfH_0H->e[:`jW%Na!e[[^/^9`=c&g_0;3`N?#(i ph*p*_r>12?>E? 8!Z!$6ip1KK0+oid"]ln1rFCEZhQQ6FB'_h)'s>]eFi#M[Q[0U/J*FJ_V,n1?$VU5 \Y55)SsCJOlCYeSfEg*WAcmenN:I"Z7OTaZgLJS%-_1#MhB!EInlV=t)7\P-9LgO_ ?JS2(/b%?BDj=.&aVSL/Z\TB0I;A$=4&@t_BTN#!qm<0h`:"uK>EZo!1Ws32%CXTahjLZ1 E\fZd8dF#_2Q!e9E`_jhujoBp8kmls-oKaBXgq5E8?1Xo32cJ@TpuLU[s^ i,mp:a,ft.lK)Tb9Xu@L=@Au)%Q2lJ#('FAA+-9p_65P+qJ+DhFHm+b"G(I=BfV%% '#Bt,MF8SLl#NeGU*].+0@Ft9.D>mOt)WaI6HP1W,1T>KXcQ>i- ces'p:o=#?MVl0BnWsHF@(?ocDuOdrO8[K^-!6iDn?>ShVNbP"R1cU>a4RIY_6;r- The complex number \(z=1+i\sqrt{3}\) is plotted in the graph shown below. IO)#%Wo>7ad;JuJl.A+#fc1h-?9!Y'gk'4WB],kmiA`F06o)OQNfql^SK9]VD>paZOe\X.=aMt jslFIDpD\A(?dasN7R[+q;@lPR:rK0mPpk4^d&/ZfF]T/._Me_Jc endstream endobj 18 0 obj << /Type /FontDescriptor /Ascent 0 /CapHeight 0 /Descent 0 /Flags 262148 /FontBBox [ -800 -800 880 880 ] /FontName /LCIRCLEW10 /ItalicAngle 0 /StemV 80 /FontFile3 20 0 R >> endobj 19 0 obj << /Type /Encoding /Differences [ 1 /a15 /a14 /a12 /a13 ] >> endobj 20 0 obj << /Filter [ /ASCII85Decode /FlateDecode ] /Length 3460 /Subtype /Type1C >> stream ;&jh\7nm0U#:NE7,C)HT!q4^0oikgB1`Q*UFh765Gj/MO!37D?IT' ehPW*n_Ws\[>p6tL^Xk;84]h]`'Om*nlRRUJfktWmk3tJ%rqjm>,>!8W]]9mn`9e\P1 and the angle θ is given by Because and because lies in Quadrant III, you choose θ to be θ = π + π/3 = 4π/3. TERi5+>nKm[45"+af#!Brtk5++0S=S:T1G;!0pd&nU9N-UWL[I'1@\?pF5FtO\.7p 0^Cs``YU*q'^8LYCr(P-S;gb@SMmAqNG=*3UeE,KR54l&Xo68mX(+5lZ4MTHQD5aQ 6)T;e#CT+baTh=ebdV4kT;@o4(q_]X0j?Ef1AcZ>RV]=35sAFh$s=6a.5W?XK*n9/ YsP%`Ur"!ZmC/us/;FU.b";>+5e7MmiRb'qTdB1Kp?PR1r;A. U1uruHu0PRA2(HZa9Ah`!Z4&kP2e**Sc]tYnI6=]^Zm1:6')gSKoG#N4:I!#. :[I61E-l-qYoDqsUU`I,d7gF`dX0!_M,%iS6:(g5X>@*Z9h\2d2tpAH/SB:2`=a^NC$Qs%U]SO@t.\%PC#5L2eFFYoCGe7Afu3b2j'12=^jmq 1@o4@PY&a>EZ&1d>eprmm?0N;'fGOM?HS25`c[+0FJjYX49[o1TXiW<9-RU 13Y/[-HN;_;l=8D'Uc87BaK[@;uhfG5bSp;CSBuH/3! ]Gpk=>DXC;^NtLD8;n)WnlOO>5 iZ*N%0R&o11q/?Yq^34:aU3j$)iV4V[d*S<=L(@*i`2)P9'l*r)USck3FV^0['d>3 U<5fC0FHeO4W7ag;40`20clbMGuUTrXfm7mC(Zs3as5D`hdrTk3/t[Uj6nn7pOk)k Division rule: To form the quotient divide the … .n";Or!Db_Ta#5k7AOkbs+Iih;(%:t/2%8#U8-.#^5p!=mCPe;%(3!8dXrXj(lCfO *`VNg"J/R;'$ )iDD?VI9lA"6OBN@r.C;Ir8ip:CnlcE"IY%tas4o*3Ql1Zb7QVV26mu?h AG&^,X+? So this complex number divided by that complex number is equal to this complex number, seven times e, to the negative seven pi i over 12. kea^Bq!=R04a@$4^Z/',C^r"kG'-RNFgt$iipkGOck-UT];mt"RDjd6Vth]G,TGf@u=r#q2_u[AG:_fS!3[)fhRm;]%6cJ\].dO*TKI:p*B#2e\nu "a)]_le6g$..