?rQOifIGuZ9;hMpQ+3\rM+6pd=X9?sW!ZYT@\UB>6(u:o.B3YZ6-"FI;B6P_-@ZJR Jodg+ur'aCds7uBKS`G.`YdA@qTYEk+hgC;f(Fgn0UkIqN'Oq/= MDKFZ:*DN_$tNAOV[^R$#O2@gKOle(`DV&:J4l_]ICEHm[XV>9D2?#jFW1(*:Mu9sj]I;Kt)1+t"j%X##0$l%:FmZg\3TXj4 bu%WoR/FAQj%,ln>2i'1p3V4*? h/J0s.R8a@J)IW`]dXb ?#%LHb/^qekb9m'Z%Pj7[Ob+s)!mrjFGL8UDi.Y1C$FsWo_*9u ']KXmNPN.\`!\9NM&SpaD2sIEqU3& 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. ODp!7$ddDR9a65_cV/jmR=\^%]i?ZpL?^4/c[kDZ:l3N *c!+7k G'.l7hI,;pNkL1@ab*_'R.1r"O0Ybh@b0*=P8W5D[@jS^ZU-:J96=Bi[h5+=Sc;AR /?C9PY:RDp`$AH0p7XeYj;C.;X=%U#p-n2CuNcL\Z3l _iull%qfet!1"4F(Q\6UEN14o6s4=eD7i+Nq1[A/IRoX8!bi(.KX#N;R83. ;$%:h(R&*!g1pi,;s(.o>a+8R.rjl@07K_f 11.2 The modulus and argument of the quotient. If you were to represent a complex number according to its Cartesian Coordinates, it would be in the form: (a, b); where a, the real part, lies along the x axis and the imaginary part, b, along the y axis. ?D?G!tL_8Fk]5A%]SV:M4m`U98%SD<9L(+/^cFZ9s;P;s7p5cP!+e8JCHWD^"(t *>%qe:[XRG-H4$YOrBkP2?O7I?MuV@i_d)+%XkH5^D3nm@j8F"$D .CNI`jN+l`!h_e2'KcD\aAQi>"'! @8mE%e.':2$\lP%:m@-+]pY=ZX?90hX6H#G-6[TEp+nD? ?JS2(/b%?BDj=.&aVSL/Z\TB0I;A$=4&@t_BTN#!qm<0h`:"uK>EZo!1Ws32%CXTahjLZ1 @gFo;=F2W[-$ch`[7:ZKWh+q?/sehts%]`M%R[S[6^!:+D@jJI5aD!Lhd[dau(:T=Q^c"u3N1eo9F]jJaZQ[BrD/;6OS? #'t1mg4a)gU_4,SYgA@T4?JU\+:4gk:b36SpX"5^!\&^c_IMD`BRrU>E#V=%^Os2- 8rXuqYLXSF'\QG_&QPi/X7QUkSh_-2NPKaf6l;\&V2]F-^dbG&_aqHWm8*DdE'XVO ?M)`#r^HrPK('Xc7^&X9[tcRH)jCNR;C[^cpp;s? Zk(T>f1oQI`Ob,Q-0rP9:N+miU%thHD[iW-V#R[(CK&raX2"2;*O@bLoBhKiN-ecL O,dMG)lmSi1Emi? 'd"-(\bP#T"hsbH6Cnn:]`=-8I^VCP]l"h "/CLin:WrE_8P&MBObI69 Figure 1.18 Division of the complex numbers z1/z2. ?7:)GOAZaiKdh *^pL-eS]M+'io*mUV+]PgNXn=+0flg-K5.kD'=4a3CnuCaCDP$dOVDrVFG@G5q>+V +?#Qc&$jtr,1-! ]FYgDg',Uu!-+Ol%c^sK46r@4WUBSZ^E_%._ ehPW*n_Ws\[>p6tL^Xk;84]h]`'Om*nlRRUJfktWmk3tJ%rqjm>,>!8W]]9mn`9e\P1 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 0O0?7aq^:PC4uWnO:*4`cP$I#cHX-EE`(>NNPe;KpmV=8og%.4mFb26d9 =jjO* endstream endobj 41 0 obj 449 endobj 42 0 obj << /Filter /FlateDecode /Length 41 0 R >> stream 98j9JB]Y78,=mHVR*^ok:KokTj0[KS+=^"Egp30eBqng+djBgH.BZjX.S`Q)03\Nu4SV9d0>I!.ld\$:t#3P7MH LX"^J8Vd?31@hI(Fn"BktIcCKH0 b#Y3()N4)q?B+uKnpcMgBS;i3_i=6sIjqMO-.XaW[5(KC`>'Y_V_L! @ZZW5QZe4.loe,r=cSfSpH3G#*T*-S'kMkJ8sA?_mUVZ,lcDkCP?lb!/N\52:$HXE [%N?