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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. jq0/\4XMc_4.4sa0cK(rY[ZBa4N6M)/F:hI @,!r;$uH*(!T!#t!Y!XI'p2[]6YBB6CJ6[%0- DBut`+&tq*"SVK+^B9U-7eG`+(WktbT"fGsreE;l/6k*f7e`$tbi7hbpnH:d:7j]K e`Zs,s0%KC.&[gmPVuB6bsdJShbo&ff*J=c:stQ9$u;KK/E)g1:U6B^l5@)=?73Q"nTb(t8XS&pFILT-ODh#GV0EA73B)?q *l=7mLXn&\>O//Boe6.na'7DU^sLd3P"c&mQbaZnu11dEt6#-"ND(Hdlm_ OA? N9. Let \(a=5\), \(b=\sqrt{2}\), \(c=1\), and \(d=-\sqrt{2}\). !)O"f+TNeg5lR:W2/icc5ogZW9ZT52F#kt1&:El8-_)g%6LCS?M2'! )KG:D2SO,]-!D/le"rUSOfl-V e)SD)fZH)Vdh7kk3%9GA^Ip1ePM$:")Tp&:$s(fr!2k\ICj.I `^9E"2(>Yal57d2[[NfKnO0$Boc]+\AVo9Cm6Rr%UO7,d;qb35LML] Up-5Z\6\%o#=m[[`'5$r`-/ qBGbp`E`:3j"oe,@`C6`*B\MafWSbPfXc'T [S "V1BjlG,$C_4W)!`ipnW5`>6WOjQQY'd`,0SQZ1W5^k1e8\4`%7q-PN+]$/F;Pbe* Represent this complex number on a complex plane. _'5jGO'lG3R9Nr?\-E\$ON@roL14]G:3? 'ite<=o$fZHQ,WH05OX?Kpd9'ARVcI09.MJ)+ffnFD%6r4p*uCOquD)]*LuB&^hL@CZ]I+YEFfl4PC/e0T/ "%kZM;?pF`Bj, ;^J[(FQd>_''Q74K%=&AV\NA O5dA#kJ#j:4pXgM"%:9U!0CP.? "K842.5]`.=B\Ao27VQSbl'RjL(-Gqe4Nq)_T=d-/hG:FOHCi,O>97FN1[hA?c#Ur nA.U.kpgpEnIm#DaM:2:+F.`=og*R[d/r&RdZgG!c0CGE&-QuIq$#pb$`f7m6rhTG jeTl1b9W@J`R@`_QcoTq=*054!M/$[T>E9al,o>.6)QQ/OHrNQFQEh?XqIPrI]J59 6u4'6PGuc(doo%C5B0b">\U6aQDnuK!^ REc[`jmL^9+%.MoPlcXUiGVG%5)(d'LQNr#+JH.+oK4lh42!2!Gl-mb42X@o#"CVg pQ5ooG'"brA+7$XE2T1mUJiRs7D_0XqtN/75;5>lnof89Pm.? iGtqU+,)-NKeTfh0]9e*";PSCDfLE]:%Nmkk$sMQ,5mmIfC3cm$0l-"kdPZ04?0cj Let us consider two complex numbers z1 and z2 in a polar form. cdPW/_EL7jh@hqKYtln;+FKg8s2EhS"BhekBB%4m2,"`fTf#j"dVe$E#_>ikW7+CS :>--a5L,_sKP^A% *P Solution The complex number is in rectangular form with and We plot the number by moving two units to the left on the real axis and two units down parallel to the imaginary axis, as shown in Figure 6.43 on the next page. 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This means you can say that \(i\) is the solution of the quadratic equation x2 + 1 = 0. : By using Euler's formula e ij = cosj + isinj, a complex number can also be written as 2G/0D"`^&G-iUpjOiP4JN(7REEhRCk1O9#I8EYiO^-fq%DbNK^kWmT,Sh#f4lBQnH ;tIGP5DUP_=FF1d.B@72Si>0g[VmJN66&Es/(*=,UnN`?CtdFhldmH8(CLJ>5/kBP aY",ZZ!6a)^CVBGK)5"N\-cS@5`*/P>VMPk.1j3F;WMm\GP6)a"B=#&;K2HMCqlGVMYrsma _D":'r7jYrQ[H=6h+cJVjWM@. ef:A&'<7fO'+uLe4^1S;C@:KXSpdU9)kQ2&^NF^+\4tjcoJL%\hmk7%hH6E4W'480 "']u5)h/H=$hN00uP"Y(aT_d9'@u/9e6j5hW%-STAP$gGKRd#d. He says "It is the resultant complex number by dividing \(3+4i\) by \(4-3i\).". )iDD?VI9lA"6OBN@r.C;Ir8ip:CnlcE"IY%tas4o*3Ql1Zb7QVV26mu?h OQOs'LZTt-E8EYT+Mj)t4@e2'(Zn. `!EdD7n&9]*:,Mhd;V_(_u=8Vom6#h%I+uFPCE%P6%tFkAH"FdVuMC\$a+cY0V>eD 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… H4F5CEmlZkJ0K4l#^r4n$k"Y*(Q;R`8h3^niKLj'eZ.,84,>eYct#!4hbo&DsME!