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The complex numbers [ 1,0] and [0,1] are written 1 and I respectively, and so each complex n umber [a,b] = " }{XPPEDIT 18 0 "a*[1,0] + b*[0,1] = a + b*I" "6#/,&*& %\"aG\"\"\"7$\"\"\"\"\"!F'F'*&%\"bGF'7$F*\"\"\"F'F',&F&F'*&F,F'%\"IGF' F'" }{TEXT -1 9 ". The " }{TEXT 267 9 "real part" }{TEXT -1 5 " of \+ " }{XPPEDIT 18 0 "a + b*I" "6#,&%\"aG\"\"\"*&%\"bGF%%\"IGF%F%" } {TEXT -1 12 " is a; the " }{TEXT 268 15 "imaginary part " }{TEXT -1 11 "is b. The " }{TEXT 269 9 "conjugate" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^`*`" "6#)%\"zG%\"*G" }{TEXT -1 5 " of " }{XPPEDIT 18 0 "z = a +b*I" "6#/%\"zG,&%\"aG\"\"\"*&%\"bGF'%\"IGF'F'" }{TEXT -1 16 " is defi ned as " }{XPPEDIT 18 0 "z^`*` = a-b*I" "6#/)%\"zG%\"*G,&%\"aG\"\"\"* &%\"bGF)%\"IGF)!\"\"" }{TEXT -1 8 ". The " }{TEXT 271 14 "absolute v alue" }{TEXT -1 2 " " }{XPPEDIT 18 0 "abs(z)" "6#-%$absG6#%\"zG" } {TEXT -1 22 " of z is defined as " }{XPPEDIT 18 0 "abs(z) = sqrt(z*z ^`*`)" "6#/-%$absG6#%\"zG-%%sqrtG6#*&F'\"\"\")F'%\"*GF," }{TEXT -1 8 " . The " }{TEXT 270 8 "argument" }{TEXT -1 17 " of z the angle " } {XPPEDIT 18 0 " theta" "6#%&thetaG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " -pi " "6#,$%#piG!\"\"" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "theta <= pi" "6#1%&thetaG%#piG" }{TEXT -1 123 " that the ray from 0 to z makes wi th the positive x-axis. So each nonzero complex number z can be writ ten in the form " }{XPPEDIT 18 0 " r*(cos(theta)+I*sin(theta))" "6#*& %\"rG\"\"\",&-%$cosG6#%&thetaGF%*&%\"IGF%-%$sinG6#F*F%F%F%" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "r = abs(z)" "6#/%\"rG-%$absG6#%\"zG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "theta = argument(z)" "6#/%&thetaG-% )argumentG6#%\"zG" }{TEXT -1 16 ". The set of " }{TEXT 277 1 "C" } {TEXT -1 31 " complex numbers is called the " }{TEXT 274 13 "complex p lane" }{TEXT -1 29 ". The x-axis is called the " }{TEXT 275 9 "real \+ axis" }{TEXT -1 30 " and the y-axis is called the " }{TEXT 276 15 "ima ginary axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }{TEXT 273 10 "Theorem. " }{TEXT -1 182 " The complex numbers form a field under addition and multiplication. The x-axis ( i.e., the complex numbers with imaginary part = 0) is a subfield isomo rphic with the real numbers." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 11 " vocabulary" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "z := 2+3*I;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG,&\"\"#\"\"\"%\"IG\"\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 " z*(3+9*I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&!#@\"\"\"%\"IG\"#F" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "z+(4-13*I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\" '\"\"\"%\"IG!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "1/z;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"\"#\"#8\"\"\"%\"IG#!\"$F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "seq(z^i,i=1..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6',&\"\"#\"\"\"%\"IG\"\"$,&!\"&F%F&\"#7,&!#YF% F&\"\"*,&!$>\"F%F&!$?\",&\"$A\"F%F&!$(f" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Re(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Im(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "abs( z) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#\"#8\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "argument(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'arctanG6##\"\"$\"\"#" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 101 "Here is a word to plot the location of a given comple x number or a list or set of complex numbers. " }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 203 "drawpts := proc(z,clr)\n if type(z,complex) \+ then plot(\{[Re(z),Im(z)]\},style=point,symbol=circle,color=clr)\n el se plot(\{seq([Re(z[i]),Im(z[i])],i=1..nops(z))\},style=point,symbol=c ircle,color=clr) fi end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "drawpts(\{seq((cos(Pi/13)+I*sin(Pi/13))^i,i=1..26)\},red);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6#7<7$$!