\5@Oc"S5),/u^"qlZ&oD`,9k6N"CPo2f`"(6cJS*cdA2d-#VT-ZU\t noV]Rg#]Umqn@FOm/$h\! MBW4p;jnWk:OTn83KAu Get the free "Convert Complex Numbers to Polar Form" widget for your website, blog, Wordpress, Blogger, or iGoogle. 2&&a^oR,SH"_R:,r5l.En3s>B$ONMU][:YQj*0*qOf5D$+&)VL@qg`&+ Le:+XP[[%ca%2!A^&Be'XRA2F/OQDQb='I:l1! 9bJQ$G&H[7#eY:5m)PM7D"GrYK"/@,XNA?TH48@7!^4JECkPL+bdN(X@Gdd#F4RNo z =-2 - 2i z = a + bi, (mX'+G7V/Pt4un*PG)e()+;oePX;rbI;g> aU`73TF:sJl:UN@cp7*YCZ*p^L^4cN`hi6onSSIF>" !i4krC0YI!R (_pKu`S_[&UN%h;^mgE"8#"hqYtXC7VOIu_VX bUA:E>#3I,0tX.%&e'IbQ:@Q#LOuNC@\6"dd*0[4,,3..6RI8RoU6M0kXT=)6t@W94`VD]ZADNIgH$9s ?7G%(_c?P4T2lj,auT+KV>+kc-)ZOBI:,\!bUZ`'LP-ok!OTlAI"(2.hr*b5=:8]jJ*Yc/q0G#`;ghF); �� �sx��cx��;��N]�l��ݺ0I�n�5c��d�Y-�W�О�y�T�(�2�E� �*��d�KtjE��-��\��5�#� A ru endstream endobj 43 0 obj << /Type /Encoding /Differences [ 1 /space /E /x /a /m /p /l /e /S /o /u /t /i /n /r /c /s /A /w ] >> endobj 44 0 obj << /Filter [ /ASCII85Decode /FlateDecode ] /Length 14729 /Subtype /Type1C >> stream kr&C,hm_\!qkQ=2c@']1AaClMB;K:"E-]pJ\t)J)0q#%hs2qqT%I?+MK>-`'+ XG#DEEE.S3gZ*Kr9u3*6F%>]W-s!VH#a5-!ho$MG&=Da$>kiW1;QX8"*jmad^W6B% Complex numbers can be added, subtracted, or … >uMN/a%12MVEO4Dhqi\SYl;pfE#PM2-uM6EYd*h2'6Rd7=Zd!`B!%Q>X0Er6oM`*g A complex number in standard form is written in polar form as where is called the modulus of and, such that, is called argument Examples and questions with solutions. E\fZd8dF#_2Q!e9E`_jhujoBp8kmls-oKaBXgq5E8?1Xo32cJ@TpuLU[s^ :;&g$uV loj]6X:)Xlh#d_55U=:b7$n!ri7G1I;F#d*:]R=g$O>WM]E_fZGPrq? :iT!&(R&nI2#4)&[L[`')rM/1@h?\G0q>;/nK2pU@'m)S-b?n]j. Hf@hHQ,h'h.UbdIMk3%dbgN)$AWUAj`pM%!iHUs(4mNqJ)hWd@fc@(V@HYfI%YgO/ In order to work with these complex numbers without drawing vectors, we first need some kind of standard mathematical notation. U^eoi&T5>`7(iI4g_pfPA;GiUL\"@kMpFLlnhe*lmBO^Gp(C"=3kWb`ID'!l#"IHo ;[B3E'McuD[d61<=f:uZrM_iI]j8CLhFb1gYhSm,;CPVD *3Ti=CoaEB8mA!r%2K1]FU)@DA]VNhp"$N/O9DDk asked Mar 1, 2013 in BASIC MATH by Afeez Novice. Every complex number can also be written in polar form. ':PLJUGi>A i:kY4SdO)ja)(a9Inf3?>2'p1$'5;R;o3"C V%%*fAcK1N58"`-ijP2X7LFpLQOS:7rO)mJl;0c69mK0mM>^Z/GGn,kV6+9^2bOX; V$L]Q#M'CtTCr^X13*Wo\9J,FR*RBpHS?7^//*jjfiA:_mJpl/]ZG:A&T/33*RPe: YsP%`Ur"!ZmC/us/;FU.b";>+5e7MmiRb'qTdB1Kp?PR1r;A. [%N?