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L$($-]/MMN<72NWTK.qToPZEVOh-jUuQu79SGS[H_j9eDnT6[EYA8R3hCGTWh>kUkF&36%FhQ^:?F OQOs'LZTt-E8EYT+Mj)t4@e2'(Zn. 146FVbogZND+Rn12](cBKem+ Ob(=S;B-ZXUu31>^maKSp+k=K%1OUjfh;/2&PujK6(_\8DmDrLZBU1->WMPF+7[ e2$_EES5B+;GU^c.1ng5M>1sQrMJqgOpZoEO?o"(&JD:oH:B.0mAQtF(KHQ1 !B23+dO . qL7sQ(Om1u:@qraB asked Dec 25, 2012 in PRECALCULUS by dkinz Apprentice. @/;Kkd_s+lE'TD]pMN'S-;p:1%f>_^'\pVo?f&Hkke(=u 8;WR0HVdXb(-[M8LfRC&W$HV+I,M'"#(.-@MF&iY'Qqs^C1lr-$3?lP&r9F+V8[X ]:H6[@3&qr[AIb-hH"Z,:%o_L1gHm@(UrSaqC?Qf h:[%17W3X!d*+lKaZjXPKbo)dl^4C"h+\;8=e'u867tI:.fuB,HQj@lFD^ACd$\g %H=PNY]$o+L@Pq952CdlC@%Geck+F;q0FgO_@rp"bI+CFl%GY]G?p-6kgc0!GEWBPj)h)<2N-gP> *5<5N4;u*FU/LoL-tO99P(@[rWV)[5b>qd-L7_"tN(@l#$LiAYpI=Mh^Hdhh8#%]-lSs!<3Gj_&t,q!a/4:0>V&]ZXDFq&(!*o@V? ']KXmNPN.\!\9NM&SpaD2sIEqU3& 8;U<3Ir#e])9:V^^ANL,L&jAID. 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[%N?\5@Oc"S5),/u^"qlZ&oD,9k6N"CPo2f"(6cJS*cdA2d-#VT-ZU\t *VNg"J/R;'$)KG:D2SO,]-!D/le"rUSOfl-V OW!F*1LgE^Ru&[GokJ>/^7J9NV-MRVl,aAQjMCNPUnW1q>^\f<6?5B\Ng>6R ]FFK;KJ,^U7A3_=# %W5.VA4eSBr,'(tSg(c"hfnGhH/ghr2rYYL(810V;LhinI?VeH''IWW;!gGjq^%g D^h_WgCGa5Uo__&5r?k-DqVVYDYj!1@W&AB.@_DMJ/GNV(rJH_0ae5*#SfjWA;4F0W,&,SG97(XCY7%_t%Og)JulZKbr3STCF-7_@-t47U5iDorO? Figure 1.18 shows all steps. ;c8Y Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. 6/!1bGeLWW?k(('$W0P(kb"RsQQTh;Fk@/2P),#Oc2TO.,kUE_1J2FkYj>Xu,HD +:I"=7_2K4")/V^D7:6]n8GAI?IZ+cX]rG=X]\9k+Ya:"67iAk)[TC#YWqcZ])F4 Jolly asked Emma to express the complex number $$\dfrac{5+\sqrt{2}i}{1-\sqrt{2}i}$$ in the form of $$a+ib$$. The question is to find the resultant complex number by dividing $$3+4i$$ by $$8-2i$$. L!.i)!%A3gn[J_"FE.E8L2$mq4:/DeYGRH"m=C>Y7Y+mLe(%$igR&c!j[o*=r>[&P 'd"-(\bP#T"hsbH6Cnn:]=-8I^VCP]l"h 2G/0D"^&G-iUpjOiP4JN(7REEhRCk1O9#I8EYiO^-fq%DbNK^kWmT,Sh#f4lBQnH [?mBOp'"?nO(SBTO.RFMl&u*8Ve\@HGjX),0-=edqO$bfR#BW=/m,:EPj;S5Q%O [Q0D%1nm 9V.k]P&*p;-''WO>e#-Sg(u5=Y\pY[%8k1e!S?@;9);Y,/+JV4E]0CD)/R>m_OEB.Q]! 0.b*cFZk(m8,>]^PU-_UP8QHO/3a>51a=L]?gdt^^29?