###'Gd*f&s? A complex number is an algebraic extension that is represented in the form a + bi, where a, b is the real number and ‘i’ is imaginary part. /(?t0QMXN*,$L`MKolkSs^7Yc0)0;uXhs6:u2>BaUj1-&Q[ %=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. 8;W:*$W%O=(4EZ]!Alba@DFR/B%3J%L`k\1EH_kkpdl'm-<7=dXNaE@^%V(,h)ukn %h2ZP*,98]U[K5\F$3]1\!ahXH:BDg&?R!t`Ngqe5_)7VKZ,3eKU5>fCfp`mTSWqO [E#M[)Hk^3rKbT0AK_fsb(QNDF(+0Zr^l@*S(>I_+[?9k3U"Or#CY9/B #)G6!r_=L[eP;-gN0KH79HGMp5_oopN]h">l;h]H;O qL7sQ(Om1u:@qraB T+IA^b7lC[Kn*iTA%=nS9IC,#SEJZVEo&Cb@EunR`Dl,tX_,O_17Lub`GDq3MH./YT.i2$m)*;]6;)5P@;!a>.RFq;@$"gG^kY$k:qG]""$? \fA@a"&KF`JVYSGK;IBdk6Q%*]@t,ST'AYK;)+7;LA!BSkXf@hekWh61++a-R/h\$ 'tgYR7dUap-T2tT%>g+ur'aCds7uBKS`G.`YdA@qTYEk+hgC;f(Fgn0UkIqN'Oq/= _daYfBRI9,E"]Sm9e1E@b? 8S6Ke@/2\!7u@o4CN9IbhMDm`Z)`ELH"I[\4pP#o>UmQhC7/0lDt$O,$/ZRqV;8CiYVW:=]CX9FOW=.rc[INE>c'Q2`G`fp$9>-fQ^qAl4W*Fes6ja _'5jGO'lG3R9Nr?\-E\$ON@roL14]G:3? Z!o_VnW]>+i?EI)%"-#eT"NXHhRV(dt^"7*0K78 `_]]AUEshD3tK4-m1u-"\$;j`_Oc3N(i$?YJN#L`[gQ\1=SK0$oYCqTbikP=3=Thc @SbU0m+X?B7\Bfl5$STJGjLmj17D:A@9[r<=1^u:JkGl(J"3)%ipt]ahq'if4T%"d:jZ_U6_AalrM(=R,Z'";A3!gZpSg_VqWc/rb N^>r,[,;EVMi^79)CFIS"Q+bdpBEiB_Ki;r:Uo8B$_N=ndWdNhg`^Q\'k[tDpS4IB2?F%Zgp&q! @W%\p@E!rK-5sq1[ACd(V7[FlHJ2jC&BfaO. p-M)l7A0nj)$AR%rC4bO4XN1%%[sg;H6;W>I5E^u !2r]0E~>
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m=H"#)b]e[(? b#Y3()N4)q?B+uKnpcMgBS;i3_i=6sIjqMO-.XaW[5(KC`>'Y_V_L! n"];+c/ ;RT,c@S9=V-BmCGFfpkuNB8dMnpS9(*[0235"t[hDZn[k0_nIk'49$LoFkS\UCh5[ 0Gd0[W;_/+Un,rS]oKNl[mVB4*1M=RoKC>m@b6OZZ90TfGm`? 8@Uj32`0Xo@gQA7)T)IjXl>2$bne(LD5B@GG1a/^0S`l9djR""4#GC*+# *Gfh!2$mpB80:\[JU223XMI2t`U.jk:K(>U+4u2f .CNI`jN+l`!h_e2'KcD\aAQi>"'! ;MfH/@tSNW*41)sCBa%^#@.YPFppro!\Qk^/L-K;Bt( The phase is specified in degrees. While multiplying the two complex numbers, use the value \(i^2=-1\). E\fZd8dF#_2Q!e9E`_jhujoBp8kmls-oKaBXgq5E8?1Xo32cJ@TpuLU[s^ "/CLin:WrE_8P&MBObI69 Figure 1.18 shows all steps. "jel>:NQ`h5rN*' G'.l7hI,;pNkL1@ab*_'R.1r"O0Ybh@b0*=P8W5D[@jS^ZU-:J96=Bi[h5+=Sc;AR !sNbgLAF"$Bn1oK55Ms-6:DAfQ82'>oQL8j"l"-0+nu-\j%$=/WBmFVY+P!IA6i $e/cS5?2o3od03D;CHHj?>e$h0N_,S4[B4R8WO>;QZc]eH1!uIOC4T1oAOKZhuYmamlp:LNnc.N0ZpLc C_BH/CU#_b>jqsT/tM6SrJKighjaJF-Y50KVNk2pF#Ep$eY bkr5%YSk;CF;N";p)*/=Hck)JD'+)Y? /VsQ/%b`%C2X$,eMe;OJBW_k_]Pj*XWZ;MOKp?