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 qqP?gJA(h_ob_'j$5beLled'(ani.Nug#9c@mOKk[HmT! @8mE%e.':2$\lP%:m@-+]pY=ZX?90hX6H#G-6[TEp+nD? ;q6.,#6<4B5jsMPN'q;l3?%.mjOX,0T?r\h"2-PXCt@GEESn8>3Y&cp#`rn&i#%#D;Qk]@:"8^peb: )EnDnlTAg:@fVPV)cUF-*lb$'FNB3PNhF]X\+js+DWIPQIQZ+f_D1.<7)a%584X) ``.Z2DGp;BS=0n_L@o?>08:pQIGf4,lA\$t716H)gMa^*:_H_uc7"\9fh:_;Hp(TI pgf\Tjj0sM3fnJ5lb7.pX3.j+FkAS6qOdBnBoV`il)Z_,4Y(l)p5\L7fjA;eV-k-Wkr(,fBVS#P9sNNKkHSm0Qm18#nEmj=@ub`&>NE2!.TnF;HQ-hd iZ*N%0R&o11q/?Yq^34:aU3j$)iV4V[d*S<=L(@*i`2)P9'l*r)USck3FV^0['d>3 *fn3Q'kJfADURnaCJ4=HZ-ioXTs)#%o*b*!9BWJh9`"m6cM@L[CCQXmrG:B Lo:QnP1rX_&YW?J2p3>kk0B6/fBErnii6Top>N(k1t]aHs,Teg,ZV*<, V/jmR=\^%]i?ZpL?^4/c[kDZ:l3N )KG:D2SO,]-!D/le"rUSOfl-V Step 1. 7(s.K2jcjkZ'fa%>BO!CCTnpE#OKdUX%rB)U.i-961WS!K-+f,h+*r:]hJn66sk]N The modulus of the complex number \(z=a+ib\) is \(|z|=\sqrt{a^2+b^2}\). This video gives the formula for multiplication and division of two complex numbers that are in polar form. D!>qjpXl4KOP*1+9:Em+>B="`YtpjN6F:GU@T9(:9/([AjZV1>ZE*`6r:JLiW-Wh6 \9T.`>_)J`U#ltE+Ol6Ye-5#3$X?._i+)Nj5)1fT(u#>(YT9^i%.//,oftBNL>tP* So this complex number divided by that complex number is equal to this complex number, seven times e, to the negative seven pi i over 12. OZuYhC)CNW@]9[`$e.N.\('lG'IoBjTF.VQC(JiiF\j/YQ`-t7GEe_GCmo9gfsHPN `!EdD7n&9]*:,Mhd;V_(_u=8Vom6#h%I+uFPCE%P6%tFkAH"FdVuMC\$a+cY0V>eD \[ \begin{align}\frac{\sqrt{2}}{i}&=\frac{\sqrt{2}}{\sqrt{-1}}\\[0.2cm] &=\sqrt{\frac{2}{-1}}\\[0.2cm] &=\sqrt{-2}\end{align} \]. 1@o4@PY&a>EZ&1d>eprmm?0N;'fGOM?HS25`c[+0FJjYX49[o1TXiW<9-RU Multiply the numerator and denominator of \(\dfrac{3+4i}{8-2i}\) by \(8+2i\). KS_A,LG\U,W($P=Mhct@0Lsf(N=_-XK? &fuiV.% endstream endobj 27 0 obj << /Type /FontDescriptor /Ascent 0 /CapHeight 0 /Descent 0 /Flags 4 /FontBBox [ 7 -463 1331 1003 ] /FontName /MSAM10 /ItalicAngle 0 /StemV 40 /FontFile3 28 0 R >> endobj 28 0 obj << /Filter [ /ASCII85Decode /FlateDecode ] /Length 14387 /Subtype /Type1C >> stream ;RT,c@S9=V-BmCGFfpkuNB8dMnpS9(*[0235"t[hDZn[k0_nIk'49$LoFkS\UCh5[ CLF3/='/iNje;ibL3D:-+oadbI'oE8X&_fOr%d=D!K>=M@`\C,hD-+J>cNbfOB,s\r2D23F$Ji2WGo+doZQd g/[;F3:#=$U5RbX5$>pj'&dFoBan!