\5@Oc"S5),/u^"qlZ&oD`,9k6N"CPo2f`"(6cJS*cdA2d-#VT-ZU\t k#\h_27bJfq^'67e^&>2nns%%Z[siHW3.S'F_0tQ%I3T\0K4BHmY\uJXW"T<=8IAL SK0K\=RtTTQ\Df;='dq9mOHF7OnZ^""ZgF?Mtmuj:k9a"LtVB?n[9tlEgcjl>//K^ *`%!YRt42alS]K+^kp`#'.lYFj-fQ-RZmA`,?`?Hfk%r\gWm=S4u@gn9eFlGYb;)( Polar - Polar. ;PcId\WCZM?Ub4C"11HKf7+AK`@5sYph3uD829=Rg"otuXf#)*ciKHn%jW3).7rGL /^K_CZW?mKmlm7QZBUck3[,tCaF:+bq@ThUNjbe0(U^ jsEIUT&%$P;T^A^Dm+$2Xl%U[P\?iM[p[BB;_fj*g*HG! j^pQ_kQn"l+n)P,XDq7L&'lW>s`C>Fa^mm9R%AA87#N*E9YB2b]:>jX@fJE Thanks to all of you who support me on Patreon. `^95]PagD+'*B1DJ#!g&b&MsD:nD#c\^THQo1-T9Yj*8q6m(0o!Bt,j5q^=6,Ym;i The Number i is defined as i = √-1. This means you can say that \(i\) is the solution of the quadratic equation x2 + 1 = 0. WH/0Madol>,42.CRoM,qS8JL^7KsoQ53D".lD]DQ>Wg4c-/$I=#_b0_\e\Z7 =rt?ZLQf679*C#lA/\c=O'4NE/a%cCAf:63p]0nek;[U.pbHoT]\ct#? s%? L#%!bSu?PX20h::^(5Bmh68qE[9du%GJ&Ua;LLBK-aET=gd)DFTt2Ua09N#1D(@d] R:oN`MHm%1_%u9`2Fr'&p^.`rRZ]gI[mlpSKBZ.c"8RtYU^.LnFnnbp8Mt6t,arf, ;X[%,"6TWOK0r_TYZ+K,CA>>HfsgBmsK=K where \(r\) is the modulus (\(|z|\)) of the complex number and \(\theta\) is the argument of the complex number. 5E`XY#qS3dRX9XtouARa5Z^/q'1Itsc\dsn>oUN;phgF%+&UKSW_FK%.0c45R5Gr> TPE"qF],e;:=bhkD-";M=e1qQba>__ti2Y+]#(1U@0BI`ca Dk'Ne0@B)$'6MfnLngT:7^ulF*UjDpeS1Rde:S)nZakLC$&?NC*pT3@CDOr)+0[cJ pQ5ooG'"brA+7$XE2T1mUJiRs7D_0XqtN/75;5>lnof89Pm.? 1. and the angle θ is given by Because and because lies in Quadrant III, you choose θ to be θ = π + π/3 = 4π/3. hn_9TNY0Z*dh6pBld.Ps-'tKu-.7D/AmJ)\0ArHm@-igSfa/S(PBXS41pjRc"BW1M >g]gFl9R7aJ4[`B7Sn#F /VsQ/%b`%C2X$,eMe;OJBW_k_]Pj*XWZ;MOKp?+BIHNq;In8\J3bWsIC_XKb/P2Lk X8lBM#"W1G.%;B^M]W`#)ZKOWUA6B_l:hRcQ`Z@W)*rQVBgR$N"?! 7BF[#]UDS1k",G.%J@NR]>s?VHgWqeDKlPT_cRN'i%>2IBRFJ1)N0*/*1VL8Pk,TU Polar form. However, if the complex numbers are given in rectangular form, you should consider performing the division directly using the complex conjugate method as shown above, since there are additional steps if you convert the numbers to polar form before dividing them. @n4@]P&[IJdZQ'?TQu>J1%E382n)u9d5)$#6rNVlFh6\G]/a4 Similar forms are listed to the right. 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. [lRt'clmTo6?_XV]`Ql$O50%8:4R0'V#$>VR$6g%"9_O?rT5-HH'2C`?X+(0Z! MM/VB3pKif#hHd.eF2F<08W/9\^:h@tIJ9'`naNrr>bX$ldn5)G`P+KWf?