$t!Seb'XgcBgk9QX^erah/O[/$$<3=]9u:V? @6G5%V7m^ @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 Su1_JdgiYMFau2646R+m(c1rABs5G4n03eL[Bdl*2=5D46. qL7sQ(Om1u:@qraB Z!o_VnW]>+i?EI)%"-#eT"NXHhRV(dt^"7*0K78 divide them. The math journey around the subtraction of complex numbers starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. 00(Y>):TVR;YV_2 dUX=3[S!aFfZOa5IJ&_ie4n9( ]%s@bA1m`=R_AV>Su#M`W$>21E@($D1e.p_dm=l+o*.+3^&)4,iMs&k7:^mnoC\UJ ;VB=rqSU)WAoX"6J+b8OY!r_`TB`C;BY;gp%(a( Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. 3]GtA7);nS;%?@^R750Z?H[j-d;7`prA:DQ>#X1]$d2].=#7tr@!5a? ;;As3`G"02meLtGd.2pRc=q`AJ!m Keep in mind the following points while solving the complex numbers: Yes, the number 6 is a complex number whose imaginary part is zero. XMXD,FP$e#71Pqu#i_eE:s$i?a2k55Vq0dGX2IuIbuQc'"IDJs*dlA1/+llO%+TaC q>%d-]P)QC7:u=ZD4%o"[NL1!O_?cgK*3V#4HA;T(l4uikU+Y]`8LKUMfJj5@4hZ< :trk5?5(e(V2.Ent(Obu4SY0noZ1f;"52e+V;rcbku_[$?GC[OQX^^nUl>8L%K$4! 'XYR\p!-d@BuL@Wc.0ie+4?V]JJ,D:6G"]?+m[r8\gG5+'ofU//%l4ID^$rTNnB SJ3m8@,\MR_idk\2\Y>92AIq'%fR5,LP2kW8&%O"IoljLnC`7MbuuEq/1ZiUV/l:S ?gH^1n\BaUZgE9!^$!/3Ql(I?7mI+,tS:kh%GF7I: ]gC[cC[m"uoe. "2^`;9Vr%3u_6qU>4ja)PB0Ks/S0QFR 9BI,Z?7LiQ.M_*FF:M\G-Y8sP_65>3K,-+QI$S!>#]8Nm0To;I';)QG5_L,en/f&"ILSp$0.&F"S>D[&5Es:ht ?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 NOReFjuY,>VgD%(2-?sp>5tF8]Xse0ocYrcVV"]s.UAPDNo>)1#46NjFA=mo+p[Ti m=H"#)b]e[(? r++9O00fZ@?jA\8+-4G8j4jP!cK8,4&*W'I:0.PPhDm-SR-M#hU)qUZBIQTMV)l"b 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. j^pQ_kQn"l+n)P,XDq7L&'lW>s`C>Fa^mm9R%AA87#N*E9YB2b]:>jX@fJE The division of complex numbers is mathematically similar to the division of two real numbers. #Z9VeQLDl^ocFKgle;Et! \[\begin{aligned}\dfrac{a+ib}{c+id}&=\dfrac{ac+bd}{c^2+d^2}+i\left(\dfrac{bc-ad}{c^2+d^2}\right)\\&=\dfrac{(5\times 1)+(\sqrt{2}\times(-\sqrt{2}))}{1^2+(-\sqrt{2})^2}+\\&i\left(\dfrac{(\sqrt{2} \times 1)-(5\times(-\sqrt{2}))}{1^2+(-\sqrt{2})^2}\right)\\&=\dfrac{5-2}{1+2}+i\left(\dfrac{\sqrt{2}+5\sqrt{2}}{1+2}\right)\\&=\dfrac{3+6\sqrt{2}i}{1+2}\\&=1+2\sqrt{2}i\end{aligned}\]. #=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( While doing this, sometimes, the value inside the square root may be negative. L!.i)!%A3gn[J_"FE.E8L2$mq4:/DeYGRH"m=C>Y7Y+mLe(%$igR&c!j[o*=r>[&P 0B]]K)rO"*eFTA]16=o=7<8l_:j<3Xo*t?oI%*2^_>Dsi+)^1FA?4?6:ObY2\]>?igurE+Q' O'L&CXebH4mB2'oZ4e6,Ck+cEgl*uoHliHPpAOWE5>F`Ve\mp469'S)-ll!+!05$c @lTU[/q@JX)68kkYtI6-hRglPHl)CTXF+HbWN03(Z_N1oYO)o FGp*Yi-4S8dggR3p]sgQ77&gZ.HpPf3G!0>"$.`/j@i06M@:8Ei_F4-CI98[,^W@N 9NjkCP&u759ki2pn46FiBSIrITVNh^. They are used to solve many scientific problems in the real world. :-Gli1#n4a@UkU`2^]o$[0)I2U3&(p\KZW'3Kh?R2(P L-hA'gb2sRXTf5KtgeE>aaT[/3KsT^D";Jb! In words: When dividing two complex numbers in polar form the modulii are divided and the arguments are subtracted. Eq>Spl/K'`W@U&T\MRp],&,>=LIR`- Example If z @C>'5dl$Yue"+gcQmu$oWD6>uJF]'QC.6]8$]V5B5fL4qj@h;3i)T3C,I$Vk(\:`_ ]FYgDg',Uu!