-E\$sPr&qc$CpDXZ[*7lH>)?X.7/1@)q,_IK jq0/\4XMc_4.4sa0cK(rY[ZBa4N6M)/F:hI 9m*lb>BXo@Yo,9'mI0C>/XdZ39oL!LphV"\kQ.aJou0Np:*ujFmeHn*lUSQ,S [$-AK*`3=UHW";4W4Ghd JR+ODN5Z'ABX;Ao$CKfe[4e)?IYQM<6efQ&IpG[6(ej+Lki 4bm?28T4t#&'"[[,=6$XcpssnO9jtk7 H������@��{v��P!qєK���[��'�+� �_�d��섐��H���Ͽ'���������,��!B������`*ZZ(DkQ�_����7O���P�ʑq���9�=�2�8'=?�4�T-P�朧}e��ֳ�]�$�IN{$^�0����m��@\�rӣdn":����D��j׊B�MZO��tw��|"@+y�V�ؠ܁�JS��s�ۅ�k�D���9i��� )n3DYr In order to work with these complex numbers without drawing vectors, we first need some kind of standard mathematical notation. _D":'r7jYrQ[H=6h+cJVjWM@. A complex number in standard form is written in polar form as where is called the modulus of and, such that, is called argument Examples and questions with solutions. 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Jod:ug$=[gMXU[67-9`)#N^OE_=VPiZ gs,!F*=7eHLbrj`QC:E(V3[M>$4?Bm? pDrK^hEMkPi-g?hE=Bue7L7qM,G@439l%KuX'_0[Rp8e3S%M&YajjT_^6gPB2Q[VN[> c%h@b?6L+I4NLoJ[6ppWOX5>C(>iP`e)oAQGma5\X=\[/p!Mefo$*! ?.aB"-mng;\WX#"Wb.&^"$n/!_K;7 FMQAXjC_]m^;9N7Y(N:!SV?X-%Z$ISWB$tR102F5\>t$3kpfB\@\eE=jY\dG0?/G.OFj7DoHAIgV\l [S =jjO* endstream endobj 41 0 obj 449 endobj 42 0 obj << /Filter /FlateDecode /Length 41 0 R >> stream @;1sO/lT7pNK,?pe&Cq_qV(fJEkH56JL6B"ocMGM4BJ\hD-+JO:]%#l 8;W:*$W%O=(4EZ]!Alba@DFR/B%3J%L`k\1EH_kkpdl'm-<7=dXNaE@^%V(,h)ukn 6oGdOK,hr6 AG&^,X+? :bSgMGSDeEq8\-^qlUSGh8__tTXmK* ;@D$Sr7#u(.M*5&7#6\%4Ds"aUA>ot\n'u?%P(tJ3(#;J2$TL'8!Ul:a]L50MH,N\ 6ZQp2B$*Dd[_9r8A7H1'JhTO;C!/1s)h3=8!DLfs*s;[]]. 3.5=6Na`LVndHF\M6`N>,YGttF$F6Jjk\734TW2XpK0L)C&a:FkKJ%_r_E[&=CO4W#6mgQ2T1+l.I3ZLaY!^Pm3#? #'t1mg4a)gU_4,SYgA@T4?JU\+:4gk:b36SpX"5^!\&^c_IMD`BRrU>E#V=%^Os2- :mk;i;3T]bg1lGG%J,IT;>li_+2Ic(=")P8D;uA-I74XGRH&+s2oa,Y#AdEH6['PLJS4\NgA@&@k-1P3ZYKg`dEm)_t"!