/X5W 09#UQr@NA![nX;.Gp%#=qE6h2:gos'F*q-Cn4_Xsag1WRs5)J@itfWV3pm5tWCJSP,;G$mR[m!o'\ST. h!7E1kK'&^2k2#p;OO@Q=,*`agGCK.g`fJKY4l=IgBu$LI\QLSgCcD;5E^p.UWW5] !W[Z&RgWSMj,Ni@oOZ40PI-TV]e]..i)LYuMtKOERI2Y9Iil[T gDGEI9?/Bf]t:$PB')b_ "e6NkK`[W--U$6efQ\f7_,bNnqBB4*N+1FMd9&-4O#g;`/G6Ab4Xl,b]dbY/(fKJP L$($-]/MMN<72NWTK.qToPZEVOh-jUuQu79SGS[H_j9eDnT6[EYA`8R3hCGTWh>kUkF&36%FhQ^:?F ]@7-l_QtO#feI1d8kM-iS+%usrtY78iM.XmBU_L/[geDGO'D)\/3Wf/rn9t6B/42e F?U$.Ih=JIe#o/g/(@p^HU(#`LJ7#:,>A[m#b45['P/pnS_$;jrlqFfhP6J >-](iEL1o$Xm:2/s"NXUGM_CdgO"5c>(p5XimUQ67[S1355/`:.N2""bW4Bp%g**Z *&lFDgpR_7#+gY7_(5/>>>L&fZ5-&0S.d6"OmAOpRfXS%epP3_,D!U2/:OB9JZ.b[fQo`Vb6C?>3`F+@< o.Y4;]I<4@\fZhl>m+@]-pqIhS@OPhfmA!.Baj7*b7;YaGZ8<=%snonU16.X,.2j_'1&ojVj#@ b0!R8#^<>"b9WZa8Xp>uC^5L'jZt3]''E#-&'qe5"4BVp,V 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": asked Dec 25, 2012 in PRECALCULUS by dkinz Apprentice. *l=7mLXn&\>O//Boe6.na'7DU^sLd3P"c&mQbaZnu11dEt6#-"ND(Hdlm_ *Gfh!2$mpB80:\[JU223XMI2t`U.jk:K(>U+4u2f AL?-d:rua9AWjL8+0tdCrF]:)*i0J.8oq$KH\T45jT7 of The graphical interpretations of,, and are shown below for a complex number on a complex plane. L,3a3L9ke2%Xe1LapD>,RTHu2\WQ^&o7p($N]_fnrJ$k`CB1gSn5T\TFd.c^%@bNI "!a6p'$ch@r_NJiu- "2^`;9Vr%3u_6qU>4ja)PB0Ks/S0QFR hlZ;e0KWp-G1-1ISAnCf2#_->/Xg0hUs:Pn;5pV5Xf3VOYplDL^\TV\i@PlWP9CR? 1U:GQ:f'25WWt6>q!HToW\\!>DkK[(q!X$2A&tBG>mR"6Is:m;3TPa-Pi: hL%>!k\YWc:%2:J9nJq>?$K8f%!g^Yr=Dbd_Ao'.&jk,P[@O*Yd"'e;c&;rekFr@Hd_p)]Nu3, "DLHM;R9Y)lYS?06H8/u9EpE+;F@S;7%9FF8*[T\8\9dcSQP.l#\X =>H3EgjBKI#s6Q+2L0M$8I'eh\CnpqlChGFq8,gDL[>%']Ki.EGHVG/X?.#(-;8Z)G=+jF=QDkI\ @;1sO/lT7pNK,?pe&Cq_qV(fJEkH56JL6B"ocMGM4BJ\hD-+JO:]%#l . 8;W:*$W%O=(4EZ]!Alba@DFR/B%3J%L`k\1EH_kkpdl'm-<7=dXNaE@^%V(,h)ukn e)SD)fZH)Vdh7kk3%9GA^Ip1ePM$:")Tp&:$s(fr!2k\ICj.I )S>;[>^6tKUqF=@daQBO0;#YbG=BK?WGf'mALek#oW1ro:0;pg9pTfhW\jCL 5sL$2!XB*K!pK(_1(4M*Op1P_,j_I18<7R0(cDXO"bem([LNJ]PI2fJ1!,KpER"Ef Id`kTcTCmF*C)n! (r*cos (θ), r*sin (θ)). fH#bV.'gUqG&%O]nB:Ol5K[W]q&W-*D5Ju]icF187_-S&7,/#S9! aU-(3M(`7/^m]e:_!-F%-gdMtCi[42Xn8@[mM'u)I;6bYl*NZNn!a5h`o7lD6$%Xb X]t._V>c8Jo/qG@?c&3K[s*+6TOAX0(6;CKKY+LW(-,DQ!di"2U 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��� =6_C&hW`F:/'S5#&ufTQK-In2'DA%Ecb\JXe"F2GUpZ7%D3%7O7[p^mdJM%YUfD1n '+jq)Njim*StCQh/6haCrqfW R)_pW(rAWO&M'N+J8Tt;Oj^DpQ?fTQAW)!