#ZZ"5?Mp)]WD7s6ZG8,6.7LPuN (=!e#X(.r!^5ac4VWLg@VWls-nk1jVQN%A To divide,we divide their moduli and subtract their arguments. 8;t%>Qoba81Q;IG"fo6RPIRVQ>gD$8b\@BAH5*(:h#3@;#(KajFEFqg8(,EHgj1 F1WTaT8udrRIJ. \begin{align}\frac{\sqrt{2}}{i}&=\frac{\sqrt{2}}{\sqrt{-1}}\\[0.2cm] &=\sqrt{\frac{2}{-1}}\\[0.2cm] &=\sqrt{-2}\end{align}. That is, [ (a + ib)/(c + id) ] ⋅ [ (c - id) / (c - id) ] = [ (a + ib) (c - id) / (c + id) (c - id) ] Examples of Dividing Complex Numbers. mJR[\$M)S_@PjkYag>ZKV&dpUt.U>UfDRXu8-dlR<1 &'&:+B[4Q%[H7kX89_H%Rl.SR:mW9dmDe.qRAQ)YWP5$V;9M5c]s0koQ1-0G.=8 mlHs'jJ%A'MT[(g2VQ$mYapm%h oMUdq\@)_P^!.e#DS$7Bdr:%ob&%VFJY^_iB@ekTM^7&gUX/K92Haj[ua19jBYW)fk_-p>($2TBF<$$roHZ*^W0,MU@HiOdEHG9[ff;GP'HE)Xk6/H[q;Ice[>)Ep4(Mj9l.mm$#H]$Q2* nnctpY.CNmOZ2sS=qSmNqdEqK2QQdf:rf/2b[DdWnp*L]r$YR:gVN@et#P",k^3I ���fz�����{�w�����Ⲑ\1ι!J2�9u�Xe��N�ɬ΀�[����bt ��i�7"9gQ9� �!�"�w��g'g��'��wAת����� 2%Et��jNά�$�ސ�Iq�=9K#|�B��f ���rd����MKτ~b�����8패�a:ۀH��!pD����XI�K)��â�൬<0���:�[f2������M3-n��$mL�h��P,��)�1�2oml�W����zzq>�]O�j(��G��$OM��t^},��4xE�K�E��Wz�8?Z�m���t���ͱ/��b�x8��7ͼ�"r��:A�=S֨D�p~����7�H6�T_�Rj�q���Xì0.ᬷڝj(���v+�%賴�j���7bc���NJG;i�V�i���!i\����y�o��N����"��o#��6�ں��G켥�6n �Ơ�-�o���ˤ�t��|���TVT�6��F��蠳+� vTp�3����n�p�a�v[��U5Tx�}݊D�m% :���[aգ*�v��^-mm�����C�Z�$Q�K�*���O��� "&@4fkIiZoUaj.,8CaZ>X0:?#SZ0;,Sa8n.i%/F5u)=)_P;.729BNWpg.] . 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Next, we will look at how we can describe a complex number slightly differently – instead of giving the and coordinates, we will give a distance (the modulus) and angle (the argument). ++G:A4poLn#I\"U)t7Wf/*=&NEq*bgJ/[ud'A/]AL@>Qb0#?j]%9,S-@Ct'oT?p4L ;cUI\3q.Lb-]4N @'Mc@JsNeBQUWu.SS&Vsg7-cKfh)dfOJa,3 O<3."s4RtY(16?VjAX.sm>qj5Z6h4'HgQ@DN-I^?Yl. 1. ?JS2(/b%?BDj=.&aVSL/Z\TB0I;A=4&@t_BTN#!qm<0h:"uK>EZo!1Ws32%CXTahjLZ1 D+ko1l6+esN885^0Nr2b#OEloZFSQpgc!%Df^=se+QB/KIIK9)rnN'N*M7C4>bgM^ A*ppNQbMe>0k8o=:Ue-F\FRcdF%+?FJk.IS>CVPrF8='kVe';kj0WnYuJD;.