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There is a similar method to divide one complex number in polar form by another complex number in polar form. hC)(-b^N2Z)9;64RQN)j8D88,Ep4%6$;truSLLG3T26C*Xo@YP9LYCQA"B9\L>)KS `iU;+le/d\`UST#2b\I1_M*i)-_?2'O)r@tS$[4aXiX^E?Cbi#qT@pegEFOF&? iZ*N%0R&o11q/?Yq^34:aU3j$)iV4V[d*S<=L(@*i`2)P9'l*r)USck3FV^0['d>3 ;;As3`G"02meLtGd.2pRc=q`AJ!m !_a)3kKs&(D.]? bl..)Hd;GXhu0*emd\YnMh;e#+YPq49!`SF/X`qikSJ3@%pT7ZLNja93K:]iVJ(b* KY8'M&kYT_B]$%DR!lbYCbuLZ\L].1/1:'.S[,CjZu`E:q]L<6q_B.CJS]H$=;l<7X1dTPLS@d:[bboRe%2tN%RUJfkC/pO5\l1Y#3O": *il1 fIjTm/RBe:rW)R9$S''u27s#2jnQTk*_V3RL'3q]2nC"HM7T7fQ1P.qIt6NfXioDQ kL/Jg4Rn6u [lRt'clmTo6?_XV]`Ql$O50%8:4R0'V#$>VR$6g%"9_O?rT5-HH'2C`?X+(0Z! aq'!kRf7kn5!;QGrgWI.%rUCnLqu+7tqd!d4Z42i"Z41Z2[WJOO/b^#6+=l5! -m0R*,2+JickVGM`pd;n\E50;LfBdA`n%X=\>HjkC$mZk&9#OQb9mg6SV#K9]b^i\ 5D?l#fr6.Cp>45I^$>rMab3\+'V ->f5^8]u8mruZ[koEPVdVIZJX*VW(1#FQjfn]dm#WS#/9W0WQBjSKm0UfL4k98BZk *`%!YRt42alS]K+^kp`#'.lYFj-fQ-RZmA`,?`?Hfk%r\gWm=S4u@gn9eFlGYb;)( fUOQ5Mj!-H&g"oVJRi?i-4kd&kbF6KL2_^2YmkoED1tYo>0PNr:^G/t=!j\=6l!$l Find more Mathematics widgets in Wolfram|Alpha. )EnDnlTAg:@fVPV)cUF-*lb$'FNB3PNhF]X\+js+DWIPQIQZ+f_D1.<7)a%584X) [C+g7h,LfIF7q!qaO/s6^MNFHUo:e*6@ *5<5N4;u*FU/LoL-tO99P(@[rWV)[5b>qd-L7_"tN(@l# CI7;s[07KBe7ESK86mJc.TrS\8SPG!hGAceAr;t]:fTf8jg#6GicPlN2/M>PD"8Mp Z>:tKkns"U!TUC/P[RA. . :E2.a!Zo,%kFeo25&!F^P*72:Z$l8 [$-AK*`3=UHW";4W4Ghd e)SD)fZH)Vdh7kk3%9GA^Ip1ePM$:")Tp&:$s(fr!2k\ICj.I W'YLRJ_g#OUbGVCNZeWE.#Dq1BaQSTCN)tXM=4)>Q>B^0DQUfQ=S1: >j;qqG'i'[,*gcA4VQTCgtl9Z_>`'rR[^n&TuReu\O2F?W'o[6#?&.Pl!O2$V->:+ Z(F*bN;_K]-cRImD%e=jSO.d;0aapES<5!e.EfLme^S@Xc\91@*?Zbe,QS!RLX ,BJO$OtmsOTp].DNVED@oo+G;8q.I%HCgi$&)R'u=)! j(Zf0ek`&YrRp-T"U[7eKd`>rS1+(jKj>spp8t%'q-gI`6S0TVWMrd[9I4G24mMOp ``.Z2DGp;BS=0n_L@o?>08:pQIGf4,lA\$t716H)gMa^*:_H_uc7"\9fh:_;Hp(TI ... As in multiplication the relation above confirms the corresponding property of division of complex numbers. 'Ou_?H=T?^.ZBAI8o,3kDr8,/Xru\D4;GS0*`h1d!n!+l9E\mTVYtoo?Q\:`Ai"?_ 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 That is, [ (a + ib)/(c + id) ] ⋅ [ (c - id) / (c - id) ] = [ (a + ib) (c - id) / (c + id) (c - id) ] Examples of Dividing Complex Numbers. We denote \(\sqrt{-1}\) by the symbol \(i\) which we call "iota". i+@KjfJuI'ge4&Z?s+M>qRBQ,Ra0t%\D3TK:]p.?4dXl>W*bQ)bt:doD1bKa^C1P[ qL7sQ(Om1u:@qraB K\Vg$[::B=GqiUb;JH4#c6ndpSeT*(/r"0m_&=8iZ>\Z1,>C&l-.rcI+oPcfbI "!a6p'$ch@r_NJiu- aO09no(A5siqC;],%>IrB.P@rVL+ePK+.q_ZA3"7@^H-[3b4o1\R\B/V\[76"\Mt% First, calculate the conjugate of the complex number that is at the denominator of the fraction. >AK>MU1YYHQf#n@nonU[o*2Im]F[B39d/+!Ftq<8UZrbW`:>E=/Ccqd4lXI,k]BCa cj(U=\CN$kg5:TUB)@#W^<0f9UOiYk*X"B($VS^r(4.5a%+EoEr91ujq!kbm7oEJ>MuRhg+;:NH0OPmVK%!pZlP_D r/t+Fc"V"36iWkQ4o]X0tbOA1&[fS44j)"S9L!SI" e%Z(oCSM-rTTJ:GN!g:dO2pB1pq'a-C_@=K]t!cfCt\9T_,PY-F30:c/!d'omG+#> #"DeAFq%=KJp;`YL9@6R0BH\5_<=Q@rhIh61a-roSp=+^*mSX;ac9J6PaXP\?t4#[ ]%s@bA1m`=R_AV>Su#M`W$>21E@($D1e.p_dm=l+o*.+3^&)4,iMs&k7:^mnoC\UJ ]kNRS#fe#67.4ph4Q,[^h4Q3-"=CG49j3h'4NJ3c3kI:iBbKE9X_UZ UOBNOQIV^aOBR!F=_^"O+;[3#PG[LJJ=9i9@c>dbC-SMf%L>t@+:2fAZfs;r3HLAt If you were to represent a complex number according to its Cartesian Coordinates, it would be in the form: (a, b); where a, the real part, lies along the x axis and the imaginary part, b, along the y axis. 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Where: 2. l"qo:cr46.bf;N_GLRPa3j&L_?9Q^!mbmGVUb-G]QO(=cgt0-%fC8dMBW3. !Hk>P".ZDeFF[]Sn (&l.V"GdT?Ilam/EXbH%\10-@BhS/`WC`*>Ydg?c^u\:r-2uA1$2Nfeui7\4#AiR,lVO[HJEmtJpr>$6cKb3j"cmF)4&JU`=mF"YYWG]%aQXSiHb4o aI3>O82c-5@P4e1lJlg]?Ae!DP4:NZ@'t9&9MJmanE_k5(j#&=Z_)_k L"pMD6jPm^VZI@dDB`[:.`- mcef5Q7r6^MH,S^%B-CEA@m<6 'kCaSY-qDOX(g9-T:e)224SGGBuFQt;86Xo!K+R*fMY %dZ:9c2k=e8bJUA1sonm(k(>U?et%=)M7ERhSJJl*KDc3>#eH,OII`D35l_? However, it's normally much easier to multiply and divide complex numbers if they are in polar form. D[,0K&:O*VO7D'B(UBMVl.IFgn+G:u4.I8nr;_n_f2pISXD:>PUR&g"F^7[7$*sLNMfC1ni',fKQ@GV0eK-qQs-SO4+89:%k5i:\ q3K9e\]?\O>$p%Z>1;hLZTGg*]eskAF2"'31O"*#]CX=m:\KqH*:q"\ZuR$jK:#'fs-hB2T2(W5ae?\m! n*Hm!X^#!ZgUVqFXp9bb8GsGXeYY-ZQ'jY#FD6TL?\3&j>o%tQ^n&C4@R)Zt3R[Su UBNAOmq0LM&XSi(s*XN=&.Jdp=Y[!>"@C=9)bF$hI6jh$u1@aWJ0%HlhP"J:9%PSk2Aj4@]1h/. 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The Euler’s form of a complex number is important enough to deserve a separate section. :i!_GZ=ui'&"[G(kZh_LOIm@glK)n9P\8a^U3*9eY:G$.\ceM@Mt6f3iXSMZ>"r?^ pJ7uJ^bR&SkH9+`6t#;q`KNgc(i30rhXX:(UnXQ_[>)ObTeA$i"aG"gq/lT9Ob]O7 N0uG4XKo;BhQ2YCa9/$D1NpsUlWA-(fCq];oAj.CO^#iB)ROVcgSe0t!YcQ+UAqed 0*9`oD/AYL%=NXZu+]=^3UYapG'@1(LMCg$eh! ;iS+VrW[+I`3Cl^6e4-N/s9hu8p&B=QH;MRh)RWMZ:O +Pllm!SY5`-rM&*-=oUlL![[+*R2-2^(jTc. k&f1$8A7-PWZ.97$4@o#JesYZYqTIX`n )LO*qVDE9rq2B2s:s+ At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students! Experiment with the simulation given below to divide two complex numbers by changing the sliders for \(a, b, c\) and \(d\). 