+ouk !o&!#5$\"+c'Q)H#)F*7$$!+b-ca))F*$\"+=7$$!+s\"=%4(*F*$\"+Sm:$R#F*7$$!+\"[2^[(F*$\"+!eE7j'F*7$$\"+\"[2^[(F* $\"+#eE7j'F*7$$\"+c-ca))F*$\"+@ " 0 "" {MPLTEXT 1 0 49 "frame := (z,n) - > drawpts(\{seq(z^i,i=1..n)\},red);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%&frameGR6$%\"zG%\"nG6\"6$%)operatorG%&arrowGF)-%(drawptsG6$<#-%$seq G6$)9$%\"iG/F6;\"\"\"9%%$redGF)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "movie := (z,len)-> plots[display]([seq(frame(z,n),n=1 ..len)],insequence=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&movieG R6$%\"zG%$lenG6\"6$%)operatorG%&arrowGF)-&%&plotsG6#%(displayG6$7#-%$s eqG6$-%&frameG6$9$%\"nG/F;;\"\"\"9%/%+insequenceG%%trueGF)F)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "movie(cos(Pi/20)+I*sin(Pi/20 ),40);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6#-%(ANI MATEG6J7%-%'CURVESG6&7#7$$\"1x8&fS$)o()*!#;$\"14BS]YMk:F.-%'COLOURG6&% $RGBG$\"*++++\"!\")\"\"!F8-%&STYLEG6#%&POINTG-%'SYMBOLG6#%'CIRCLEG-%+A XESLABELSG6$%!GFD-%%VIEWG6$%(DEFAULTGFH7%-F(6&7$7$$\"+1M)o()*!#5$\"+^Y Mk:FP7$$\"+j^c5&*FP$\"+X*p,4$FPF1F9F=FAFE7%-F(6&7%7$$\"+T_15*)FP$\"+** \\!*RXFPFMFSF1F9F=FAFE7%-F(6&7&Ffn7$$\"+V*p,4)FP$\"+DD&y(eFPFMFSF1F9F= FAFE7%-F(6&7'Ffn7$$\"+5y1rqFP$\"+9y1rqFPF_oFMFSF1F9F=FAFE7%-F(6&7(FfnF ho7$$\"+@D&y(eFP$\"+Y*p,4)FPF_oFMFSF1F9F=FAFE7%-F(6&7)FfnFhoFap7$$\"+% *\\!*RXFP$\"+X_15*)FPF_oFMFSF1F9F=FAFE7%-F(6&7*FfnFhoFapFjpF_oFMFS7$$ \"+S*p,4$FP$\"+l^c5&*FPF1F9F=FAFE7%-F(6&7+FfnFhoFap7$$\"+XYMk:FP$\"+3M )o()*FPFjpF_oFMFSFcqF1F9F=FAFE7%-F(6&7,FfnFho7$$!+x)fl#e!#>$\"+++++5! \"*FapF\\rFjpF_oFMFSFcqF1F9F=FAFE7%-F(6&7-FfnFhoFerFapF\\r7$$!+dYMk:FP FNFjpF_oFMFSFcqF1F9F=FAFE7%-F(6&7.7$$!+^*p,4$FP$\"+i^c5&*FPFfnFhoFerFa pF\\rF`sFjpF_oFMFSFcqF1F9F=FAFE7%-F(6&7/Fgs7$$!+0]!*RXFP$\"+S_15*)FPFf nFhoFerFapF\\rF`sFjpF_oFMFSFcqF1F9F=FAFE7%-F(6&70FgsF`tFfnFhoFer7$$!+J D&y(eFP$\"+T*p,4)FPFapF\\rF`sFjpF_oFMFSFcqF1F9F=FAFE7%-F(6&71FgsF`tFfn FhoFerFitFapF\\rF`sFjpF_o7$$!+?y1rqFP$\"+2y1rqFPFMFSFcqF1F9F=FAFE7%-F( 6&72FgsF`tFfnFhoFerFitFapF\\rF`sFjp7$$!+^*p,4)FP$\"+FgsF ixFf[lF`tFfnFfwFho7$$!+H*p,4$FP$!+s^c5&*FPFerFbyFitFapF_xF[zF\\rF`sFjp F[vF_oFbuF]wFdzF_\\lFdvFMFSF][lFcqF1F9F=FAFE7%-F(6&7?FgsFixFf[lF`tFfnF fwFhoFh\\lFerFbyFitFapF_xF[zF\\rF`sFjpF[vF_oFbu7$$!+MYMk:FP$!+8M)o()*F PF]wFdzF_\\lFdvFMFSF][lFcqF1F9F=FAFE7%-F(6&7@FgsFixFf[lF`tFfnFfwFhoFh \\lFerFbyFitFapF_xF[zF\\rF`sFjpF[vF_oFbuFa]lF]w7$$\"+\\O`[ C by p(z) is the compl ex number obtained by substituting x=z into p and computing the result ing complex number p(z). We want to investigate these 'polynomial fun ctions' from C to C by seeing where they take certain curves in C. A lot can be learned by looking at the image of circles centered at 0 u nder the function p." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 274 "ima ge := proc(p,rs,trng,clr)\n local x,y,z,pz,r;\nz:= unapply(r*cos(theta )+r*sin(theta)*I,r);\npz := unapply(evalc(p(z(r))),r);\n x := unapply( Re(pz(r)),r); y := unapply(Im(pz(r)),r);\n plot(\{seq([x(rs[i]),y(rs[i ]),theta=trng],i=1..nops(rs))\},color=clr,scaling=constrained); end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&imageGR6&%\"pG%#rsG%%trngG%$clrG6 '%\"xG%\"yG%\"zG%#pzG%\"rG6\"F1C'>8&-%(unapplyG6$,&*&8(\"\"\"-%$cosG6# %&thetaGF;F;*(%\"IGF;F:\"\"\"-%$sinGF>F;F;F:>8'-F66$-%&evalcG6#-9$6#-F 46#F:F:>8$-F66$-%#ReG6#-FFFPF:>8%-F66$-%#ImGFWF:-%%plotG6%<#-%$seqG6$7 %-FR6#&9%6#%\"iG-FZFbo/F?9&/Ffo;F;-%%nopsG6#Fdo/%&colorG9'/%(scalingG% ,constrainedGF1F1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "image(z->z^2+1,[1,1.5,2],0.. 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Make some conjectures. Prove one." }}}}}{MARK "7 5 0 1" 21 }{VIEWOPTS 1 1 0 1 1 1803 }