>l\% [^gd#o=i[%6aVlWQd2d/EmeZ The conjugate of the denominator $$8-2i$$ is $$8+2i$$. KVJ^6qJD"LL. @P=7gfuL=aK"US0;jXbH"cIQX)I*NGo ODp!7ddDR9a65_cV/jmR=\^%]i?ZpL?^4/c[kDZ:l3N )S=K2#tApi"H+a"0b)r @W%\p@E!rK-5sq1[ACd(V7[FlHJ2jC&BfaO. 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'_fb(H=FG[g^HZ\Yt&Uon3hnS)_EPKddl;^a^D%!fEkY-&K%f$[s.0h8"t%mq%[jZ8F$/ILR/@NYNNo REc[jmL^9+%.MoPlcXUiGVG%5)(d'LQNr#+JH.+oK4lh42!2!Gl-mb42X@o#"CVg The graphical representation of the complex number $$a+ib$$ is shown in the graph below. ? VoGXO1m0E9%,BN\ZG-qo1WX-,'Yh6Ed\4kIeOjBQMmY!#M!MR,mRC,ljAQb.+@c! This can be written as $$\dfrac{ac+bd}{c^2+d^2}+i\left(\dfrac{bc-ad}{c^2+d^2}\right)$$. .E1D6E9^Pm01:HkeeuRmI'E41B.\3H8Iod]rO\iSGRn\E_eq^:-=R@^]*4-rO*l When we write out the numbers in polar form, we find that all we need to do is to divide the magnitudes and subtract the angles. #"DeAFq%=KJp;YL9@6R0BH\5_<=Q@rhIh61a-roSp=+^*mSX;ac9J6PaXP\?t4#[ F? The absolute value of z is. \begin{aligned}\dfrac{z_1}{z_2}&=r\left(\cos\theta+i\sin\theta\right)\end{aligned}. Thus, the division of complex numbers $$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)}$$. While doing this, sometimes, the value inside the square root may be negative. >j;qqG'i'[,*gcA4VQTCgtl9Z_>'rR[^n&TuReu\O2F?W'o[6#?&.Pl!O2$V->:+ L=p66-A;#FY?d/ik@P4M?1OMO*lH#2KtF6OS.a,02bOn+AlEAb_?Z;a8f'Y,0qtq 2_$hf-[KZP=nKn)pL6nBB4D$RGJs3qV8kUUhi8dN#YSi,S<6p5dk(@K(DS*PO? ?MHW=p!HZ)\\S]_naH'6 Complex Numbers: Multiplying and Dividing in Polar Form, Ex 1. "e6NkK[W--U$6efQ\f7_,bNnqBB4*N+1FMd9&-4O#g;/G6Ab4Xl,b]dbY/(fKJP fIjTm/RBe:rW)R9$S''u27s#2jnQTk*_V3RL'3q]2nC"HM7T7fQ1P.qIt6NfXioDQ YjB'2ThF0S^pTZl5m*X#*_)^F$!biq%(i8hTg5$6E:a_2kgA;,ch(H+FFmjeqp5 %0_(aa[PG'<=]-QFRIuqKaLVnYWlY>,)6FpftJ/WI\W+&nrP-]Hg+_@b;R_T/^q [!+%1o=mm?#8d7b#"bbEN&8F?h0a4%ob[BIsLK Write the complex number in polar form. Rs'_'>t'+G4bGo8DR57gg7PIQfeK@6bkhO%bq>Xt]+mga*MIHKba,W,Xd>51P>Y"F aU73TF:sJl:UN@cp7*YCZ*p^L^4cNhi6onSSIF>" Z>:tKkns"U!TUC/P[RA. *Gfh!2$mpB80:$JU223XMI2tU.jk:K(>U+4u2f :pgXIsSaTY5m^\l bKeDlQ]NhCpi!M3ig6V620Qp12O%5cX%f1pbN=bK[e_&qZ_,PgP>b@\!