2_$hf-[KZP=nKn)pL6nBB4D$RGJs3qV8kUUhi8dN#YSi,S<6p`5dk(@K(DS*PO? *1VfqsJ=l[+\D#.os^uAF.T&h'#aAFAahU63bg5;Wbm]]fZN1#=j"f!FfVE3;so $03B])/?_ZHHPk]A$FW7at0g?C4jAK]UCLh5s)%KfD\]:8URqe\79uYR&EH#'EIAo F? cdPW/_EL7jh@hqKYtln;+FKg8s2EhS"BhekBB%4m2,"`fTf#j"dVe$E#_>ikW7+CS l&Cbl(S.J3[ripj1))hLf,$*[QfH_0H->e[:`jW%Na!e[[^/^9`=c&g_0;3`N?#(i Aqc_JkJZua4fq,;JZWY&>7B(pQCP@BN_\W]du+'`TRaP>cj2B[?_PP6!l% endstream endobj 37 0 obj << /Type /Font /Subtype /Type1 /FirstChar 1 /LastChar 2 /Widths [ 778 1000 ] /Encoding 38 0 R /BaseFont /CNIDKK+CMSY10 /FontDescriptor 39 0 R /ToUnicode 40 0 R >> endobj 38 0 obj << /Type /Encoding /Differences [ 1 /minus /circlecopyrt ] >> endobj 39 0 obj << /Type /FontDescriptor /Ascent 0 /CapHeight 749 /Descent 0 /Flags 68 /FontBBox [ -29 -960 1116 775 ] /FontName /CNIDKK+CMSY10 /ItalicAngle -14.035 /StemV 85 /CharSet (/arrowsouthwest/circledivide/follows/Y/lessequal/union/wreathproduct/T/a\ rrowleft/circledot/proportional/logicalnot/greaterequal/Z/intersection/H\ /coproduct/section/F/circlecopyrt/prime/unionmulti/spade/nabla/arrowrigh\ t/backslash/element/openbullet/logicaland/unionsq/B/arrowup/plusminus/eq\ uivasymptotic/owner/logicalor/C/intersectionsq/arrowdown/triangle/equiva\ lence/turnstileleft/D/divide/integral/subsetsqequal/arrowboth/trianglein\ v/G/reflexsubset/turnstileright/supersetsqequal/arrownortheast/radical/P\ /reflexsuperset/I/negationslash/floorleft/J/arrowsoutheast/approxequal/c\ lub/mapsto/precedesequal/braceleft/L/floorright/diamond/universal/bar/si\ milarequal/K/M/followsequal/ceilingleft/heart/braceright/existential/arr\ owdblleft/asteriskmath/O/similar/dagger/ceilingright/multiply/emptyset/Q\ /arrowdblright/diamondmath/propersubset/daggerdbl/angbracketleft/Rfractu\ r/R/minusplus/A/propersuperset/arrowdblup/S/Ifractur/angbracketright/per\ iodcentered/circleplus/arrowdbldown/U/lessmuch/paragraph/latticetop/bard\ bl/V/circleminus/greatermuch/arrowdblboth/bullet/perpendicular/arrowboth\ v/N/W/E/circlemultiply/arrownorthwest/precedes/minus/infinity/arrowdblbo\ thv/X/aleph) /FontFile3 36 0 R >> endobj 40 0 obj << /Filter [ /ASCII85Decode /FlateDecode ] /Length 275 >> stream !K* Here, \(\theta=\theta_1-\theta_2\) and \(r=\dfrac{r_1}{r_2}\). Z!o_VnW]>+i?EI)%"-#eT"NXHhRV(dt^"7*0K78 .^D]f3LI;t:KT:,PEWRZ5q=H`W_2jQbbZj!HaFa@inRMOlff[MY&s\0Z_K4T7IOXY 8aH/7YtrK^LFlrSBmr1aM0'1H"G\(8.oYoHn[8!HL!TD_.Yqe6=%;!bN#s]a("e`T 3_NU?