{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times " 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } {PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 22 "Module 6 : Precalculus" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 40 "601 : Complex N umbers - A Geometric View" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 17 "O B J E C T I V E" }}{PARA 0 "" 0 "" {TEXT -1 254 "In \+ this project we will examine at complex numbers from both an algebraic and geometric point of view. We will look at where the come from, how to define them in Maple, how to perform mathematical operations, and \+ what these operations mean geometrically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 9 "S E T U P" }}{PARA 0 "" 0 "" {TEXT -1 252 "In this project we will use the following command packag es. Type and execute this line before begining the project below. If y ou re-enter the worksheet for this project, be sure to re-execute this statement before jumping to any point in the worksheet." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "resta rt; with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name cha ngecoords has been redefined\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________________________________________ ____________________________________" }}{PARA 4 "" 0 "" {TEXT -1 27 "A . Defining Complex Numbers" }}{PARA 0 "" 0 "" {TEXT -1 83 "___________ ______________________________________________________________________ __" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 74 "Complex numbers come about naturally as s olutions to polynomial equations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 119 "Note that Maple uses a capital I for the imaginery unit, whereas we normally write this as a small letter i in by hand." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve( x^2 + 1 = 0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$^#\"\"\"^#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve ( x^2 + x+ 1 = 0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&#!\"\"\"\"#\" \"\"*&^##F'F&F'-%%sqrtG6#\"\"$F'F',&F$F'*&^#F$F'F+F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 132 "To define a complex numbers directly, enter the number using capital I for the imaginery \+ unit and Maples usual * for multiplication." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "Note that we use := to assign a value to a variable. The sqrt command is the square root function." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " z := 4 + I; w := 1 + 3*I; \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG^$\"\"%\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG^$ \"\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " u := 5 - 4* I; v := -5 + sqrt(5)*I;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG^$ \"\"&!