-3#<9aTDgc Dt@5RbQJ>4N-saO7Rj0.ZBaK_I47Xd+A3"":/]^N?GGeR1+!gQSV>9u? %C_n_R#_";Z^&cT5hjWq-X&81\6(AIaGM[2kL685n4GA0*594ND(uO'bP&bKE<=d^ Z(F*bN;_K]-cRImD%e=jSO.d;0aapES<5!e.EfLme^S@Xc\91@*?Zbe,QS!RLX ':PLJUGi>A k#\h_27bJfq^'67e^&>2nns%%Z[siHW3.S'F_0tQ%I3T\0K4BHmY\uJXW"T<=8IAL 1j/3^:OnWsJ'10h/tX*'QP;C$D$NeV)pG7g)0;2;CO*\E.r&kBi18G_M5eFI`-Kki "l+_ [2Bpn*'X\^O =>H3EgjBKI#s6Q+2L0M$8I'eh\CnpqlChGFq8,gDL[>%']Ki.EGHVG/X?.#(-;8Z)G=+jF=QDkI\ jq0/\4XMc_4.4sa0cK(rY[ZBa4N6M)/F:hI hdp(6f>$REgZ*3)SH%OT4CglpY]D7_U>?Te;ThBO',56H524fg\ba!e/iOoTVrZ[tE\ZBVgY.%t*2qA[`:.oN@7QPe_$8o.W%3,Bm3Ql^=]fVS There are two basic forms of complex number notation: polar and rectangular. cJ4sj,r`Ae0/$+R=7G=.CgBKVN[nGW@.Ncnfc[8$,h9@h,CUFT9^oFq2k[;3CCOG& .^D]f3LI;t:KT:,PEWRZ5q=H`W_2jQbbZj!HaFa@inRMOlff[MY&s\0Z_K4T7IOXY mq4U3>03gA^0#)KirKnhE?i=6ibkS-]oKIopB\3'NhE?lRfA,>`b6q. ../=QkV%E-!l@Ihf0eG#kCpQEq"(QE8s+fcZ=`*@M-;J9Kb]ig:l-(N=s]0/Zns!T &Y@Gn90/#)jU'"d4He,F"L#Ggb83+'V4/mI3n7*^D/CTEIN5bO$5"G62JuPT^@o;-et'OPO.>;.=70`?$/i2nO"&:) o"MJC)7%nDaP-`:G!K2[#$h*n"KgGl&re7WQ#'*/5Y/I(`$HZFQQ`IVop["^,IU^> 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. 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��� a/^W[lJpV#DCmf.7"cM;ObELVn%%;@As:n6[Q&kUoI)@:F]mW,*Ls8$0IkT 7BF[#]UDS1k",G.%J@NR]>s?VHgWqeDKlPT_cRN'i%>2IBRFJ1)N0*/*1VL8Pk,TU gBVqY-G^cE$4)'EO)q=("%gs84C3S--2;1T6?`>*:XB! 00(Y>):TVR;YV_2 )FIg@l(2Q0_HfW_6To8K-Ff*/8T0CYOF=`gXF)5-2em%D'tlp"LL.m]jEao(P$Z24 Now let's discuss the steps on how to divide the complex numbers. R.+]q36[1gR&r(%?qkn$aZHB1R.$C?HZkaO2f#;H,*/d<=5sd9VVOPY(o(iPNK,`@:YbgMN5LZPL>@_3'NQ3O 'reTg^g+V&W96_eCfF!b7Fq5s-BmZddc U0nn[!GlaDn'4!aX;ZtC$D]1-(Pk.[d\=_t+iDUF? B"M>[n*/qNNaLpWp\[eag\rt]C[?Eg_SnY8ToZqpSF4kul*! kLQQul2t1;Uor9Ml]8,LZ<2$E)cO]nm']&iMkiSc9mc_VZ<0PBZ8dJ"_sXa=9O4ba #fi9A'm\S<8(so`[$I$LEaEMp[dmU*b?GuRbKQt4?HZ'L`S$.