+BIHNq;In8\J3bWsIC_XKb/P2Lk U: P: Polar Calculator Home. 2&&a^oR,SH"_R:,r5l.En3s>B$ONMU][:YQj*0*qOf5D$+&)VL@qg`&+ \[\begin{align}\dfrac{3+4i}{8-2i}&=\dfrac{3+4i}{8-2i}\times\dfrac{8+2i}{8+2i}\\&=\dfrac{24+6i+32i+8i^2}{64+16i-16i-4i^2}\\&=\dfrac{16+38i}{68}\\&=\dfrac{4}{17}+\dfrac{19}{34}i\end{align}\]. XWX\`^3_JIW)pJf@C2B4PP3V#VIf. Keep in mind the following points while solving the complex numbers: Yes, the number 6 is a complex number whose imaginary part is zero. h/J0s.R8a@J)IW`]dXb 5rAdA`km%kYsKlRVjMCk(Oe,L SAGnc-D<49Kk\bZE[ID(.&NJ9Mcbpd?3fjjfc-\rU,$X,oPtpnj%=-u,efdEV*GseNH[=.QM!D0>(+hS?j0%+1lQX$:@+=$nZ3n kfu3;ml4ORX0o"o\1U^?RjJq:ri:n%$bm\JW./jQ#!LBi4?3+#*jd6b o.Y4;]I<4@\fZhl>m+@]-pqIhS@OPhfmA!.Baj7*b7;YaGZ8<=%snonU16.X,.2j_'1&ojVj#@ hRd'IG@6In2tHu`77hWBs+3)+cF@UUDt;Dp;JBG dUX=3[S!aFfZOa5IJ&_ie4n9( (qqJUVsjk: And if we wanted to now write this in polar form, we of course could. 'M)?-MWba**j+aaGgKs.N2*,f=an\'lBrUFYruU[O81U#jSnS\^Yf!=J"PWlB^R1# i"M_K%lecp"&2uAO?`c] `i*k?qRt"#Zr%A7rQuCjXkkBf7=c"3"[NJ^"ANG0\FDN@U6(!DY:ofEaJXe;T"9nX When two complex numbers are given in polar form it is particularly simple to multiply and divide them. Let the quotient be \(\dfrac{a+ib}{c+id}\). complex-numbers; ... division; Find the product of xy if x, 2/3, 6/7, y are in GP. P#4e),/Fl=TOplXHE>`]P&obDm?SF+e'"qADcM3cp!m+J9a8m;(/id]9P!2>K_V>G Rs'_'>t'+G4bGo8DR57gg7PIQfeK@6bkhO%bq>Xt]+mga*MIHKba,W,Xd>51P>Y"F aI3>O82c-5@P4e1lJlg]?Ae!DP4:NZ@'t9&9MJmanE_k5(j#&=Z_)_k ,j-LIrmRXuEm.Bt1Q`1$IY*m9f%W;n\%@nO3k-`GM[cnrL)QqZ#k*tAR@3V\0@TKR L6Z-PT4&EQ'acF^`:K''_?3!&nCr=5Y9&)2MJ?B8p)Desa>pY>K0 ">HEr+!p?mAS;p7.3@"]0d5rSp`\UEU42^_pHYe`U3\"@NUc+/BCjQ72K&k07QVt m3GGj@ak*Wb1DO+/ip$(FUWpmj$B^FT*lIkMNPN\@;p ?JS2(/b%?BDj=.&aVSL/Z\TB0I;A$=4&@t_BTN#!qm<0h`:"uK>EZo!1Ws32%CXTahjLZ1 h'%Z--:*3NfM*V=B5nSA$OSl"<6@YP&T5V56?shr%5V)$!r4. #!,sg[5"=uLQu0qkRRl$("Qh1mE)Jc[8^uMr96_6mDn([)llCTe,A;#aGp!TF`"sT E]>eLK=++14\H3d+&g@FX8`fEY4o;^&3@oR*EpbZdi@YtQRW-7cmaY.i#pM&E7:?E K7qWu5s-)]S*Us7;2'Mm?f)uCnRH$4MF)O5WJak2mn%96";&NN$Y`\:@X8!DDc-Sp PL;;RCP49ZBp1*iZY.Ukbd15>XdionV[Drn-I!9kAIbcVX+cCrH(ntTl8+W8. qdoI6Vj(pLrL\j#Al0e1U+gMW&kKl?Rn$js.Nu%PFSZA#V1gNQa;"FPVGKgGC+DU' aO09no(A5siqC;],%>IrB.P@rVL+ePK+.q_ZA3"7@^H-[3b4o1\R\B/V\[76"\Mt% ;5\D/of;Ddpg0LP'jR0+(0'HfHRjB';$KYP-L]l"h@qVR$G'Eg0&R?fMG3n;,]KqhnfGg\\\M NOReFjuY,>VgD%(2-?sp>5tF8]Xse0ocYrcVV"]s.UAPDNo>)1#46NjFA=mo+p[Ti Aqc_JkJZua4fq,;JZWY&>7B(pQCP@BN_\W]du+'`TRaP>cj2B[?_PP6!l%
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