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG,&!\"&\"\"\"*&^#F'F '-%%sqrtG6#\"\"&F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "we can view these complex numbers by using the complex plot command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 " complexplot( \{z,w,u,v \}, x = -6..6, \n y = - 6..6, style = point, color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 349 262 262 {PLOTDATA 2 "6'-%'CURVESG6#7&7$$\"\"&\"\"!$!\"%F*7$$!\"&F*$\"3 \")*y*\\xz1OA!#<7$$\"\"\"F*$\"\"$F*7$$\"\"%F*F4-%&STYLEG6#%&POINTG-%+A XESLABELSG6$Q\"x6\"Q\"yFC-%'COLOURG6&%$RGBG$F*F*FI$\"*++++\"!\")-%%VIE WG6$;$!\"'F*$\"\"'F*FP" 1 5 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 61 "We can also define complex numbers in a trigonomet ric format." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 31 " z := cos(Pi/7) + sin(Pi/7) *I;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"zG,&-%$cosG6#,$%#PiG#\"\"\"\"\"(F,*&^#F,F,-%$sinG F(F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " w := 5*cos(2*Pi/ 3) + 5*sin(2*Pi/3)*I;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG,&#! \"&\"\"#\"\"\"*&^##\"\"&F(F)-%%sqrtG6#\"\"$F)F)" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 71 " complexplot( \{z,w\}, x = -6..6, y = 0..6,\n \+ style = point, color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 349 262 262 {PLOTDATA 2 "6'-%'CURVESG6#7$7$$!3++++++++D!#<$\"3w#>A*=q7IVF*7$$ \"3Y\">C!z')o4!*!#=$\"3@\"ev6RP)QVF0-%&STYLEG6#%&POINTG-%+AXESLABELSG6 $Q\"x6\"Q\"yF;-%'COLOURG6&%$RGBG$\"\"!FBFA$\"*++++\"!\")-%%VIEWG6$;$! \"'FB$\"\"'FB;FAFL" 1 5 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 83 "_______________________________________________________ ____________________________" }}{PARA 4 "" 0 "" {TEXT -1 32 "B. Proper ties of Complex Numbers" }}{PARA 0 "" 0 "" {TEXT -1 83 "______________ _____________________________________________________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 48 "Lets explore some properties of complex n umbers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "What does the negative of a complex number look like?" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " z := 4 + 3*I;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG^$\"\"%\"\"$" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 239 " display( complexplot( \{0 ,z\}, x = -6..6, \n color=red), \n comp lexplot( \{0,-z\}, x = -6..6, \n color=blue),\n \+ polarplot( abs(z), scaling=\n constrained, col or = gold));" }}{PARA 13 "" 1 "" {GLPLOT2D 349 262 262 {PLOTDATA 2 "6( -%'CURVESG6$7$7$$\"\"!F)F(7$$\"\"%F)$\"\"$F)-%'COLOURG6&%$RGBG$\"*++++ \"!\")F(F(-F$6$7$F'7$$!\"%F)$!\"$F)-F06&F2F(F(F3-F$6$7S7$$!\"&F)$\"3Q6 L:&GM50#!#E7$$!3cp0E(=\"=`\\!#<$!3]M+Z#*3REo!#=7$$!3*G9J*\\5!p$[FL$!3Y JL\"yZWmE\"FL7$$!36[0@SKHCYFL$!3i]7`g^b,>FL7$$!3&\\\\E/YNlK%FL$!3K#)>& zY5i]#FL7$$!3/2RpFL$!39#\\YZv#y?YFL7$$!3+(G&4SJpi7FL$!3_FFsmS$z$[FL7$$!3id&* RLToDnFO$!3G2ff\"oeX&\\FL7$$\"3[ls)f$G`&)H!