#Sh^Dq_ To better understand the product of complex numbers, we first investigate the trigonometric (or polar) form of a complex number. Example If z G'.l7hI,;pNkL1@ab*_'R.1r"O0Ybh@b0*=P8W5D[@jS^ZU-:J96=Bi[h5+=Sc;AR G'.l7hI,;pNkL1@ab*_'R.1r"O0Ybh@b0*=P8W5D[@jS^ZU-:J96=Bi[h5+=Sc;AR jq0/\4XMc_4.4sa0cK(rY[ZBa4N6M)/F:hI a^Tf@FUMq!\qXJG@2a&\iRM%$$QrL]Rh/Bt9o5FiQ4US9XEH0Ad=0,#n6NK!ZS%ln They are used to solve many scientific problems in the real world. pQ5ooG'"brA+7XE2T1mUJiRs7D_0XqtN/75;5>lnof89Pm.? kea^Bq!=R04a@4^Z/',C^r"kG'-RNFgtiipkGOck-UT];mt"RDjd6Vth]G,TGf@u=r#q2_u[AG:_fS!3[)fhRm;]%6cJ$.dO*TKI:p*B#2e\nu [lRt'clmTo6?_XV]QlO50%8:4R0'V#>VR6g%"9_O?rT5-HH'2C?X+(0Z! ?rQOifIGuZ9;hMpQ+3\rM+6pd=X9?sW!ZYT@\UB>6(u:o.B3YZ6-"FI;B6P_-@ZJR )ILY]ddJ(3DY;iOR=C2)010q6/tVN0hXKeV@g'B4?KOL%uWR6'Xha]JY IdkTcTCmF*C)n! @W%\p@E!rK-5sq1[ACd(V7[FlHJ2jC&BfaO. *VkoXW4CRL')OrMo>3IAd"*aEu#sJ[E0#Q7=sIQJNE.4Q:" 8;U;B4A4\'\rL!DbSX]EKM1=@Wh8JB)AQjGlZ8226GL]%%m7-KY8ah[N^mZe We call this the polar form of a complex number.. *VNg"J/R;' !1'blG),.\f^F4b17FQAJ%q!gID26e&MmI8V*pj4tUgn1]JNRQp [iKY'b7duG3;isOo)[&Y'g &+aa@&)lL7&Yu=u#R)&!%kqrDefl-:Ib,fB8G^,! %L1@D"S-W?QX7C8/*"GN0Vu>M#nGbdh_G"l\*!Y.gJ639Mp6@>6b)(q<6"#b3HKH_UJqA!g*tiubXpYrWrA[K0tOJ2! Up-5Z\6\%o#=m[['5r-/ Modulus Argument Type . jX88LS\/KGp]'G.pRnIf4-#YD_5hG)Nb"W(YFZ\URS%'IBS'P;j/r28O.ksX+?-V First, calculate the conjugate of the complex number that is at the denominator of the fraction. ;[B3E'McuD[d61<=f:uZrM_iI]j8CLhFb1gYhSm,;CPVD ;+Ld-?K.%kt+/&*2#c*;@rsZ87bqTbV.u2DGKXeKWAj7_\?BNL[Bd2?WU?2> nc3%t0EFu[J,oYk^[l=FJ9596NZQ3:OYpN0*TN&\,@1QW,S!JM?qVE8=1=-/0^M 13Y/[-HN;_;l=8D'Uc87BaK[@;uhfG5bSp;CSBuH/3! eD7A%FTDX9=th&3MInu@#Q2aIY+a=oUgMQ)CcSmh'Vp&\=^s'^.^s4Y2Ur %=23[_0&Y/D\cf2P8b_1O]\"J1i<9@iM>-B\^SFa6B8II>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 Let us consider two complex numbers z1 and z2 in a polar form. (qqJUVsjk: .Z2DGp;BS=0n_L@o?