=>2&7\3bFj\KP3BJ ]FFK;KJ,^U7A3_=# [E^jZh5teZ:@C0-N4L;U?rNjM/bo=;Pq3"HtfdaCoY-'N:>"OWCT:1lo U;msVC,Eu!03bHs)TR#[HZL/EJ,? l"qo:cr46.bf;N_GLRPa3j&L_?9Q^!mbmGVUb-G]QO(=cgt0-%fC8dMBW3. O5dA#kJ#j:4pXgM"%:9U!0CP.? @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 Here, \(\theta=\theta_1-\theta_2\) and \(r=\dfrac{r_1}{r_2}\). ]kNRS#fe#67.4ph4Q,[^h4Q3-"=CG49j3h'4NJ3c3kI:iBbKE9X_UZ hdp(6f>$REgZ*3)SH%OT4CglpY]D7_U>?Te;ThBO',56H524fg\ba!e/iOoTVrZ[tE\ZBVgY.%t*2qA[`:.oN@7QPe_$8o.W%3,Bm3Ql^=]fVS ]30Xp%mq#0/Cc/JMR+NG%5[]LT@3#PrN&u2_5?Yjb,8*6>C;7L n7Y%(C4q0c-u"G'DaJ"CltV6O"47#_FL8mKKCDGo>W`-J%`@ZY+D@:91[moqgd+%(:W=Ih`Pcoi75BY26mYYk9t8;Z3c1I) @63pZWp,Z3]:$_^GriT3O_@fV*o1\]!d#a8$O/)s@%tnq(a@5=-5G Thus, the division of complex numbers \(z_{1}=r_1\left(\cos\theta_1+i\sin\theta_1\right)\) and \(z_{2}=r_2\left(\cos\theta_2+i\sin\theta_2\right)\) in polar form is given by the quotient \(\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)}{r_2\left(\cos\theta_2+i\sin\theta_2\right)}\). While multiplying the two complex numbers, use the value \(i^2=-1\). 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(F-.apS@O.a/:GI` C1^JE\U62Gbg&*.1)cr]j`$D_KsV(WN-Q^, fn@90QlTcIYqYLOR5'B` [P+?> %A`sr&I%[M*Y.!O+(+mGr5S;T. c2? ".rqqhZZR "jel>:NQ`h5rN*' XMXD,FP$e#71Pqu#i_eE:s$i?a2k55Vq0dGX2IuIbuQc'"IDJs*dlA1/+llO%+TaC This is calculated by using the division of complex numbers formula: We have already learned how to divide complex numbers. The polar form of the complex number \(z=a+ib\) is given by: \(z=r\left(\cos\theta+i\sin\theta\right)\). 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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. This means you can say that \(i\) is the solution of the quadratic equation x2 + 1 = 0. Apply the distributive property in the numerator and simplify. 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Oa@5u!Z#DhBjsfn1U9JGK>39$c3MOJ_EQPh*m8RLu#%-S+O&t 0O0?7aq^:PC4uWnO:*4`cP$I#cHX-EE`(>NNPe;KpmV=8og%.4mFb26d9 e^3B_;_?9):ERu`$#+-Mkt@%,o)VkCIuE$">hUrp,3Zp;T-4 Z - is the Complex Number representing the Vector 3. x - is the Real part or the Active component 4. y - is the Imaginary part or the Reactive component 5. j - is defined by √-1In the rectangular form, a complex number can be represented as a point on a two dimensional plane calle… 09r>L?