#?$!3U*Gf'3\"*****\\FL7$$\" 3u6%))ex6C\"oFO$!3o%=*3WOP`\\FL7$$\"3Ac1?t54B8FL$!3#=qL^Fm<#[FL7$$\"3! QvKOYal)=FL$!3&pE&ooKVIYFL7$$\"3lS+/Nw:@DFL$!3*p^tv`UyJ%FL7$$\"355LaOZ u>IFL$!3?,yV?]6&)RFL7$$\"3vub$>:5-@&RFL $!3!R.xPpBG1$FL7$$\"3l'Gz)oLlHVFL$!3K+F_1-#3]#FL7$$\"3#*[%zd8oah%FL$!3 RIw'=YwG#>FL7$$\"3'*eW#fG\"FL7$$\"3Ag%H%4]^`\\FL$! 3Iu)zb/D@!oFO7$$\"3mo_jaf(***\\FL$!3[a&[L#\\l]:!#>7$$\"3?U%Q'4!>U&\\FL $\"3m&)o)Q.tRFL$ \"3_1^)G8Ac.$FL7$$\"3M;y)*pHGbNFL$\"304dd/Rn:NFL7$$\"3%yA#p;KHCIFL$\"3 _HKm#Hk;)RFL7$$\"3!*fW!4@W`\\#FL$\"3]=Jy/<\"GL%FL7$$\"3/j(fFu!R')=FL$ \"36TQHp**\\IYFL7$$\"3,cii#zKVI\"FL$\"3.D4XZV(o#[FL7$$\"3;4Ya'*Q$yX'FO $\"3/#Hpi*47e\\FL7$$\"3F4ky>>pG:F^u$\"3YKb&4jw***\\FL7$$!3#ed@yC;'QkFO $\"3Q*[1G#4Pe\\FL7$$!3m4!p3n?vF\"FL$\"32*Ge_\")RS$[FL7$$!3@'z%)3>**y\" >FL$\"3X(3sofQvh%FL7$$!30\\HF'G];]#FL$\"374UI1QvY0$FL$ \"3!f_xrk.%eRFL7$$!3u:$*y\">]'\\NFL$\"3Hw.vJ5O@NFL7$$!3;tMg&>*3]RFL$\" 3'o+&[x#>a1$FL7$$!30sX3d(orL%FL$\"3CY!p!)[jx[#FL7$$!3eBi\"eZ+Dh%FL$\"3 (pnX-H%)*H>FL7$$!3Y,r2MWMF[FL$\"3vRx\"=#>f-8FL7$$!3A^OqnxZ`\\FL$\"3-uV 0^l$[!oFO7$FD$!3Q6L:&GM50#FH-F06&F2$\")+++!)F5$\")AR!)\\F5$\")Vyg>F5-% (SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%Q\"x6\"Q!6\"%(DEFAULTG-%%VIEW G6$;$!\"'F)$\"\"'F)Fh]l" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Every complex number has a modu lus and an argument." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 304 "The modulus or absolute value of a complex number, |z| given by the maple command abs(z), is the distance from the number to the origin. The argument of a complex number, given by the Maple comm and argument(z), is the counter-clockwise angle from the x-axis to the number. This angle is given in radians." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " z := 4 + 3*I;\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG^$\"\"%\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " abs(z);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " evalf(argument( z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)36]V'!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Lets take a look at \+ what is going on." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " z := 4 + 3*I; w := -1 + 2*I; \n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG^$\"\"%\"\"$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"wG^$!\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "display( complexplot( \{0,z\}, x = -6..6,color=red), \n complexplot( \{0,w\}, x = -6..6,color=blue),\n po larplot( \{abs(z),abs(w)\}, scaling=constrained, color = gold));" }} {PARA 13 "" 1 "" {GLPLOT2D 349 262 262 {PLOTDATA 2 "6)-%'CURVESG6$7$7$ $\"\"!F)F(7$$\"\"%F)$\"\"$F)-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6$7 $F'7$$!