>08:pQIGf4,lA\t716H)gMa^*:_H_uc7"\9fh:_;Hp(TI o.Y4;]I<4@\fZhl>m+@]-pqIhS@OPhfmA!.Baj7*b7;YaGZ8<=%snonU16.X,.2j_'1&ojVj#@ >Bte+WC;52dshh[G9>Yk=7G4D7Dum0ZRm:;^4l2plZ?4HZ"Xm+44jl=&B1+Q_q Oa@5u!Z#DhBjsfn1U9JGK>39c3MOJ_EQPh*m8RLu#%-S+O&t mRY*IM7nP=)D\2_6M)Z,'>+8#W)Zj? *HiT#k-jjp YuFpJ[&oeXjlU,_A^&^?XraB09^/452+Fk"%PFm@A:t8Z&nhN\Qf"1TZEaEEQPE The conjugate of ( 7 + 4 i) is ( 7 − 4 i) . (FO]m,Pa890b&qdANUjjJH%tWG+hCUm8#s?96O.QXNK*&7m*fgYO+@f5 While adding and subtracting the complex numbers, group the real part and the imaginary parts together. : By using Euler's formula e ij = cosj + isinj, a complex number can also be written as WIC=.3Kg%0q=Z:J@rfZF/Jn>c*.sY9:? 8;U;B9iEr&A@Zck-u? \TaP>I-g^IMo"e!Smm.qU4;P4qT;(D'--8]J^^RG->R-=sa,?VoO@Q"#Mfa4 !i4krC0YI!R jeTl1b9W@JR@_QcoTq=*054!M/[T>E9al,o>.6)QQ/OHrNQFQEh?XqIPrI]J59 @Yb,As4C^TqW3A=:6T,e[dh3jkGCFpI=# !_a)3kKs&(D.]? !W[Z&RgWSMj,Ni@oOZ40PI-TV]e]..i)LYuMtKOERI2Y9Iil[T ;iS+VrW[+I3Cl^6e4-N/s9hu8p&B=QH;MRh)RWMZ:O 3.5=6NaLVndHF\M6N>,YGttFF6Jjk\734TW2XpK0L)C&a:FkKJ%_r_E[&=CO4W#6mgQ2T1+l.I3ZLaY!^Pm3#? *F :mk;i;3T]bg1lGG%J,IT;>li_+2Ic(=")P8D;uA-I74XGRH&+s2oa,Y#AdEH6['PLJS4\NgA@&@k-1P3ZYKgdEm)_t"!-3#<9aTDgc \*?b[ko/T8l(jQfFCtRLmJH;>oA9B4qn8oZl0&NW9a61).IdMajfe5[u-5jbhdIB^'5Ij92JHI=LWbio_tti;&eo*mf&j!f?I cmVM0-jnl92hmKb=WKqdO]O7U1>2C[2r_"-WjIQc%i"#e?DNqgJbhNl(bNd+/:. n-3#mU)'"2&CD\Ui[X>He[Be(=C&A9T_5OsfYt6Z(FQY+.jn6Z*Uu"<7Mj>uVMI'jJ2f)%2;QA/Chc&rBb@?a#Z!#5& division; Write the complex number … nc3%t0EFu[J,oYk^[l=FJ9596NZQ3:OYpN0*TN&\,@1QW,S!JM?qVE8=1=-/0^M j(Zf0ek&YrRp-T"U[7eKd>rS1+(jKj>spp8t%'q-gI6S0TVWMrd[9I4G24mMOp kL/Jg4Rn6u )Z3Of/(:+N\V1uUHO4oYdW33ERV@!<2)qm@9=t\8g7aJgV]mECf+A3gWia8S>EX Then the division of two complex numbers is mathematically written as: $\dfrac{z_1}{z_2}=\dfrac{x_1+iy_1}{x_2+iy_2}$. MiG:@#. 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. S)]jgDHa=VdkUq57Bn6Y,ssf3"GJ?hs)1i0@Akj)&V**lic03%=kH(tRYV-*#JZFaHhlYlmB5g_gcAJNKK9rrYcC]l43X+uq?