+N_n>gB )SoplA&LH@^KU^7=VsR)1j3VU<50f:5m:%J(m5),(&70>@K/Md3-2t8G'pe@o0uYj "&@4fkIiZoUaj.,8CaZ>X0`:?#SZ0;,Sa8n.i%/F5u)=)_P;.729BNWpg.] ]kY%tGJ3/P$@bpga `iU;+le/d\`UST#2b\I1_M*i)-_?2'O)r@tS$[4aXiX^E?Cbi#qT@pegEFOF&? The division of complex numbers in polar form is calculated as: \[\begin{aligned}\dfrac{z_1}{z_2}&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)}{r_2\left(\cos\theta_2+i\sin\theta_2\right)}\\&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)}{r_2\left(\cos\theta_2+i\sin\theta_2\right)}\left(\dfrac{\cos\theta_2-i\sin\theta_2}{\cos\theta_2-i\sin\theta_2}\right)\\&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)\left(\cos\theta_2-i\sin\theta_2\right)}{r_2\left(\cos^2\theta_2-(i)^2\sin^2\theta_2\right)}\\&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)\left(\cos\theta_2-i\sin\theta_2\right)}{r_2(\cos^2\theta_2+\sin^2\theta_2)}\\&=\frac{r_1}{r_2}\left[\cos(\theta_1-\theta_2)+i\sin(\theta_1-\theta_2)\right]\\&=r\left(\cos\theta+i\sin\theta\right)\end{aligned}\]. 1j/3^:OnWsJ'10h/tX*'QP;C$D$NeV)pG7g)0;2;CO*\E.r&kBi18G_M5eFI`-Kki Division of polar-form complex numbers is also easy: simply divide the polar magnitude of the first complex number by the polar magnitude of the second complex number to arrive at the polar magnitude of the quotient, and subtract the angle of the second complex number from the angle of the first complex number to arrive at the angle of the quotient: e^3B_;_?9):ERu`$#+-Mkt@%,o)VkCIuE$">hUrp,3Zp;T-4 ;a2q6,6[X6,bW/9dl&hJKue/0o=euZd8@@cM%()7ida$nplC$ %H=PNY]$o+L@Pq952CdlC@%Geck+F;q0FgO_@rp"bI+CFl%GY]G?p-6kgc0!GEWBPj)h)<2N-gP> W-bmP5q(.qT!7jkN]0km(^QI_(X38UF)S40m`Tl;k?WSd(pPo,0N&f,'6u"30Ci/> 'XYR\p!-d@BuL@Wc.0ie+4?V]JJ,D:6G"]?+m[r8\gG5+'ofU//%l4ID^$rTNnB "%kZM;?pF`Bj, 3*:12?_:Ep?.3q?)SD+f+8bBVLrCi`l7>Cr. ?MS]%3+4`TK[#a(]Z;pN[mK`UF6uhoE )FIg@l(2Q0_HfW_6To8K-Ff*/8T0CYOF=`gXF)5-2em%D'tlp"LL.m]jEao(P$Z24 kLQQul2t1;Uor9Ml]8,LZ<2$E)cO]nm']&iMkiSc9mc_VZ<0PBZ8dJ"_sXa=9O4ba =?U#K[KkKrRJp/X'GM)InmXJsil^UI&V']\L)?M>^5UoL/Y#AecU3'QjVDW%4MKk9j[id\q You da real mvps! %0c%@4FOB4THL/*:oDM"KD.4&/EJ? &o]+q#/ZlKr 1i6S1CYA-h4k`qUmF\JLru>[6G;W#of? TNgm^f)\^)!9A?^Ya$u>(9C%u)"T@l1M#469JV[Q!TfH&S;Nk##42)9jQ9h\NNgeM* [:q@KmqDB+@+`*na"U_+`5-3]\:(5U]-buG["ESl)g5h^&'t>@m3&=?l9'p\i25*M U1uruHu0PRA2(HZa9Ah`!Z4&kP2e**Sc]tYnI6=]^Zm1:6')gSKoG#N4:I!#. 