\"\"F)$\"\"#F)-F06&F2F(F(F3-F$6$7gu7$$!3%=E#QXwN&>%!#<$\"3a\\oW ab5?FFF7$$!3[Q![)eDLmWFF$\"3s2Z@4tjZAFF7$$!35PZr=$4Vo%FF$\"3;)zK=F)\\[ h[\\$FF7$$!3#=q?/!Rq!=$FF$!37Kq,W]'y&QFF7$$!391 8y.R/\\FFF$!33&>S]v_k<%FF7$$!3Abjv\"yLdG#FF$!3#*Heu$QcpW%FF7$$!3_<7Wt# )[)z\"FF$!3+fi:nXMlYFF7$$!3mN;!4pK2H\"FF$!3tW;_l)G0$[FF7$$!3Qh\"G(4^d# o(Fgn$!3[wrj1biS\\FF7$$!3w6C6P2@qBFgn$!3mL<0$ FF$!3!R]'3*z#ogRFF7$$\"3[bGT]+?TMFF$!3fJ#HEO:ui$FF7$$\"3e?o)4K5f!QFF$! 3I!eEKE\"pUKFF7$$\"3<%z3)o\"Qy7%FF$!3A;H&Q\"f^@GFF7$$\"341qI(ykLS%FF$! 3'3!\\1nPioBFF7$$\"3A()>QoHRHYFF$!3a^#za!e5*)=FF7$$\"3&f.@=N4H![FF$!3a z^)QC')**Q\"FF7$$\"3f&)z379yA\\FF$!3-o)o+WiP/#4NFgn7$$\"3w8%G1@^o*\\FF$\"3+;Tg]M?u\"G&zdjn% FF$\"3!QL_Pqa'pm9=AhGFF$ \"3r%p=$*z:/5%FF7$$\"3YZQIZVD.CFF$\"3#[!\\,X-c%Q%FF7$$\"3R(*4RIPO<>FF$ \"3\"z68X'4w49FF$\"3KC`9C%3tz%FF7$$\"3p>!*3/quC))Fg n$\"392_xOx]@\\FF7$$\"33E4TnD;aMFgn$\"3eA/W\"\\a!))\\FF7$$!3gn#[ZM&)o& >Fgn$\"3?%p#>K\"ph*\\FF7$$!3M'RX-]4]M(Fgn$\"3!**o03gcd%\\FF7$$!3!z_w4* onX7FF$\"3omDGYLMU[FF7$$!30$)RK4,IVA#F F$\"3]v6RxQ;zWFF7$$!31Tk7d&Qkn#FF$\"3jPy4&)*[LA%FF7$$!3S5'*3)>lTU$FF$ \"3%\\AJ71+Nk$FF7$$!3YL'z(3HE\\SFF$\"3l#)fM7s;LHFF7$$!3u;zww-S`VFF$\"3 _Gs_y!\\#fCFF7$$!3NeH.v%>Bg%FF$\"3%eS:C.QT&>FF7$$!3S\\I,LK'Gz%FF$\"3yW 3#3'3CC9FF7$$!3)Q(>T4ZhA\\FF$\"33ZKBdtyi()Fgn7$$!3/IzpOiX**\\FF$!3CiOr 3.$QP(!#>7$$!3q4F5C,j%*[FF$!3:71!y(e2@5FF7$$!3y%oLg6S8v%FF$!3%3#*G$pj; d:FF7$$!3!RakQ%y_\\XFF$!33(4+,CxS2#FF7$$!3%3w'oH!z;H%FF$!3/w.1)pTac#FF 7$$!3O%zsxfp4)RFF$!3=5]O1w?DIFF7$$!3\"f&3#=9PMJ$FF$!3^;xK70[WPFF7$$!3u ^;5/tF=DFF$!3*[E\\Y0B&>VFF7$$!3z'>^*[HuQ?FF$!3=F8:!>rac%FF7$$!3#)y!fL* H_N:FF$!3Eqj%\\$)y$eZFF7$$!32&H='>PY95FF$!3!>4gn$[+'*[FF7$$!3)\\XDri'= ;[Fgn$!39i!y9H]n(\\FF7$$\"3zskAy_\"p5$Fa_l$!3gJF%)pM!***\\FF7$$\"3H0g \"\\U'HMaFgn$!3!olD?p!Qq\\FF7$$\"3M7?Y')[1]5FF$!3m.=9#*H\\))[FF7$$\"3# 4GjCGPca\"FF$!3MO]U*3.^v%FF7$$\"3E\\)QxFx_/#FF$!3C*)Q$ziZDc%FF7$$\"3K, 7j#eg9_#FF$!3*y,sVjlwJ%FF7$$\"3G\\'y#*)fsoHFF$!3?i!=\"*elK-%FF7$$\"3!4 )eySR%>Q$FF$!3e6po!)Qs#o$FF7$$\"3e.k5C4qOSFF$!3MpxVM9V]HFF7$$\"3#yF'f# )ykNXFF$!3^M7J\"=dU5#FF7$$\"3E3aK*fIIt%FF$!3-s'fa&>'>h\"FF7$$\"3uy;i0X :x[FF$!3QjT0(*z_,6FF7$$\"3#R>[b$yRm\\FF$!3k]`#e0!*py&Fgn7$$\"3+\\'HMFF$\"3TH%)*ps5M'Fgn$\"3W)H2c!yif\\FF7$$\"3!=,\"R=ba/5Fgn$\"3#=G]ky!**)*\\FF 7$$!3:\\YN6r[VVFgn$\"3G2uj#R)4\")\\FF7$$!3P%QPUnzx_*Fgn$\"3aXAAH?Q3\\F F7$$!3q>!*\\@Ewg9FF$\"3__&)R$pe=y%FF7$$!3]M9A7St_>FF$\"3Iq**=n`\"Hg%FF 7$$!3$p'z/?!*HBCFF$\"3%=j;Vn8NP%FF7$$!3e6>A$**fL&GFF$\"3brR/%\\!*e5%FF 7$$!3#R\")4w(p8aKFF$\"3Wc<8C'Ghz$FF7$$!3%=%oJ0o^@OFF$\"3sTZyXrSZMFF7$$ !3cID%H1H<&RFF$\"3S=q0#)\\IjIFF7$$!3W4)3a'fsqUFF$\"3l*>K*=I<+EFF7$$!3z EedV\")pNXFF$\"3qP\\DU)[T5#FF7$$!3#f4cop$HVZFF$\"3c,Kgir]\"e\"FF7$$!3; &e:&Rl)3*[FF$\"3F/@A-%f)Q5FF7$$!3uWJ/7-xq\\FF$\"3%pX#yA2b)R&Fgn7$$!3C> E2O^()**\\FF$\"3U3MVjpeLNFa_l7$$!3o;y:\"*R!z(\\FF$!3+cz'[0Vap%Fgn7$$!3 Zhi%GA\"30\\FF$!3SX>f`'yip*Fgn7$$!3E[Y^-P/sZFF$!3%H1r%)G9D\\\"FF7$$!31 b&z/vQMe%FF$!3YkKOE8-)*>FF7$$!3`\"[e@bi9M%FF$!3Ucc>pEE![#FF7$$!3;#)pQn F$*[SFF$!3G/pV@HiLHFF7$$!3?fH,:iM,MFF$!3')***)=\"e6[m$FF7$$!3\"[uMPejR i#FF$!3QHP()o-:cUFF7$$!3mJTV***GY:#FF$!3>c'*y?o$>^%FF7$$!3/e]On6ng;FF$ !3Z))4r\\1;;ZFF7$$!3SO=Yr]tZ6FF$!37WUtry[m[FF7$$!3&fGwb!e#o@'Fgn$!3-h2 2P0?h\\FF7$$!3T70fGP6,7Fgn$!3_M+iOm;#\\FF7$$\"3[g8.