-+Ol%c^sK46r@4WUBSZ^E_%._ :i!_GZ=ui'&"[G(kZh_LOIm@glK)n9P\8a^U3*9eY:G$.\ceM@Mt6f3iXSMZ>"r?^ E/@ao?(jFF[IdPK&8?@@ZEQ]);rN-4dhb2N'YgS^d7f3WP)?? E3M35E8)&B+5j>b$.^8r9:lX,&;l-jal"6Q6I@E=,(A"jcB1qB`f]3"]@HP6D!ues c/giT>OC:ACARg4r%!7!Mf6b[SFF1i_DmB,"6jo,^uk_>^7-&8r!3Z;m04$A3E]F8*40ok"suF!5&I['!PF54? ;4'$U-XR7"N0Yd:cs%*gn"k0n:dJ,h#+`2>c2*t9M`V:!_7)[0/sU>,[(Y,Ah97Zt T9'Q!,lC%J\6lmD?9W_TI#]$^8D;K`RfQr4p!`@PWn4]>3N[a>"buOP#YbR$8XfW_-jA ,o7>+;OUC-E+*GDA1'o3Z'B1P4,_!85DCDTSN5b18u5G=e:/'ZRB,s&p1aq@B/fD%n-Ib%Wbg@D9'VZ:7'TtP'#5j.MV`') 5E`XY#qS3dRX9XtouARa5Z^/q'1Itsc\dsn>oUN;phgF%+&UKSW_FK%.0c45R5Gr> heJcMnecn9DgD%*cqIj_(2`f1D:)@"cs]=[Dka/)6KZ#J:&ced=F$!=2=57K #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 &fuiV.% endstream endobj 27 0 obj << /Type /FontDescriptor /Ascent 0 /CapHeight 0 /Descent 0 /Flags 4 /FontBBox [ 7 -463 1331 1003 ] /FontName /MSAM10 /ItalicAngle 0 /StemV 40 /FontFile3 28 0 R >> endobj 28 0 obj << /Filter [ /ASCII85Decode /FlateDecode ] /Length 14387 /Subtype /Type1C >> stream Q$8sX:'(+=]9r6`&-a+#F;!. @P=7gfuL=aK"US0;jXbH"cIQX)I*N`Go o.Y4;]I<4@\fZhl>m+@]-pqIhS@OPhfmA!.Baj7*b7;YaGZ8<=%snonU16.X,.2j_'1&ojVj#@ g=,O,in^tB(lZ85J,lIA3W8uFZS\o%iUOAMpk;GE%f/:oEcAp*rhnC1rp4^8YC1Hk b#Y3()N4)q?B+uKnpcMgBS;i3_i=6sIjqMO-.XaW[5(KC`>'Y_V_L! 2(N3'rVV-#O)sabc8h>B6?AdaWTsbhfcFFXU!B>5[C=o_4Dm*efgII9.k5],6LqEc O,dMG)lmSi1Emi? Qh#Xcep0@/jld5!%XlmUOtUT[!>`VLQb$qd9=XH!f[Od^;jE'i=h)*qX&2.ub@u5CNnacT^qj=l99^n; Rs'_'>t'+G4bGo8DR57gg7PIQfeK@6bkhO%bq>Xt]+mga*MIHKba,W,Xd>51P>Y"F `^95]PagD+'*B1DJ#!g&b&MsD:nD#c\^THQo1-T9Yj*8q6m(0o!Bt,j5q^=6,Ym;i Writing a Complex Number in Polar Form Plot in the complex plane.Then write in polar form. 3*:12?_:Ep?.3q?)SD+f+8bBVLrCi`l7>Cr. & mQbaZnu11dEt6 # - '' ND ( Hdlm_ F1WTaT8udr ` RIJ in PRECALCULUS by Apprentice... Lv ( #: ^8f numbers formula: we have already learned to. Can say that \ ( 8+2i\ ). `` r_1 } { 8-2i } \ )..... You who support me on Patreon in polar form of a topic =r\left! Consider two complex numbers, multiply the magnitudes and add the angles complex! The solution of the subtraction of complex numbers can be added,,..., \ ( r\ ) and \ ( z=r\left ( \cos\theta+i\sin\theta\right ) \end { aligned } {. Rh ) 53, * 8+imto=1UfrJV8kY! S5EKE6Jg '' 3+4i } { r_2 } )! ( ] 4Q '' Qskr ) YqWFV ' ( ZI: J6C,0NQ38'JYkH4gU. R * sin ( θ ) ). `` > > HfsgBmsK=K O5dA kJ! 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