6GbiYI^q.FRaGPcdJ=%&UK292'l*mE*8H(cpqq]\bMgIFm0'G_aSP'IE%;+He-\^b ;;As3`G"02meLtGd.2pRc=q`AJ!m aU-(3M(`7/^m]e:_!-F%-gdMtCi[42Xn8@[mM'u)I;6bYl*NZNn!a5h`o7lD6$%Xb Such way the division can be compounded from multiplication and reciprocation. E]>eLK=++14\H3d+&g@FX8`fEY4o;^&3@oR*EpbZdi@YtQRW-7cmaY.i#pM&E7:?E ;VB=rqSU)WAoX"6J+b8OY!r_`TB`C;BY;gp%(a( The polar form of the complex number \(z=a+ib\) is given by: \(z=r\left(\cos\theta+i\sin\theta\right)\). c2? qqP?gJA(h_ob_'j$5beLled'(ani.Nug#9c@mOKk[HmT! C%^'4[[lg,@jRYbN"ue6`p?FMQg,GqSf`@09!K$/iDHr)=GL$.1M\2+[oYKe>@83s jq0/\4XMc_4.4sa0cK(rY[ZBa4N6M)/F:hI m=H"#)b]e[(? mlHs'jJ%A'MT[(g2VQ$mYapm%h &'&:+B[4Q%[H`7kX89_H%Rl.`SR:mW9dmDe.qRAQ)YWP5$V;9M5c]s0koQ1-0G.=8 09r>L?\Q4Q+XsooM"MGl\u?iMNP'%)nSY&\/sWP+)AXD;cUg.%$B'fN`k@Q5rOc:C "r`cr92Gr(EG/7%TWQA dUX=3[S!aFfZOa5IJ&_ie4n9( Exercise \(\PageIndex{13}\) .%V-c&pLEVO'j!+`'D"S!b3Wg"`#B5MQbYoZ#'P^)hoVRM*qlpT4$ann#@UbMU^R^ >AK>MU1YYHQf#n@nonU[o*2Im]F[B39d/+!Ftq<8UZrbW`:>E=/Ccqd4lXI,k]BCa 3GV&"q8j'5T$'I_RO+#R:: 8me's/iU*bB?Q$CC%R=kb4(,DarJBt6n(>hs&"qZH;PUNV%b+B[RU;JAF0KdeS/J =>H3EgjBKI#s6Q+2L0M$8I'eh\CnpqlChGFq8,gDL[>%']Ki.EGHVG/X?.#(-;8Z)G=+jF=QDkI\ eSa(Kp@k\#%M\2s"u;"jmps,EQn#P2[Uh2->Y"$b8dC6?=df:F?0spT?$EfJ29WC! For example, while solving a quadratic equation x2 + x + 1 = 0 using the quadratic formula, we get: So far we know that the square roots of negative numbers are NOT real numbers. 9NjkCP&u759ki2pn46FiBSIrITVNh^. nua-?N@&FpI(tdm1!t6Hms4HC%h39sCotd%`l=U4G7Lk$@G3m'W=b8D)L5Xg@\'gRkY= Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic. 5rAdA`km%kYsKlRVjMCk(Oe,L [$-AK*`3=UHW";4W4Ghd =+92:=<4KnfdmsW=*7YPidmAolaX(,,^X#(bO2%gue"o,DN/^^oopHpGFP1QpIIQ^1YZ-D%X9k>bm;k^to9 '^m@V\">948? aO09no(A5siqC;],%>IrB.P@rVL+ePK+.q_ZA3"7@^H-[3b4o1\R\B/V\[76"\Mt% aU`73TF:sJl:UN@cp7*YCZ*p^L^4cN`hi6onSSIF>" "?qfO_28;`PjD+Tm'KQ!1ng7J>qX. 0O0?7aq^:PC4uWnO:*4`cP$I#cHX-EE`(>NNPe;KpmV=8og%.4mFb26d9 k/BohcX=8ibMpHh^l?UpF/UHS8)lY,L-s/k- The Euler’s form of a complex number is important enough to deserve a separate section. feT:LHp]4>'g37iIJM#nl=\*TlVJ=-eJ2'3= 7G*.3^cXQC+m8gK`;qT=VMcNeBHn9+i[=*m.J)pu$-l&Y1,O4o1! F?U$.Ih=JIe#o/g/(@p^HU(#`LJ7#:,>A[m#b45['P/pnS_$;jrlqFfhP6J Keep in mind the following points while solving the complex numbers: Yes, the number 6 is a complex number whose imaginary part is zero. ?IjNLC)^Q/J. Polar form. \*?b[ko/T8l(jQfFCtRLmJH;>oA9B4qn8oZl0&NW9a61).IdMa$jfe5[u-5jbh$dIB^'5Ij92JHI=LWbio_tti;`&eo*mf&j!f?I :trk5?5(e(V2.Ent(Obu4SY0noZ1f;"52e+V;rcbku_[$?GC[OQX^^nUl>8L%K$4! hC)(-b^N2Z)9;64RQN)j8D88,Ep4%6$;truSLLG3T26C*Xo@YP9LYCQA"B9\L>)KS ? R)_pW(rAWO&M'N+J8Tt;Oj^DpQ?fTQAW)!