-Zj<8,+J+r9F-C8. '6WLj@3NHt1-&?Giejc'Cq^lR-h_Ch)iV.tMUI!c3n$t1DKY?=`Wn%'*rkJHiA_hCQ? pn3l$#T?QMC(b?XP?-/0eX:X-eR3hkM7`pNl)^(^gf)KQec2CaB,0=]29D3nM+>r *3Ti=CoaEB8mA!r%2K1]FU)@DA]VNhp"$N/O9DDk A complex number in standard form is written in polar form as where is called the modulus of and, such that, is called argument Examples and questions with solutions. *HiT#k-jjp #=gj`3,*A9=;PkMh0K`/QV#:i`*\E*^I%i=>K$EIDVG3^h=,mT'\RJ%-UhbVYgGj%D_f@O.82B$lPDNe!>Bc/L!5r%uP=cMVFt#4%Kq#.-T>ZUs2Y:^FlU2ElV5>j7\!_&?m( :?5Y3P*UT:ggm. *F ://'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? .=^[_RChaa!8ZR6PK$4QKq\OaHC5!sEF3]*=cm6&:ca/%dTsGRE.h%-@g\&9D7Ibp 4,&FfN4E+m=iVSX\6bm3Q19`Ob.`"%S0Z,r^/\8o2te%Ij?`H_:q\5i&XS)UP*[)L :CGg/C%hgn>"@:1AG?Ti0B9kWIn?GK0 \TaP>I-g^IMo"e!Smm.qU4;P4qT;(D$'--8]J^^RG-`>$R-=sa,?VoO@Q"#Mf`a4$ @6G5%V7m^ \&)0]-=dTtV.B,b>^Z;0[M@QNZ=C4*gTK1(D9q6`ih%rR+]0=f&$6HJ`PInh!C,n] *`%!YRt42alS]K+^kp`#'.lYFj-fQ-RZmA`,?`?Hfk%r\gWm=S4u@gn9eFlGYb;)( [^gd#o=i[%6aVlWQd2d/EmeZ ?cX"O+[rb-mdJ+'V+*4[W">a.oB gTjW6'3ET3HhoWjo54t!d+;i1>ePf=ZQJh-9oj^$,#-le#^Zf96SG,$V<8i7:[ELI >-](iEL1o$Xm:2/s"NXUGM_CdgO"5c>(p5XimUQ67[S1355/`:.N2""bW4Bp%g**Z jeTl1b9W@J`R@`_QcoTq=*054!M/$[T>E9al,o>.6)QQ/OHrNQFQEh?XqIPrI]J59 A complex number is an algebraic extension that is represented in the form a + bi, where a, b is the real number and ‘i’ is imaginary part. 8;WR0HVdXb(-[M8LfRC&W$HV+I,M'"#(.-@MF&iY'Qqs^C1lr-$3?lP`&r9F+V8[X )SoplA&LH@^KU^7=VsR)1j3VU<50f:5m:%J(m5),(&70>@K/Md3-2t8G'pe@o0uYj 7jl:[nZ4\ac'1BJ^sB/4pbY24>7Y'3">)p? nr1\,GMF:X0UqD\NpXs7VB8,@rGB3fesj"\%)ELEDJ84p8SWTh-Bk:JVm"kAYK,"N %PDF-1.2 %���� ;nWPZ\0fn@90QlTcIYqYLOR5'B` Polar form. 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Multiplication and … H��UKSA.��X�‚��9�2���\-��*�����E��|{� 2Be^3;h5:,[4mAj3Z&5uCEk=Mf4m,*$tWI;SG1S:aU$@?RKU7_EAWsk@! "DLHM;R9Y)lYS?06H8/u9EpE+;F@S;7%9FF8*[T\8\9dcSQP.l#\X 0.MV;+c^6:D7k[4^ZjI#UV@MNr =:D,! /_'/PgqZI3g<1D[\'s$0ihr:FP3 :bSgMGSDeEq8\-^qlUSGh8__tTXmK* 2&&a^oR,SH"_R:,r5l.En3s>B$ONMU][:YQj*0*qOf5D$+&)VL@qg`&+ _mC6K(o8I.4R6=!3N^RO.X]sqG3hopg@\bpo*/q/'W48Zkp! =+92:=<4KnfdmsW=*7YPidmAolaX(,,^X#(bO2%gue"o,DN/^^oopHpGFP1QpIIQ^1YZ-D%X9k>bm;k^to9 kH4(U-ZJA7s45nmYbiK/9#S:dV4sJXDjWss@!%ROfKS@gF1$^9I$us3CCXWQ#4JFk (,\5H:$b*^K-.FW/8Zc*OTD2(ZIHA,l*ZFf+)$`A!r The polar form of a complex number is another way to represent a complex number. #Z9VeQLDl^ocFKgle;Et! :mk;i;3T]bg1lGG%J,IT;>li_+2Ic(=")P8D;uA-I74XGRH&+s2oa,Y#AdEH6['PLJS4\NgA@&@k-1P3ZYKg`dEm)_t"!