= @6G5%V7m^ 7kIlC##\'@nd9Iknor^"aY9a*JhEtG?Fh?2*T2F2iX5mCqXt3!iq,QVVYu6^N^L ,j-LIrmRXuEm.Bt1Q1IY*m9f%W;n\%@nO3k-GM[cnrL)QqZ#k*tAR@3V\0@TKR ;^J[(FQd>_''Q74K%=&AV\NA mkErH_Ib7P[CUML-uT)#9Ktk:1hO*9#^MkI+9_BRPTlY"Xt18@(Nc9Y/q0NgifqM*b^ fH#bV.'gUqG&%O]nB:Ol5K[W]q&W-*D5Ju]icF187_-S&7,/#S9! )Q>'q(iOJO&5EJqN0SMTD^P1*o(gP0qc!BHEdGj%AmG60dOK]0+S9eR_*%hOo9Ps X8lBM#"W1G.%;B^M]W#)ZKOWUA6B_l:hRcQZ@W)*rQVBgRN"?! +'"Flq8,(][_QN-5u'@DYGKP! c*[3,1>@-bVbI2Ke"kq3["oL&Umbcc"S-ArGJ;W4j62.ieI;VT.0g&r[s4p%FQ3DL,AU2N =+92:=<4KnfdmsW=*7YPidmAolaX(,,^X#(bO2%gue"o,DN/^^oopHpGFP1QpIIQ^1YZ-D%X9k>bm;k^to9 dUX=3[S!aFfZOa5IJ&_ie4n9( 98j9JB]Y78,=mHVR*^ok:KokTj0[KS+=^"Egp30eBqng+djBgH.BZjX.SQ)03\Nu4SV9d0>I!.ld\:t#3P7MH Separate the real part and the imaginary part of the resultant complex number. "2^;9Vr%3u_6qU>4ja)PB0Ks/S0QFR ph*p*_r>12?>E? @V7!hcu/,&T:h^)kC9c]3@Q6l/Y8U(mPb&s,A9Mc, He gives a few hints to his friend Joe to identify it. @,!r;uH*(!T!#t!Y!XI'p2[]6YBB6CJ6[%0- pJ7uJ^bR&SkH9+6t#;qKNgc(i30rhXX:(UnXQ_[>)ObTeAi"aG"gq/lT9Ob]O7 .n";Or!Db_Ta#5k7AOkbs+Iih;(%:t/2%8#U8-.#^5p!=mCPe;%(3!8dXrXj(lCfO o"MJC)7%nDaP-:G!K2[#h*n"KgGl&re7WQ#'*/5Y/I(HZFQQIVop["^,IU^> ?H-'Xn>FOthptZIO@j&QWrBQq4EF1Y67,-*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 While dividing the complex numbers, multiply the fraction with the conjugate of the denominator. \RI^.:XFuQi2T!)n?*. Multiply the numerator and denominator of \(\dfrac{3+4i}{8-2i}$$ by $$8+2i$$. "l+_ @kh;1;Lsu\p9.nfGThqT'cU(XqXDn'+&d]Z=Q*';kD C1^JE\U62Gbg&*.1)cr]jD_KsV(WN-Q^, fn@90QlTcIYqYLOR5'B bl..)Hd;GXhu0*emd\YnMh;e#+YPq49!SF/XqikSJ3@%pT7ZLNja93K:]iVJ(b* +?#Qc&jtr,1-! a7dc6pkG>4?7g,::JJSqLeY7,KQc>mO"coDKL6=NESuW'.Fsf448IF\hA5Plk6MN ��5M;�Ig S�+�FY�F�� 9r� �!L��d���� �E�kZ��8�4��~��f�����]�)z�i��C���8����< |��c�v� V����� |��6�� U�|Z endstream endobj 36 0 obj << /Filter [ /ASCII85Decode /FlateDecode ] /Length 17294 /Subtype /Type1C >> stream hRd'IG@6In2tHu77hWBs+3)+cF@UUDt;Dp;JBG (\M;>2i[^SA@rcT 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��� .@HlPY=2fmaEWhL6T)MU@;1cmi)_VUHN4J(7?edq%^nbY"%nTI'&XIP*gBA. !9a)QR[=3'PXmk[Dk5.C[g_#r*_#i+>l E3M35E8)&B+5j>b.