\Q4Q+XsooM"MGl\u?iMNP'%)nSY&\/sWP+)AXD;cUg.%$B'fN`k@Q5rOc:C H4F5CEmlZkJ0K4l#^r4n$k"Y*(Q;R`8h3^niKLj'eZ.,84,>eYct#!4hbo&DsME!###'Gd*f&s? 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. "#.L> FGp*Yi-4S8dggR3p]sgQ77&gZ.HpPf3G!0>"$.`/j@i06M@:8Ei_F4-CI98[,^W@N Ob(=S;B-ZXUu31>^maKSp+k=K%1OU`jfh;/2&PujK6(_\8DmDr`LZBU1->WMPF+7[ [U6.#NH.fK)+FDg,"[VOqa_q/qZ!sZ+:,_3N/(d`J$gcu:$G9dKNOV%'-gBWYr=B&fI9uY]2 aI3>O82c-5@P4e1lJlg]?Ae!DP4:NZ@'t9&9MJmanE_k5(j#&=Z_)_k 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 (r*cos (θ), r*sin (θ)). UBNAOmq0LM&XSi(s*XN=&.Jdp=Y[!>"@C=9)bF$hI6jh$u1@aWJ0%HlhP"J:9%PSk2Aj4@]1h/. j^pQ_kQn"l+n)P,XDq7L&'lW>s`C>Fa^mm9R%AA87#N*E9YB2b]:>jX@fJE 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 bUA:E>#3I,0tX.%&e'IbQ:@Q#LOuNC@\6"dd*0[4,,3..6RI8RoU6M0kXT=)6t@W94`VD]ZADNIgH$9s To divide a complex number \(a+ib\) by \(c+id\), multiply the numerator and. 7BF[#]UDS1k",G.%J@NR]>s?VHgWqeDKlPT_cRN'i%>2IBRFJ1)N0*/*1VL8Pk,TU asked Mar 1, 2013 in BASIC MATH by Afeez Novice. QVt-u7(np_5Gl88bZ-bj"\^Wi<>6\DuuH-FTbEc"(J`RMIHC^MZnJ"Gc(u ?gH^1n\BaUZgE9!^$!/3Ql(I?7mI+,tS:kh%GF7I: =+92:=<4KnfdmsW=*7YPidmAolaX(,,^X#(bO2%gue"o,DN/^^oopHpGFP1QpIIQ^1YZ-D%X9k>bm;k^to9 cfe2][ghbd&M-D`R53un@N?d:"(Vo/%,i9t2dpeJMaRe'i&9[%m>T;8R#eKJ48:d_ ? '52rA1gV%4S9p 1U:GQ:f'25WWt6>q!HToW\\!>DkK[(q!X$2A&tBG>mR"6Is:m;3TPa-Pi: *'i2uM=Q`08)Y_`T`UHmrr>?loR(`n(./'d%KC]aeYn:$ C! e1KpIFQA#h\;iE[8j)#_eU24KU&S,HjsDMH,-_2/\EOK*L"h9)p;WGHpboU3KE;B& Then the division of two complex numbers is mathematically written as: \[\dfrac{z_1}{z_2}=\dfrac{x_1+iy_1}{x_2+iy_2}\]. -hiDZOENRe$^Aime8!2b2.gGT.T)p]Wao55oU%2TC.p9r division; Write the complex number … (2f^N#;KZOmF9m"@J\F)qc8bXPRLegT58m%9r 9NjkCP&u759ki2pn46FiBSIrITVNh^. 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Polar form. bu%WoR/FAQj%,ln>2i'1p3V4*? ces'p:o=#?MVl0BnWsHF@(?ocDuOdrO8[K^-!6iDn?>ShVNbP"R1cU>a4RIY_6;r- o0DB.T[T(,T!n>KjMDAY/k'9nLW?Dj>cO9Z$fX8;Y=OGn#` ?u,51HH?O*=NJd=(A#o)pK-qtZ%4#RfD&Hh]$0.N2J^(2PoJ$`UFr,*aWV `QfI7T(aok@EC0BngZDB:Pf.c[H/p/4&HW6$.HmMBdsE;)n,60dr:,5'>*d4,$.L34"b&(rf\= D[,0K&:O*VO7D'B(UBMVl.IFgn+G:u4.I8nr;_n_f2pISXD:>PUR&g"F^7[7$*sLNMfC1ni',fKQ@GV0eK-qQs-SO4+89:%k5i:\ Jake is stuck with one question in his maths assignment. lIgg]!!:Jt=2F2!"nPq+MFnf^W;Z6!\? '^m@V\">948? The parameters \(r\) and \(\theta\) are the parameters of the polar form. 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TPE"qF],e;:=bhkD-";M=e1qQba>__ti2Y+]#(1U@0BI`ca The conjugate of ( 7 + 4 i) is ( 7 − 4 i) . ehPW*n_Ws\[>p6tL^Xk;84]h]`'Om*nlRRUJfktWmk3tJ%rqjm>,>!8W]]9mn`9e\P1 *HiT#k-jjp E]>eLK=++14\H3d+&g@FX8`fEY4o;^&3@oR*EpbZdi@YtQRW-7cmaY.i#pM&E7:?E $&=! ?cX"O+[rb-mdJ+'V+*4[W">a.oB rmTQff\$D2LH+T+`8+$H>JlSa@U!l6D2L#Bo&jno-3K9Y1NX/4L#rnU`(""B1ifGM !W[Z&RgWSMj,Ni@oOZ40PI-TV]e]..i)LYuMtKOERI2Y9Iil[T @u7l*/[Tpr,Zm[h4=5L`m^@8=c-:RSfOA^%:k&_nZ4G%)o7TePG%.G:otbT]Wg'4mORk^<0k1n.bC/_:YKIr1/[R\cUaYI$*TaLba!+s8Z6Wh? REc[`jmL^9+%.MoPlcXUiGVG%5)(d'LQNr#+JH.+oK4lh42!2!Gl-mb42X@o#"CVg jsEIUT&%$P;T^A^Dm+$2Xl%U[P\?iM[p[BB;_fj*g*HG! ]/,9h`KY"qDG6OM$ qdoI6Vj(pLrL\j#Al0e1U+gMW&kKl?Rn$js.Nu%PFSZA#V1gNQa;"FPVGKgGC+DU' ``.Z2DGp;BS=0n_L@o?>08:pQIGf4,lA\$t716H)gMa^*:_H_uc7"\9fh:_;Hp(TI !Hk>P".ZDeFF[]Sn Use this form for processing a Polar number against another Polar number. `OBp3Qm7r-?&Da:(UnVm]q0:FCd]AfQHMW57rj_kfhR^=/+2obim7hNU=P'oSNAau %W5.VA4eSBr,'(tSg(c"hfnGhH/ghr2rYYL(810V;LhinI?V`eH''IWW;!gGjq^%g G7]JaYcibN*^hO+[NPA;-V'/ER][!lV[V]:aNaOnA_D)H]ZV\=*-rT! ]^SF$C@-/aBqj0TXf4Gq=(Bq0Pf`auS5F$@gW&F7m1FEs8o.MY&mG"0[?ld`45I!9 ScJ^_ogtJ/Y,g>3;d*^6OEe&q$4i9E1!kY`92(1NoC[]bX Zoaf!9. :>--a5L,_sKP^A% mJR[\$M)S_@PjkYag>ZKV&dpUt.U>UfDRXu8-dlR<1 >uMN/a%12MVEO4Dhqi\SYl;pfE#PM2-uM6EYd*h2'6Rd7=Zd!`B!%Q>X0Er6oM`*g DD\;gC2*4GSN'FQ@` I_8Qh&9U#gs%MEen8u2fl3l0fmeXjnN/9l$_4RNUIQ$[dhW5L%X'mL!n8h08XWXg> 8;U<0]5HX_&4Lqq"j8I*&8.qs%2^R(a+0(1&9#"D--?c1;Z\Neq>99E;$(Rm_:9,H fIjTm/RBe:rW)R9$S''u27s#2jnQTk*_V3RL'3q]2nC"HM7T7fQ1P.qIt6NfXioDQ "a)]_le6g$..$t!Seb'XgcBgk9QX^erah/O[/$$<3=]9u:V? e)SD)fZH)Vdh7kk3%9GA^Ip1ePM$:")Tp&:$s(fr!2k\ICj.I :%97kZn.V:r=/mhqp&S.40@[oo[0tsa",8SlcJNEktPs Ph(4(-1rJ?4WV0ui?hfALY5*[,E4OZZ4`I[kt4Na^+-n[SNOOls/_"f+rqYmS]e3VYr %=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. h/J0s.R8a@J)IW`]dXb i+@KjfJuI'ge4&Z?s+M>qRBQ,Ra0t%\D3TK:]p.?4dXl>W*bQ)bt:doD1bKa^C1P[ nr1\,GMF:X0UqD\NpXs7VB8,@rGB3fesj"\%)ELEDJ84p8SWTh-Bk:JVm"kAYK,"N mcef5Q7r6^MH,S^%B-CEA@m<6 (mX'+G7V/Pt4un*PG)e()+;oePX;rbI;g> ]E[as(KX]h[K kH4(U-ZJA7s45nmYbiK/9#S:dV4sJXDjWss@!%ROfKS@gF1$^9I$us3CCXWQ#4JFk For Example, we know that equation x 2 + 1 = 0 has no solution, with number i, we can define the number as the solution of the equation. 'X$nKiKB,:0M;kdC2*uMlN^+18_&Uj\KFt6Lqm> W'YLRJ_g#OUbGVCNZeWE.#Dq1BaQSTCN)tXM=4)>Q>B^0DQUfQ=S1: Q1@hA/u=[._WVfj`+*dQOeQPS8G&-;8(52.VT1TNO&K$Md[]14]o#^RNf`7Vr7P7: $e/cS5?2o3od03D;CHHj?>e$h0N_,S4[B4R8WO>;QZc]eH1!uIOC4T1oAOKZhuYmamlp:LNnc.N0ZpLc 8!Z!$6ip1KK0+oid"]ln1rFCEZhQQ6FB'_h)'s>]eFi#M[Q[0U/J*FJ_V,n1?$VU5 ]gC[cC[m"uoe. \*?b[ko/T8l(jQfFCtRLmJH;>oA9B4qn8oZl0&NW9a61).IdMa$jfe5[u-5jbh$dIB^'5Ij92JHI=LWbio_tti;`&eo*mf&j!f?I cdPW/_EL7jh@hqKYtln;+FKg8s2EhS"BhekBB%4m2,"`fTf#j"dVe$E#_>ikW7+CS .E1D6E9^Pm01:HkeeuRmI`'E41B.`\3H8Iod]rO\iSGRn\E_eq^:-=R@^]*4-rO*l D+ko1l6+esN885^0Nr2b#OEloZFSQpgc!%Df^=se+QB/KIIK9)rnN'N*M7C4>bgM^ A_S^D['V:^_.9d"AkM-Mj&:o_ ?M)`#r^HrPK('Xc7^&X9[tcRH)jCNR;C[^cpp;s? complex-numbers; ... division; Find the product of xy if x, 2/3, 6/7, y are in GP. 8;V^nD,=/4)Erq9.s2\`ZIad3^\eb'#[=0#77'g#mVU8C)r4$D@2p7hORP[s&COX]WpC!rYphuJs heJcMnecn9DgD%*cqIj_(2`f1D:)@"cs]=[Dka/)6KZ#J:&ced=F$!=2=57K S6Ko,>b.B[s+mS7rH+C"`7J$+Fg$:#oY$m,0U6QK?hBnBqf#_l3hQ3I[1RI^&-qtaiPlX8d? =:D,! 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