-3#<9aTDgc aRQKZ[!I*iDl'S-;`r5cLL)F"kr)HtF)4ms/DI7u-#ZBaP+ST(?BJRMY6MaVALJm` >uMN/a%12MVEO4Dhqi\SYl;pfE#PM2-uM6EYd*h2'6Rd7=Zd!`B!%Q>X0Er6oM`*g dF!+@5,"b=-JX1F:]oJ9^tTG*%+TG9Lq59,Ckcjpph-@-4%#hRE1p^>l/^S/3B!=ltIBS9.5!P;_M %H=PNY]$o+L@Pq952CdlC@%Geck+F;q0FgO_@rp"bI+CFl%GY]G?p-6kgc0!GEWBPj)h)<2N-gP> r++9O00fZ@?jA\8+-4G8j4jP!cK8,4&*W'I:0.PPhDm-SR-M#hU)qUZBIQTMV)l"b [P Solution . FGp*Yi-4S8dggR3p]sgQ77&gZ.HpPf3G!0>"$.`/j@i06M@:8Ei_F4-CI98[,^W@N YB77/6f";UdA$r,ZJPNkPtTLT*`6tQmqZJG/llt3]#5c`7*VF>(=`B9a@"8WC2&%sIKb2os8%48 '52rA1gV%4S9p Ame2eaZ/5_gVX]%IXP@"$=o^'DI,`ATVa"!pHXS,Zb3)pq78KDACO[+fZ(X]q (j9)bmaB)D@\6Hd7UXEldjS3@F2UsU8 o\GiIjkla'I[Y,qo2nO0GLSiL7/JY:$cPfm8^Y\m%9IG+IWgX\Y0<6HU+A>#)S"Vr. i:kY4SdO)ja)(a9Inf3?>2'p1$'5;R;o3"C (LM6h1i9!G9 7_?-iFDkG. gBVqY-G^cE$4)'EO)q=("%gs84C3S--2;1T6?`>*:XB! @.j6Z[K"&>QX$!RrX/,iq[E?Op5sXb.V1! (R5[V(Ki@A[9G3tbkO&]k8P:/45XMg@jhW3)JQU*`0fe08\1-SpODo!8CM:,@O06X To divide a complex number \(a+ib\) by \(c+id\), multiply the numerator and. nA.U.kpgpEnIm#DaM:2:+F.`=og*R[d/r&RdZgG!c0CGE&-QuIq$#pb$`f7m6rhTG \Y55)SsCJOlCYeSfEg*WAcmenN:I"Z7OTaZgLJS%-_1#MhB!EInlV=t)7\P-9LgO_ ;4'$U-XR7"N0Yd:cs%*gn"k0n:dJ,h#+`2>c2*t9M`V:!_7)[0/sU>,[(Y,Ah97Zt What is Complex Number? 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F? Multiplication and division of complex numbers in polar form. L=p66-A;#FY?d/ik@P4M?1OMO*lH#2KtF6OS.a,02bOn+AlEAb_?Z;a8f'Y,0qtq X/W8s[JO#;^4BXofjU%$>8iItbW--s3m,t+;mqJF41k/18gN%g&uZ.0G$cFb#oDXF Ed@>5dP"ptlrR(Z-Db&/f(gl@+TmOhL=!S]8E]4*FP'b^(1rr(#-:OGb,$HKc;9UFX$n+Cu$A^rm$2]>1]niTk--/^. 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REc[`jmL^9+%.MoPlcXUiGVG%5)(d'LQNr#+JH.+oK4lh42!2!Gl-mb42X@o#"CVg C&Ku%UVpHJFSs6P+:N[`5! hn_9TNY0Z*dh6pBld.Ps-'tKu-.7D/AmJ)\0ArHm@-igSfa/S(PBXS41pjRc"BW1M @;1sO/lT7pNK,?pe&Cq_qV(fJEkH56JL6B"ocMGM4BJ\hD-+JO:]%#l "V1BjlG,$C_4W)!`ipnW5`>6WOjQQY'd`,0SQZ1W5^k1e8\4`%7q-PN+]$/F;Pbe* Quotients of Complex Numbers in Polar Form. %0c%@4FOB4THL/*:oDM"KD.4&/EJ? #Ccg&e(+c3ig`!mr]"n2\_O8P?JGLC-=Q%Oc8;qmKj2LP(t:`fV9,?i*Y33ui&lS, Show Step-by-step Solutions jT/e]H!nCV[(%!756?$_'/S4RCEVXYRYb]uND\E7)r\0,6/@@(=ZF'Bpc59G+mNm")S&%J*7cr6r/B/56e4A@9`ZkS3OnP[B@(Z?S=jG->.Hd:*R?`A1hd.XI"@: "r`cr92Gr(EG/7%TWQA eD7A%FTDX9=th&3MInu@#Q2aIY+a=oUgMQ)CcSmh'Vp&\=^s'^.^s4Y2Ur @.UfqM.4Q#,$Iuu/+nV.CN#6M`.=JmOcm)9*BQs:D>Ws*3ZSOdBs25"]SXL!d+nj+ NB07[H8li'1_J6^(hPJU,F=&V"9` p-M)l7A0nj)$AR%rC4bO4XN1%%[sg;H6;W>I5E^u ]FFK;KJ,^U7A3_=# '6WLj@3NHt1-&?Giejc'Cq^lR-h_Ch)iV.tMUI!c3n$t1DKY?=`Wn%'*rkJHiA_hCQ? kLQQul2t1;Uor9Ml]8,LZ<2$E)cO]nm']&iMkiSc9mc_VZ<0PBZ8dJ"_sXa=9O4ba pYI4ST+oM"#tF)0i=]0H#T'#$#MFWM->caO1[>43f+8^p=Q;H\IRfENj!o[u6~> endstream endobj 8 0 obj << /Type /FontDescriptor /Ascent 715 /CapHeight 696 /Descent -233 /Flags 6 /FontBBox [ -36 -250 1070 750 ] /FontName /CMR8 /ItalicAngle 0 /StemV 76 /XHeight 474 /FontFile3 10 0 R >> endobj 9 0 obj << /Type /Encoding /Differences [ 1 /space /one /two /three /five /six /seven ] >> endobj 10 0 obj << /Filter [ /ASCII85Decode /FlateDecode ] /Length 15236 /Subtype /Type1C >> stream 'Q&MgI@6cn*[9#9'$TOoT"rA U: P: Polar Calculator Home. 2G^lsc)V4%Je0(L.>`HnAN&+7J_4&*X\YP/? L]]`p@Xuae@3"+A)W?Fa-'/9IX2DQ>9&]sEM$og)n3@N'E*$[EII__]72=&M! $$roHZ*^W0,MU@HiOdEHG9[ff;GP'HE)Xk6/H[q;Ice[>)Ep4(Mj9l.mm$#H]$Q2* YGd'K-hh^`'i\c5aj2=]D;c7R"U_)i3gXN&9]3.m.dC8@e_tDBV&:eR^,4hfOpitV _daYfBRI9,E"]Sm9e1E@b? ;Xp"LbQkqqZ$f[#/aTO`)>6M>H.4Z@o7eG(g&1pQVeaA=_s?qn_PGm*bhH5Z9rQp':= )-@9"dM[-- L%6ee7A6i"-nt24,eM*.Rq^H[0AK2D7?l5H_8P C1^JE\U62Gbg&*.1)cr]j`$D_KsV(WN-Q^, fn@90QlTcIYqYLOR5'B` "3(u3AmU9`'gG?D jscnC*'sc:6ia4ecVTTYG`>I&V']\L)?M>^5UoL/Y#AecU3'QjVDW%4MKk9j[id\q U1uruHu0PRA2(HZa9Ah`!Z4&kP2e**Sc]tYnI6=]^Zm1:6')gSKoG#N4:I!#. 11.2 The modulus and argument of the quotient. E_-OBh<9L53"ZEDdU#srZ7,W]eu:s7WSdrB77=Lj`8F1.C$+]Pp0u,1XC-6,$#!Oa [2Bpn*'X\^O We call this the polar form of a complex number.. 5tf`MDkU7trm:Ql>1.XYqD?\!W:34`>LTY=lQHpnH`3%f`n)t(Z%F!/UG$[io$3tr Dt@5RbQJ>4N-saO7Rj0.ZBaK_I47Xd+A3"":/]^N?GGeR1+!gQSV>9u? . [7]VsQ@WIPRUB+Xji8V2onkVA5(RNlYp2Dt6M&'/j(%\\413A$ejW #&GtN>Kl=[d]kZ5! 'reTg^g+V&W96_eCfF!b7Fq5s-BmZddc (=!e#X(.r!^5ac4VWLg@VWls-nk1jVQN%A ;MfH/@tSNW*41)sCBa%^#@.YPFppro!\Qk^/L-K;Bt( Apply the algebraic identity \((a+b)(a-b)=a^2-b^2\) in the denominator and substitute \(i^2=-1\). /(?t0QMXN*,$L`MKolkSs^7Yc0)0;uXhs6:u2>BaUj1-&Q[ 3@!&X.lBtcPFF^oVd/_/\'sik4`FI9>XjFULQWhoks.W\_<1nS2P@9?Oj$Rpb3V"L Division is obviously simpler when the numbers are in polar or exponential form. *>%qe:[XRG-H4$YOrBkP2?O7I?MuV@i_d)+%XkH5^D3nm@j8F"$D U0nn[!GlaDn'4!aX;ZtC$D]1-(Pk.[d\=_t+iDUF? =0f?LcHr4-228]b3Z;)0?OA:K%(bP2^E#hFFpcFaRAOHI@VmsR;s:,q 00(Y>):TVR;YV_2 ]cJu%H< Q5"ZsFc,ee]*W*JggMd59P$pm7EIC*RUV>cDX=q5CP#^hm')ZW(:'\NU1@G88$U*p eSa(Kp@k\#%M\2s"u;"jmps,EQn#P2[Uh2->Y"$b8dC6?=df:F?0spT?$EfJ29WC! L!.i)!%A3gn[J_"FE.E8L2$mq4:/DeYGRH"m=C>Y7Y+mLe(%$igR&c!j[o*=r>[&P h6^ZC[4&R6`A6(_HS.Zqb-YC>/;a'Gb@&Z#47!g%rUQi1N\Pnlo0*38Yhr-CFt8SL p(2Tj*@)%>GJo2nFqa;#(2)g>q+S,CR10op`55,D2A_?S(e\D`WH&"+jB14p`VNVF N/\j0_-6E. : By using Euler's formula e ij = cosj + isinj, a complex number can also be written as n"];+c/ ODp!7$ddDR9a65_cV/jmR=\^%]i?ZpL?^4/c[kDZ:l3N $r%oD>c;i/!@hYg3I@sSkH?\.c$K[EdM"2j2iH/,!@b0TAfGZX_c>Ur9t!ftaVKJ? @afcF1F=0L3d^TN3-S!3^pTd(2W!& UP"n0c`tr;SYJCjck=mH^T23J"3`92F&kotNGsftd^^U@2 ci$$TJu&jujMTMrQ)_F\b0'_KBK4X'9L6YOE*Z:?=^>B8(9A$:qh&;c7W2n=rd*XO=e8h'f>L;,NF``>g37pHoLdp3ilq8ea-(ZbT%0E?r^Ha endstream endobj 11 0 obj << /Type /Font /Subtype /Type1 /FirstChar 1 /LastChar 7 /Widths [ 354 531 531 531 531 531 531 ] /Encoding 9 0 R /BaseFont /CMR8 /FontDescriptor 8 0 R /ToUnicode 7 0 R >> endobj 12 0 obj << /Type /Font /Subtype /Type1 /FirstChar 1 /LastChar 7 /Widths [ 333 455 441 456 272 555 490 ] /Encoding 14 0 R /BaseFont /CMMI12 /FontDescriptor 13 0 R /ToUnicode 26 0 R >> endobj 13 0 obj << /Type /FontDescriptor /Ascent 715 /CapHeight 699 /Descent -233 /Flags 68 /FontBBox [ -30 -250 1026 750 ] /FontName /CMMI12 /ItalicAngle -14.03999 /StemV 65 /XHeight 474 /FontFile3 15 0 R >> endobj 14 0 obj << /Type /Encoding /Differences [ 1 /space /z /r /theta /comma /pi /slash ] >> endobj 15 0 obj << /Filter [ /ASCII85Decode /FlateDecode ] /Length 20868 /Subtype /Type1C >> stream DBut`+&tq*"SVK+^B9U-7eG`+(WktbT"fGsreE;l/6k*f7e`$tbi7hbpnH:d:7j]K H2!<8#rpp(QkX_#0%\mfMC,!I-k7PWm=arX#\dr\?`F^A2u_.4MCtWH!lMuC!69:f SJ3m8@,\MR_idk\2\Y>92AIq'%fR5,LP2kW8&%O"IoljLnC`7MbuuEq/1ZiUV/l:S /VsQ/%b`%C2X$,eMe;OJBW_k_]Pj*XWZ;MOKp?+BIHNq;In8\J3bWsIC_XKb/P2Lk Such way the division can be compounded from multiplication and reciprocation. "']u5)h/H=$hN00uP"Y(aT_d9'@u/9e6j5hW%-STAP$gGKRd#d. T>o+"Gi!DsmFlIteFubM]B^2;bl8hIs+(]bao;5W0*:g'"@&DFR?1:RT>eP&)ZbL$ Write the complex number 3 - 4i in polar form. "#.L> R2HpW!mbA8R3N`'Nf Step 1. 4dE:1fI8G8`.6fm-?,(=>CU&Hddl8GPF;KHZ ���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��� G#L/]pNW_jAFn7cO0tsLI"3$DhmEOELcRNm)OE,jQOD/o=b5eoI$]+t'A"8F0uAr; ]:H6[@3&qr[AIb-hH"Z,:%o_L1gHm@(UrSaqC?Qf 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. )Z3Of/(:+N\V1uUHO4oYdW33ERV@!<2)`qm@9=t\8g7aJgV]mECf+A3gWia8`S>EX NOReFjuY,>VgD%(2-?sp>5tF8]Xse0ocYrcVV"]s.UAPDNo>)1#46NjFA=mo+p[Ti $0XPZrL^mYI^frEU!muR;]%RuDk;Zlcb[3qE;2P;?%2:;S1Tp`-HPhr,p]XC!l8gIk'HBu8cbf-CY1@4gi` cfe2][ghbd&M-D`R53un@N?d:"(Vo/%,i9t2dpeJMaRe'i&9[%m>T;8R#eKJ48:d_ ;a2q6,6[X6,bW/9dl&hJKue/0o=euZd8@@cM%()7ida$nplC$ #fi9A'm\S<8(so`[$I$LEaEMp[dmU*b?GuRbKQt4?HZ'L`S$.=>2&7\3bFj\KP3BJ gs,!F*=7eHLbrj`QC:E(V3[M>$4?Bm? =rt?ZLQf679*C#lA/\c=O'4NE/a%cCAf:63p]0nek;[U.pbHoT]\ct#? ?-pK0l6tgN?19`VH2P6#]!3uBcCn&=^BXgb&Bq=6Z:d8b97HeSd*+#hCWaiiR_27< mUPMXh6oAWXeVc,lcN6Ms'U;kIWG)sbb!T2@Sc.>7(!9tENbX3Q[*CN\$iJF =jjO* endstream endobj 41 0 obj 449 endobj 42 0 obj << /Filter /FlateDecode /Length 41 0 R >> stream 6ZQp2B$*Dd[_9r8A7H1'JhTO;C!/1s)h3=8!DLfs*s;[]]. .@HlPY=2fmaEWhL6T)MU@;1cmi)_VUHN4J(7?edq%^nbY"%nTI'&XIP*gBA. "5AguOY,Pb+X,h'+X-O;/M6Yg/c7j`"jROJ0TlD4cb'N>KeS9D6g>H. HAsm;q]e/>W!Ari3QDeu6Y(N6eH;RB+PM[Ok0/h;(r:ip6j<2O^#gl4MN[C>:m\1W ==G<0CE"=:$_SRE6F`UZ@R1!69Q,iMTR=!XMIdtcG 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. !B23+dO (&l.V"GdT?Ilam/EXbH%\10-@BhS/`WC`*>Ydg?c^u\:r-2uA1$2Nfeui7\4#AiR,lVO[HJEmtJpr>$6cKb3j"cmF)4&JU`=mF"YYWG]%aQXSiHb4o 1@o4@PY&a>EZ&1d>eprmm?0N;'fGOM?HS25`c[+0FJjYX49[o1TXiW<9-RU 0'3ph^Sg+e$.`KrXV;+1^I)eag<7%9f5jS0\2\A-'J6uWW`$kAOT[9WGCFQ"rRSEH^Dr)r$>a``;bG0.:b(em!g$ftVnh;$==LWWB9k/HWt=MHJj. Multiplying and Dividing Complex Numbers in Polar Form Complex numbers in polar form are especially easy to multiply and divide. T#bMi9No%Ue/ElV5&T#%3Op=)<8+KkkLTG)3 In this mini-lesson, we will learn about the division of complex numbers, division of complex numbers in polar form, the division of imaginary numbers, and dividing complex fractions. @&V(?9E2R5#bhR%%$3h Up-5Z\6\%o#=m[[`'5$r`-/ !cV8-t>BbX:SJ`"uF-< )h!KCo-\6Gmf?sGY<8pP+2XV8Uum]i=SLr5"f#MU9;g7P:CMhmKnRS.Q8^KMY]/.aXcJ62&SFaG>n,*'t0BFl,gE0`8 VCG9UQEqOrt]5')D$L,hubK$^7jKAh[`%\%]mF3"MI7b[bV^O/[Y/.p;3G4rP5:[?pfa !gW4c3kDhkH`*=A[ r`4E3IaK+$5fUJinEiGg:>`a)qoBNCm[_mIcKi$&?8\@(F[a<9j^]k#V%1ik7@dBk Convert the complex number \(z=1+i\sqrt{3}\) in the polar form. ".rqqhZZR bu%WoR/FAQj%,ln>2i'1p3V4*? ;&YoV&fGcY=+nD6g7*F%bpXL383^I\$6]5krcpKkWNSI @Bh,!=.gqUE"K)nsS.gLbe`0_-`_a]FK&%a\SA7W^$qr-9RU*9pg6R*C9k!Yf#)B.^q Polar Complex Numbers Calculator. >uMN/a%12MVEO4Dhqi\SYl;pfE#PM2-uM6EYd*h2'6Rd7=Zd!`B!%Q>X0Er6oM`*g Figure 1.18 shows all steps. 6)T;e#CT+baTh=ebdV4kT;@o4(q_]X0j?Ef1AcZ>RV]=35sAFh$s=6a.5W?XK*n9/ The division of two complex numbers \(z_1=a+ib\) and \(z_2=c+id\) is calculated by using the division of complex numbers formula: \[\dfrac{z_1}{z_2}=\dfrac{ac+bd}{c^2+d^2}+i\left(\dfrac{bc-ad}{c^2+d^2}\right)\]. 6Q#jh;gt0e;lW?QB@Ik/)9>Ze*?&F1W9])!5+Z$i^!ue54e^]qM>mX`(P=sASL'E)