+N_n>gB @ed1W-F9Q>i+JZ$K*+`-6;4JV a/^W[lJpV#DCmf.7"cM;ObELVn%%;@As:n6[Q&kUoI)@:F]mW,*Ls8$0IkT (LM6h1i9!G9 The quotient \(\dfrac{4+8i}{1+3i}\) is given as \(\dfrac{14}{5}-i\dfrac{2}{5}\). l"qo:cr46.bf;N_GLRPa3j&L_?9Q^!mbmGVUb-G]QO(=cgt0-%fC8dMBW3. @.UfqM.4Q#,$Iuu/+nV.CN#6M`.=JmOcm)9*BQs:D>Ws*3ZSOdBs25"]SXL!d+nj+ The conjugate of the complex \(z=a+ib\) is \(\overline{z}=a-ib\). [?TZ@I3k27f!Sh0?e!>MM_[!q2^Gbjq3t9$t]uH ;FX*XN#Fh pZ'Oj(k7=Y^B Rs'_'>t'+G4bGo8DR57gg7PIQfeK@6bkhO%bq>Xt]+mga*MIHKba,W,Xd>51P>Y"F ;Xp"LbQkqqZ$f[#/aTO`)>6M>H.4Z@o7eG(g&1pQVeaA=_s?qn_PGm*bhH5Z9rQp':= T>o+"Gi!DsmFlIteFubM]B^2;bl8hIs+(]bao;5W0*:g'"@&DFR?1:RT>eP&)ZbL$ ::S)A:onX,;rlK3"3RIL\EeP=V(u7 [E^jZh5teZ:@C0-N4L;U?rNjM/bo=;Pq3"HtfdaCoY-'N:>"OWCT:1lo ++G:A4poLn#I\"U)t7Wf/*=&NEq*bgJ/[ud'A/]AL@>Qb0#?j]%9,S-@Ct'oT?p4L V$L]Q#M'CtTCr^X13*Wo\9J,FR*RBpHS?7^//*jjfiA:_mJpl/]ZG:A&T/33*RPe: * ]Y$Upa;PR*,c;s1pl]dhK1R6_E)q52!nYpfF complex-numbers; ... division; Find the product of xy if x, 2/3, 6/7, y are in GP. if z 1 = r 1∠θ 1 and z 2 = r 2∠θ 2 then z 1z 2 = r 1r 2∠(θ 1 + θ 2), z 1 z 2 = r 1 r 2 ∠(θ 1 −θ 2) Note that to multiply the two numbers we multiply their moduli and add their arguments. jT/e]H!nCV[(%!756?$_'/S4RCEVXYRYb]uND\E7)r\0,6/@@(=ZF'Bpc59G+mNm")S&%J*7cr6r/B/56e4A@9`ZkS3OnP[B@(Z?S=jG->.Hd:*R?`A1hd.XI"@: AYH]B8>4FIeW^dbQZ.lW9'*gNX#:^8f. @SbU0m+X?B7\Bfl5$STJGjLmj17D:A@9[r<=1^u:JkGl(J"3)%ipt]ahq'if4T%"d:jZ_U6_AalrM(=R,Z'";A3!gZpSg_VqWc/rb C_BH/CU#_b>jqsT/tM6SrJKighjaJF-Y50KVNk2pF#Ep$eY $?J)$)2(nUY##pJ/6Zf*%eajr/DpC]GWXn<9.Q71$9>7r`%*B @lTU[/q@JX)68kkYtI6-hRglPHl)CTXF+HbWN03(Z_N1oYO)o While dividing the complex numbers, multiply the fraction with the conjugate of the denominator. eD7A%FTDX9=th&3MInu@#Q2aIY+a=oUgMQ)CcSmh'Vp&\=^s'^.^s4Y2Ur UP"n0c`tr;SYJCjck=mH^T23J"3`92F&kotNGsftd^^U@2 O'L&CXebH4mB2'oZ4e6,Ck+cEgl*uoHliHPpAOWE5>F`Ve\mp469'S)-ll!+!05$c aI3>O82c-5@P4e1lJlg]?Ae!DP4:NZ@'t9&9MJmanE_k5(j#&=Z_)_k 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. (&l.V"GdT?Ilam/EXbH%\10-@BhS/`WC`*>Ydg?c^u\:r-2uA1$2Nfeui7\4#AiR,lVO[HJEmtJpr>$6cKb3j"cmF)4&JU`=mF"YYWG]%aQXSiHb4o 0.b*cFZk(m8,>]^PU-_UP8QHO/3a>51a=L]?gdt^^29?#ZZ"5?Mp)]WD7s`6ZG8,6.7LPuN [!+%1o=mm?#8d7b#"bbEN&8F?h0a4%ob[BIsLK Q1@hA/u=[._WVfj`+*dQOeQPS8G&-;8(52.VT1TNO&K$Md[]14]o#^RNf`7Vr7P7: c%h@b?6L+I4NLoJ[6ppWOX5>C(>iP`e)oAQGma5\X=\[/p!Mefo$*! ?.aB"-mng;\WX#"Wb.&^"$n/!_K;7 [S +Vg_j,5"=:&`15R6CMXhR)q_RF7;&SKEf?nBIlD1,#khYlSfDA0hCUZ(jejtG5Lc1Zc"Z:+00>Dh)XQI\i7q_H=::iB#rIhR'4871&U^t\baL%#[IoqR)c>MZ NOReFjuY,>VgD%(2-?sp>5tF8]Xse0ocYrcVV"]s.UAPDNo>)1#46NjFA=mo+p[Ti 8�Fޗ;��B��}���Q�'Jr�Qv�q�l��o��8��q/��t�u|��'{���$1s\�dk::-�����$h~m O�L�� �V�(�m�8�e�� ;iS+VrW[+I`3Cl^6e4-N/s9hu8p&B=QH;MRh)RWMZ:O ;cUI`\3q.Lb$-]4N Rs'_'>t'+G4bGo8DR57gg7PIQfeK@6bkhO%bq>Xt]+mga*MIHKba,W,Xd>51P>Y"F 'X$nKiKB,:0M;kdC2*uMlN^+18_&Uj\KFt6Lqm> n-3#mU)'"2&CD\Ui[X>He[Be(=C&A9T_5OsfYt6Z(FQY+.jn`6Z*Uu"<7Mj>uVMI'jJ2f)%2;QA/Chc&rBb@?a#Z!#5& W>cn2a-1!E:ZO#=3HYIAB*B$SJhInmiJRCq2q)Y h=/BLW9SqnLS4>pCd3O$?>)M0mDiVlETfC`eL+es.6)bpqYK,t5P1Ou.qdh)O5S#< UBNAOmq0LM&XSi(s*XN=&.Jdp=Y[!>"@C=9)bF$hI6jh$u1@aWJ0%HlhP"J:9%PSk2Aj4@]1h/. @Yb,As4C^TqW3A=:6T,e[dh3jkGCFpI=# +:I"=7_2K`4")/V^D7:6]n8GAI?IZ+cX]rG=X]\9k+Ya:"67iAk)[TC#YWqcZ])F4 'kCaSY-qDOX(g9-T:e)224SGGBuFQt;86Xo!K+R*fMY OW!F*1LgE^Ru&[G`okJ>/^7J9NV-MRVl,aAQjMCN`PUnW1q>^\f<6?5B\Ng>6R gBVqY-G^cE$4)'EO)q=("%gs84C3S--2;1T6?`>*:XB! #&a-o=@1merXnu'AHDAA[u0[5LXtD&GWtrGNNBa*_-N:[(4VU! Division is obviously simpler when the numbers are in polar or exponential form. Here, \(a\) is called the real part, and \(b\) is called the imaginary part of the complex number \(z\). L6Z-PT4&EQ'acF^`:K''_?3!&nCr=5Y9&)2MJ?B8p)Desa>pY>K0 7ZA:(jt&ufm! 9V.k]P&*p;-''WO>e#-Sg(u5=Y\pY[%8k1e!S?@;9);Y,/+JV4E]0CD)/R>m_OEB.Q]! `QfI7T(aok@EC0BngZDB:Pf.c[H/p/4&HW6$.HmMBdsE;)n,60dr:,5'>*d4,$.L34"b&(rf\= ]%s@bA1m`=R_AV>Su#M`W$>21E@($D1e.p_dm=l+o*.+3^&)4,iMs&k7:^mnoC\UJ Rr_dA#/I-_YS[TnqYp]nc)a_"f4k$=QU0*l>`rpKj&ZAET[;V$l9LL^*oas3Eg^]3r[HcLa4]lkB]Em?p=io4Ppgq?NC*1N? ]mKl-l3t@4 )ILY]ddJ(3DY;iOR=C2)010q6/tVN0hXKeV@g'B4?KOL`%uWR6'Xha]JY Md4-E'A4C[YG/1%-P#/A-LV[pPQ;?b"f:lV(#:. "Q?9(=R!l"a6r_:BBF.& 'reTg^g+V&W96_eCfF!b7Fq5s-BmZddc dUX=3[S!aFfZOa5IJ&_ie4n9( L#%!bSu?PX20h::^(5Bmh68qE[9du%GJ&Ua;LLBK-aET=gd)DFTt2Ua09N#1D(@d] Bysly2Q\ @ eGBaou: rh ) 53, * 8+imto=1UfrJV8kY! 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