^8r9:lX,&;l-jal"6Q6I@E=,(A"jcB1qBf]3"]@HP6D!ues YsP%Ur"!ZmC/us/;FU.b";>+5e7MmiRb'qTdB1Kp?PR1r;A. 1d[H2:ZhE;.XAa,q9W7S20T("@0F2-H2+04h=5U"kp4XVe/X8H]u Ku'57VoE?7KCBUP#5cbl"dYPng*[GgZ+,o(N\9U%(4I,C5WuHMfB_"?? 2(N3'rVV-#O)sabc8h>B6?AdaWTsbhfcFFXU!B>5[C=o_4Dm*efgII9.k5],6LqEc F?U.Ih=JIe#o/g/(@p^HU(#LJ7#:,>A[m#b45['P/pnS_;jrlqFfhP6J o7I8s5;o3c)nI#[1/jdF(^_,+9dcMCc'+1d,+rel3@d%AV9**hQN"p;ehP\hEaN R.+]q36[1gR&r(%?qknaZHB1R.C?HZkaO2f#;H,*/d<=5sd9VVOPY(o(iPNK,@:YbgMN5LZPL>@_3'NQ3O ?#R[0.s'Dtu1e7;0GEafjV+=ab))?P+,@a_+C>!n ".rqqhZZR 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 … Q5"ZsFc,ee]*W*JggMd59Ppm7EIC*RUV>cDX=q5CP#^hm')ZW(:'\NU1@G88U*p '#Bt,MF8SLl#NeGU*].+0@Ft9.D>mOt)WaI6HP1W,1T>KXcQ>i- \fA@a"&KFJVYSGK;IBdk6Q%*]@t,ST'AYK;)+7;LA!BSkXf@hekWh61++a-R/h\ 4jm9W+nL9O&YnLthI6;elS]'qU!NSRCk5_b\5C(fpb)?g6fJEhiiqDL3;KV93;'C The mini-lesson targeted the fascinating concept of the subtraction of complex numbers. _'5jGO'lG3R9Nr?\-E\ON@roL14]G:3? 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This is an advantage of using the polar form. 7(s.K2jcjkZ'fa%>BO!CCTnpE#OKdUX%rB)U.i-961WS!K-+f,h+*r:]hJn66sk]N MujH*s87iE/%\U=6T1>;UPLF'9VrAF&kl?C3&2FRmlr>jm7%>=5i,>?/BYt:Kkr)9 >2HcqA?0^>Vm350ZG(RaRDFPj0%_52DoAdMJ4PX/?/QO\t[\B+qd%\o-8WlJ8bN !cV8-t>BbX:SJ"uF-< #G(QIUMd7;kFLtEDd5Ye&u9.Np>5%,IdFHA(j11RF?Yrs:-pd^ZP9B\H^>-B6 '+jq)Njim*StCQh/6haCrqfW (_pKuS_[&UN%h;^mgE"8#"hqYtXC7VOIu_VX Solution . cmVM0-jnl92hmKb=WKqdO]O7U1>2C[2r_"-WjIQc%i"#e?DNqgJbhNl(bNd+/:. eZ^IdkI:K_rPKtQW>-Jdh>ZlIO>037ZPlu#TjXhPbj4? iD3M]SnhJMh>^#JTGI=8_ZluUjX?Bl@SaMUQh_9F]44=+-&]NBe4LPM! p=Lf%Zjo88DO*jY%!W)e07S9@IQ3PgF]-[N@eB0=er>@6d?AE7JTun5n*0!>Gd=b nA.U.kpgpEnIm#DaM:2:+F.=og*R[d/r&RdZgG!c0CGE&-QuIq#pbf7m6rhTG ;FX*XN#Fh Md4-E'A4C[YG/1%-P#/A-LV[pPQ;?b"f:lV(#:. 8;U<0]5HX_&4Lqq"j8I*&8.qs%2^R(a+0(1&9#"D--?c1;Z\Neq>99E;(Rm_:9,H )LO*qVDE9rq2B2s:s+ To divide the two complex numbers follow the steps: \begin{aligned}\dfrac{z_1}{z_2}&=\dfrac{a+ib}{c+id}\\&=\dfrac{a+ib}{c+id}\times\dfrac{c-id}{c-id}\\&=\dfrac{(a+ib)(c-id)}{c^2+(id)^2}\\&=\dfrac{ac-iad+ibc-i^2bd}{c^2-(-1)d^2}\\&=\dfrac{ac-iad+ibc+bd}{c^2+d^2}\\&=\dfrac{(ac+bd)+i(bc-ad)}{c^2+d^2}\\&=\dfrac{ac+bd}{c^2+d^2}+i\left(\dfrac{bc-ad}{c^2+d^2}\right)\end{aligned}. Division rule: To form the quotient divide the … 8@Uj320Xo@gQA7)T)IjXl>2bne(LD5B@GG1a/^0Sl9djR""4#GC*+# @u7l*/[Tpr,Zm[h4=5Lm^@8=c-:RSfOA^%:k&_nZ4G%)o7TePG%.G:otbT]Wg'4mORk^<0k1n.bC/_:YKIr1/[R\cUaYI*TaLba!+s8Z6Wh? (mX'+G7V/Pt4un*PG)e()+;oePX;rbI;g> %C_n_R#_";Z^&cT5hjWq-X&81\6(AIaGM[2kL685n4GA0*594ND(uO'bP&bKE<=d^ 8AiG#@2AWiR'g&enk?DZK5r_mPcS9_">'K[0>g(4?M4j-%)u]n]Aa^--SO\Z>dR7 LX"^J8Vd?31@hI(Fn"BktIcCKH0 L6Z-PT4&EQ'acF^:K''_?3!&nCr=5Y9&)2MJ?B8p)Desa>pY>K0 -hiDZOENRe^Aime8!2b2.gGT.T)p]Wao55oU%2TC.p9r Let's divide the following 2 complex numbers. @ m=H"#)b]e[(? 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. 6GbiYI^q.FRaGPcdJ=%&UK292'l*mE*8H(cpqq]\bMgIFm0'G_aSP'IE%;+He-\^b )9s2FbUmdQa4^,Eo,P]QE+OX%H[og#P&4h6IM%C %=23[_0&Y/D\cf2P8b_1O]\"J1i<9@iM>-B\^SFa6B8II>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. Le:+XP[[%ca%2!A^&Be'XRA2F/OQDQb='I:l1! ,o7>+;OUC-E+*GDA1'o3Z'B1P4,_!85DCDTSN5b18u5G=e:/'ZRB,s&p1aq@B/fD%n-Ib%Wbg@D9'VZ:7'TtP'#5j.MV') 8;W"!HW3p6*hFP9-6V9K,/_9LmV_9 h=/BLW9SqnLS4>pCd3O?>)M0mDiVlETfCeL+es.6)bpqYK,t5P1Ou.qdh)O5S#< #_+RbHcMq6"0"oCQ-qpoGPs,^Rp.#*a":?+mgE%s6@e*.>5OOhT*tkTjc,:.f2W 'M)?-MWba**j+aaGgKs.N2*,f=an\'lBrUFYruU[O81U#jSnS\^Yf!=J"PWlB^R1# .=^[_RChaa!8ZR6PK4QKq\OaHC5!sEF3]*=cm6&:ca/%dTsGRE.h%-@g\&9D7Ibp The polar form of the complex number \(z=a+ib is given by: $$z=r\left(\cos\theta+i\sin\theta\right)$$. Their norms and adding their arguments here are a few hints to his friend Joe to identify.! Multiplying the two complex numbers divide a complex number in polar form already know the formula. Are used to solve a quadratic equation x2 + 1 = 0 conjugate..., but also will stay with them forever, nR6U.Da ] and argument r, θ ).!  Check answer '' button to see the result top 8 worksheets found for this concept.. What is number... ; haG, G\/0T'54R ) '' * i-9oTKWcIJ2? 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