%Q$er8FF$!3^# \\'\\lq>3[FF7$$\"3a&e!R&*Q&)o=FF$!3QRQ4PbgPYFF7$$\"3!\\M)p&*QYXBFF$!32 -Q]dWu:WFF7$$\"37-j#pPWhz#FF$!3avTGSl1XTFF7$$\"3kI`Z')H\"f@$FF$!3s-^yx TcGQFF7$$\"3A3%G*fO(>f$FF$!3Ajz4N%y\"yMFF7$$\"3iaO3Mb2IRFF$!3!))o=PWO5 4$FF7$$\"3&)*o[#fckEUFF$!3?j4@$RH7n#FF7$$\"3K[+jz*\\&yWFF$!3JHwwmO>BAF F7$$\"3YcT=MA_\"p%FF$!3L;fb5:0HvwOFgn7$$\"3,7@OV\"=;#\\FF$\"3U,?>xue=))F gn7$$\"35zkd(H(*Q![FF$\"3i(3=o(zc'Q\"FF7$$\"3!Q;!)pPlXj%FF$\"3;gQ>'\\! 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The blue line indicates the position of w, and the length of the blue segment is |w|." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Lets compare the argument of z and z2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " a1 := evalf( argument(z)) ; \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a1G$\"+)36]V'!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " a2 := evalf( argument(z^2)) ; \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a2G$\"+=A+(G\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " a2 / a1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+,+++? !\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_____ ______________________________________________________________________ ________" }}{PARA 4 "" 0 "" {TEXT -1 24 "C. The Complex Conjugate" }} {PARA 0 "" 0 "" {TEXT -1 83 "_________________________________________ __________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "Every complex number has a complex conjugate. The conjugate of a+ib i s a-ib, and vice-versa." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " z := 5 + 2*I;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG^$\"\"&\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " conjugate(z);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#^$\"\"&!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 136 "The product of a complex number and its \+ conjugate is always a real number. In fact, not just any real number, \+ the square of the modulus." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The evalc command forces Maple to evaluate this e xpression as a complex number." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " z := 5 + 2*I;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG^$\"\"&\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " z*conjugate(z); evalc(%); \n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#H" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"#H" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " abs(z)^2;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#H" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 38 "But what does the conjugate look like ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 " display( complexplot( \{0,z\}, x = -6..6, color=blu e),\n complexplot( \{0,conjugate(z)\}, x = -6..6, color=green ),\n polarplot( abs(z),scaling = constrained ));" }}{PARA 13 "" 1 "" {GLPLOT2D 349 262 262 {PLOTDATA 2 "6(-%'CURVESG6$7$7$$\"\"!F)F (7$$\"\"&F)$\"\"#F)-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-F$6$7$F'7$F+$! \"#F)-F06&F2F(F3F(-F$6$7S7$$!3u.X82[;&Q&!#<$\"3fxn7I:.4A!#E7$$!3;j#*f- %RZL&FD$!3wCYx)*zC_t!#=7$$!3+Di$>'=]4_FD$!3uAV,`y@k8FD7$$!3Ml.,Vi^!)\\ FD$!3UyN'eeP![?FD7$$!3[Z;m*H@)fYFD$!3GAuD#Gr#*p#FD7$$!3/t\"p())*plD%FD $!3KGMeN)G()H$FD7$$!3LHrCPtFM$!3$RLZ\\oY\\L &FD7$$\"34*z1op7]U\"FD$!3y2ch57?$>&FD7$$\"35B&f3M\")=.#FD$!3_P<1c#Hr) \\FD7$$\"3?;7*Q()p`r#FD$!3q\\y@N(e/l%FD7$$\"3[PDN;YO_KFD$!3U\\*)3K-5#H %FD7$$\"3BOe;)o8x\"QFD$!3cS$*y!*R.)z$FD7$$\"3-c9EIUacUFD$!3kdrBt?w)H$F D7$$\"3K;g$)*QzJm%FD$!3Ixt-$zlMp#FD7$$\"3aO+l98,r\\FD$!3uuL=I8+r?FD7$$ \"3/EelA0-/_FD$!3-$)ye)G!)\\Q\"FD7$$\"3n8%y+%*)4N`FD$!3i$)p$=)G6EtFM7$ $\"3m=T[4*Q^Q&FD$!3W(HXUY1,n\"!#>7$$\"3$*Q$3:%))\\FD$\"3GRGze>rG?FD7$$ \"3iZ,,!>#>rYFD$\"3gdG-&RX&zEFD7$$\"3u\"RTIX)3zUFD$\"3+\"Hab4l%pKFD7$$ \"3#[gwe%p:HQFD$\"3)4Xcon'\\'y$FD7$$\"3U&FD7$$\"3oi%)*f'**HbpFM$\"3H6[sR(f+M&FD 7$$\"3kIpZ&e^kk\"F\\u$\"3;NL!zjR^Q&FD7$$!3$=j#)[#>gMpFM$\"3i2L%)=*G.M& FD7$$!3II,Vr=$fP\"FD$\"35$=$)y@?k?&FD7$$!3%*3pFX1kl?FD$\"3%*f@\\p7Ct\\ FD7$$!3K3!zi\")fVp#FD$\"3!)*eN^)GmiYFD7$$!3;w)G[(e)**G$FD$\"3_bEk-7LjU FD7$$!3)GT#4e-4BQFD$\"3g%Rz-!>i#z$FD7$$!3_87CkiPaUFD$\"3p(yWCgd:I$FD7$ $!39F5l5PFrYFD$\"37@Z)oF.%zEFD7$$!3/O80Z]\"y'\\FD$\"3m'pIcpc'y?FD7$$!3 1E&FD$\"3P=h3NW$HS\"FD7$$!3Cx@^H)e]L&FD$\"315\"3EEL!HtFM7$FB$! 3fxn7I:.4AFG-F06&F2$\"#5!\"\"F(F(-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLAB ELSG6%Q\"x6\"Q!6\"%(DEFAULTG-%%VIEWG6$;$!\"'F)$\"\"'F)Fc]l" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C urve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 183 "The blue line indicates the position of z, and the green line ind icates the position of w. The conjugate of a complex number is a refle ction of the original number through the x-axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "How are the arguments of \+ the number and its conjugate related?" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " z := 5 + 2*I;\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG^$\"\"&\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " evalf(argument(z));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+rP10Q!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " evalf(argument(zcon));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%)argu mentG6#%%zconG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }