{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 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 22 "Module 6 : Precalculus" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 37 "602 : Operation s with Complex Numbers" }}{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 t his 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 w hat 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 83 "_________ ______________________________________________________________________ ____" }}{PARA 4 "" 0 "" {TEXT -1 44 "A. The Sum and Difference Of Comp lex Numbers" }}{PARA 0 "" 0 "" {TEXT -1 83 "__________________________ _________________________________________________________" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 40 "Its easy to add complex numbers in Maple" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " z := - 4 + I; w := 1 + 3*I; \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG^$! \"%\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG^$\"\"\"\"\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " z+w;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$!\"$\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 " complexplot( \{z,w,z+w \}, x = -6..6, \n style = po int);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 349 262 262 {PLOTDATA 2 "6&-%'CURVESG6$7%7$$\"\"\"\"\"!$\"\" $F*7$$!\"%F*F(7$$!\"$F*$\"\"%F*-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*F<-%+A XESLABELSG6$Q\"x6\"Q!6\"-%&STYLEG6#%&POINTG-%%VIEWG6$;$!\"'F*$\"\"'F*% (DEFAULTG" 1 5 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cu rve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Complex numbers are added geometrically using the parallelogram ru le." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 189 "T he blue line shows the position of z, the green line shows the positio n of w. The yellow lines complete the parallelogram. The diagonal of t he parallelogram indicates the position of z+w.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 411 " display( complexplot( \{ 0, z \}, x = -6..6 ,\n color=blue, thickness = 2),\n complexplot( \{ 0, w \}, x = -6..6,\n color=green, thickness = 2),\n complexplot( \{ z,z+w\}, x = -6..6,\n color=ye llow),\n complexplot( \{w, z+w \},x = -6..6,\n \+ color=yellow),\n complexplot( \{0, z+w \},x = -6..6,\n \+ color=red, thickness = 3) );" }}{PARA 13 "" 1 "" {GLPLOT2D 349 262 262 {PLOTDATA 2 "6)-%'CURVESG6%7$7$$\"\"!F)F(7$$!\"%F)$\"\"\"F )-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%*THICKNESSG6#\"\"#-F$6%7$F'7$F- $\"\"$F)-F06&F2F(F3F(F6-F$6$7$F*7$$!\"$F)$\"\"%F)-F06&F2F3F3F(-F$6$7$F =FEFJ-F$6%7$F'FE-F06&F2F3F(F(-F76#F?-%+AXESLABELSG6%Q\"x6\"Q!6\"%(DEFA ULTG-%%VIEWG6$;$!\"'F)$\"\"'F)Fgn" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Y ou can also compute the difference of two complex numbers" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " z := 4 + I; w := 1 + 3*I;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG^$ \"\"%\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG^$\"\"\"\"\"$" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " z+w, z-w, w-z;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%^$\"\"&\"\"%^$\"\"$!\"#^$!\"$\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 296 " display( complexplot( \{z ,w\}, x = -6..6, style = point, color=blue),\n complexplot( \+ \{-z,-w\}, x = -6..6, style = point, color=red),\n complexpl ot( \{z+w\}, x = -6..6, style = point, color=green), \n \+ complexplot( \{z-w, w-z \}, x = -6..6, style = point, color=black) ); " }}{PARA 13 "" 1 "" {GLPLOT2D 349 262 262 {PLOTDATA 2 "6(-%'CURVESG6% 7$7$$\"\"\"\"\"!$\"\"$F*7$$\"\"%F*F(-%'COLOURG6&%$RGBG$F*F*F4$\"*++++ \"!\")-%&STYLEG6#%&POINTG-F$6%7$7$$!\"\"F*$!\"$F*7$$!\"%F*F@-F16&F3F5F 4F4F8-F$6%7#7$$\"\"&F*F.-F16&F3F4F5F4F8-F$6%7$7$F+$!\"#F*7$FB$\"\"#F*- F16&F3F*F*F*F8-%+AXESLABELSG6%Q\"x6\"Q!6\"%(DEFAULTG-%%VIEWG6$;$!\"'F* $\"\"'F*F]o" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 " Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 226 "Both z and w are blue. Their negati ves are red. Their sum is green, and their differences are black. You can see which difference is which by thinking of z -w as z + (-w), w \+ - z as (-z) + w, and using the parellelogram rule.\n" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_________________________________________________________ __________________________" }}{PARA 4 "" 0 "" {TEXT -1 33 "B. The Prod uct Of Complex Numbers" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________ ____________________________________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 62 "The product of two complex numbers can be easily computed also" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " z := 2 \+ + I; w := -1 + 3*I;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG^$\" \"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG^$!\"\"\"\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " z*w;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$!\"&\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 " display( complexplot( \{ 0,z\}, x = -6..6, color=blue ),\n \+ complexplot( \{ 0,w\}, x = 0..6, color=green ),\n comple xplot( \{ 0,z*w \}, x = 0..5, color=red) );" }}{PARA 13 "" 1 "" {GLPLOT2D 349 262 262 {PLOTDATA 2 "6'-%'CURVESG6$7$7$$\"\"!F)F(7$$\"\" #F)$\"\"\"F)-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-F$6$7$F'7$$!\"\"F)$\" \"$F)-F06&F2F(F3F(-F$6$7$F'7$$!\"&F)$\"\"&F)-F06&F2F3F(F(-%+AXESLABELS G6%Q\"x6\"Q!6\"%(DEFAULTG-%%VIEWG6$;$!\"'F)$\"\"'F)FQ" 1 2 0 1 10 0 2 9 1 4 2 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 39 "The red line is the product of z and w." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "How is \+ the product zw related to z and w geometrically?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "The modulus of the produc t is the product of the moduli." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " abs(z), abs(w), abs(z)*abs( w), abs(z*w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&*$-%%sqrtG6#\"\"&\"\" \"*$-F%6#\"#5F(*&F$F(F*F(,$*$-F%6#\"\"#F(F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "The argument of the product is the sum of the arguments (sometimes plus or minus a multiple of 2), s omewhat like a logarithm." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " angle_z:= evalf( argument(z)); \n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(angle_zG$\"+!4wkj%!#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " angle_w := evalf( argument( w)); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(angle_wG$\"+#)oa#*=!\"* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " angle_zw := evalf(argu ment(z*w));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)angle_zwG$\"+!\\%> cB!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " angle_z + angle_ w;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\"\\%>cB!\"*" }}}{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 28 "C. Powers 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 -1 139 "Taking a power o f a complex number is repeated multiplication. For numbers with modulu s 1, the results are other points on the unit circle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 " z := cos (Pi/7) + sin(Pi/7)*I; a := abs(z);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"zG,&-%$cosG6#,$%#PiG#\"\"\"\"\"(F,*&^#F,F,-%$sinGF(F,F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG*$-%%sqrtG6#,&*$)-%$cosG6#,$%#Pi G#\"\"\"\"\"(\"\"#F2F2*$)-%$sinGF.F4F2F2F2" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 156 " display( complexplot( \{0,z\}, x = -1..1, color=b lue,scaling=constrained ),\n seq( complexplot( \{ 0, z^k \}, x = -1..1, color=green ), k = 2..11 ) );" }}{PARA 13 "" 1 "" {GLPLOT2D 349 262 262 {PLOTDATA 2 "60-%'CURVESG6$7$7$$\"\"!F)F(7$$\"3Y \">C!z')o4!*!#=$\"3@\"ev6RP)QVF--%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-F $6$7$F'7$$\"+:!)*[B'!#5$\"+F[J=yF=-F16&F3F(F4F(-F$6$7$F'7$$\"+L$4_A#F= $\"+B\"z#\\(*F=F@-F$6$7$F'7$$!+[$4_A#F=$\"+>\"z#\\(*F=F@-F$6$7$F'7$$!+ E!)*[B'F=$\"+=[J=yF=F@-F$6$7$F'7$$!+%o)o4!*F=$\"+!QP)QVF=F@-F$6$7$F'7$ $!+**********F=$!+[Kbh9F-F@-F$6$7$7$$!+r')o4!*F=$!+1u$)QVF=F'F@-F$6$7$ 7$$!+.!)*[B'F=$!+N[J=yF=F'F@-F$6$7$F'7$$!+>$4_A#F=$!+D\"z#\\(*F=F@-F$6 $7$7$$\"+i$4_A#F=$!+:\"z#\\(*F=F'F@-%(SCALINGG6#%,CONSTRAINEDG-%+AXESL ABELSG6%Q\"x6\"Q!6\"%(DEFAULTG-%%VIEWG6$;$!\"\"F)$\"\"\"F)F_r" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "For numbers with modulus less than one, each multip lication by z creates a number closer to the origin. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " z := .8 \+ + .35*I; a := abs(z);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG^$ $\"\")!\"\"$\"#N!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$\"+)fC@ t)!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 234 " display(complexp lot( z, x = -1..1, color=blue, style = point, scaling = constrained ), \n complexplot( \{ z^k $ k = 2..18 \}, x = -1..1, color=green, style = point ) ,\n polarplot( a, scaling=constrained, color \+ = gold) );" }}{PARA 13 "" 1 "" {GLPLOT2D 349 262 262 {PLOTDATA 2 "6(- %'CURVESG6%7S7$$\"3U+++++++!)!#=$\"3w*************\\$F*F'F'F'F'F'F'F'F 'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F 'F'F'F'F'F'-%'COLOURG6&%$RGBG$\"\"!F2F1$\"*++++\"!\")-%&STYLEG6#%&POIN TG-F$6%737$$!3&)*****4$y.QR!#>$!3\"******4c))f@#F*7$$!37++++]PzXF@$\"3 /++++++'z&F*7$$!39+++++&\\R#F*$\"3?+++v=_wWF*7$$!3A+++cEu#[$F*$\"3++++ +])Hu#F*7$$\"3g***********\\<&F*$\"3a+++++++cF*7$$\"3+++++++!=#F*$\"3W +++++D\"H'F*7$$!3u*****\\()Qiu$F*$\"3y*****Hq!Ga(*F@7$$!3!******\\#4RQ LF*$!37++++:T3`F@7$$!3)******pL=\\[#F*$!3++++W(4Jf\"F*7$$\"3W+++<1%*\\ uF@$\"3j*****z,?vj'F@7$$\"3p*****p[?oj$F@$\"3R+++I_\\/tO\"F*7$$\"37+++* pHbg%F@$!3.+++t@i5>F*7$$!3!*******eiMI9F*$!36+++8#4U9#F*-F.6&F0F1F3F1F 6-F$6$7S7$$!3_*****zfC@t)F*$\"3I:da'Qx>e$!#F7$$!3msdsc!f.l)F*$!39:Sb;z <#>\"F*7$$!3o^\"Rs_%GZ%)F*$!3VJpK![*47AF*7$$!3=H:C]4)f2)F*$!3FadM=L#4K $F*7$$!3)**)=iV$pfb(F*$!3I>P*\\T3pP%F*7$$!36],qi84-pF*$!3e(30ysP*[`F*7 $$!3O=A/9?f%='F*$!33#RsVk![khF*7$$!30AF*$!3ARj]8')3\\%)F*7$$!3yJF>P-fu6F*$!3k8'frukFl)F*7$$\"3))>A0V*3S@& !#@$!3aq.MTI7K()F*7$$\"3C,))zolt*=\"F*$!3i7s7o^p]')F*7$$\"3tLz8@#z1J#F *$!3pm.QzF&3U)F*7$$\"3-`3ZFds%H$F*$!3qKo(*[Sq'3)F*7$$\"3cGP2?D,.WF*$!3 I*GFl!yySvF*7$$\"3ws,teuvt_F*$!3UQ[;zTqfpF*7$$\"3oo?$HOz/>'F*$!3kY;yT% o&ehF*7$$\"3<#)y*[f\\?!pF*$!3gIN:B;**[`F*7$$\"3/y3kOXThvF*$!3-i/6GZ\\n VF*7$$\"3m&)\\%zco01)F*$!3c3zJq$f\"eLF*7$$\"3!\\Y1VX'RQ%)F*$!3Q\\6_xgw XAF*7$$\"3EQ-3_?%4l)F*$!3m(f$eo+%z=\"F*7$$\"3jKed/E3K()F*$!3GVj$)*R-\" 3F!#?7$$\"3k3')eN:<_')F*$\"3Wpnlb>&*y6F*7$$\"3YoP$Q]9vW)F*$\"3!\\/l&)3 @7@#F*7$$\"3)=[kT)fz)3)F*$\"3#41Zu_'e*G$F*7$$\"3y**[&4T2Wd(F*$\"39KMI' fA\\M%F*7$$\"3M^:Z'Q0'QpF*$\"3D_)f$Rh[,`F*7$$\"39@i,vZ.4iF*$\"33]%Qh^g )RhF*7$$\"3E[&3y.,zF#F*$\"30%GmTkt (H%)F*7$$\"3T2jN/A\"y7\"F*$\"3+pmFig)*e')F*7$$\"3E\\h5jcupEFaz$\"3<%R= dy$3K()F*7$$!3![q$H')fXC6F*$\"3e+$4-aA%f')F*7$$!31[(4\\$R4JAF*$\"3Xccq &f(GU%)F*7$$!3u%)=CSpY\\LF*$\"3#=B#zsW=k!)F*7$$!3g^:2+W%*oVF*$\"3S`#4m ,x0c(F*7$$!3Oj9G()3wM`F*$\"3.djErZ08pF*7$$!3)*zP!3b(>*>'F*$\"3kBsvcEz \\hF*7$$!3;CY%e?M&)*oF*$\"3#G/A_hCNN&F*7$$!3UbI)yfRXd(F*$\"3**)zM'\\@p WVF*7$$!3q4BjsdQb!)F*$\"3ap:#*eEdqLF*7$$!3G3Y;:YfI%)F*$\"3_b@L#**y[F#F *7$$!3/OA?7q(3l)F*$\"3iFml7OT)=\"F*7$Fcr$!3I:da'Qx>e$Fgr-F.6&F0$\")+++ !)F5$\")AR!)\\F5$\")Vyg>F5-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%Q !6\"Fgbl%(DEFAULTG-%%VIEWG6$;$!\"\"F2$\"\"\"F2Fibl" 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 96 "Lookin g at a number of powers of z at once, the numbers seem to spiral inwar d toward the origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "For numbers with modulu s greater than one, the powers spiral away from the unit circle" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " z := .8 + .9*I;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG^$$\" \")!\"\"$\"\"*F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " a := a bs(z); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$\"+e%fT?\"!\"*" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 196 " display( complexplot( \{ 0,z\}, x = -1..1, color=blue),\n polarplot( 1,scaling=const rained, color = gold), \n seq(complexplot(\{z^(k-1), z^k \} ,x= -8..8, color=green ),k =2..11));" }}{PARA 13 "" 1 "" {GLPLOT2D 349 262 262 {PLOTDATA 2 "61-%'CURVESG6$7$7$$\"\"!F)F(7$$\"3U+++++++!)! #=$\"3A+++++++!*F--%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-F$6$7S7$$!\"\"F )$\"30BmIq&o?5%!#F7$$!3KR6_uBO1**F-$!3%o+%\\y\"y_O\"F-7$$!3Y'Gi)*4-Qn* F-$!3,jmib*)GLDF-7$$!3+'4@/['e[#*F-$!38,D1@.6.QF-7$$!3!**)H&3#42`')F-$ !34lR!f$4U7]F-7$$!3H9yQN?D/zF-$!3.t0ymceDhF-7$$!3%>AN:8\"zF-7$$!3xot&)f:x5]F-$!3cSPrOh-a')F -7$$!3yx!G'QDDF-$!3QaaWL\"oen*F-7$ $!3e6*zm#o8X8F-$!3\"R\"=>jt64**F-7$$\"3%4`u>nl5(f!#@$!2'y&=t@)******!# <7$$\"3S#ox^N#[i8F-$!39p$y\")GZn!**F-7$$\"3W78SY@=YEF-$!33/uE]D`V'*F-7 $$\"3s2bEF*3Jx$F-$!3cM0PPl'3E*F-7$$\"32\"3!3q_JU]F-$!3([.Z^2&oN')F-7$$ \"3)*>m3t%*[RgF-$!3;-c(3/I-(zF-7$$\"3G\\6(QM;$*3(F-$!3AXcFj?x_qF-7$$\" 3l*zRI?/U!zF-$!3NnSb(QZc7'F-7$$\"3=u&ext1$f')F-$!3j+a/8/k,]F-7$$\"3H)* )e:FO4B*F-$!3cg_tBHvXQF-7$$\"3e=*[$3Nij'*F-$!3Iy(y0$Q%=d#F-7$$\"34@*e) =+.2**F-$!3uuf64]Ug8F-7$$\"3+Q0F4>&*****F-$!394rpY)485$!#?7$$\"31&)oF> !Q%3**F-$\"3U$eM@&=8]8F-7$$\"3=7BHoa1u'*F-$\"3cnrICIGKDF-7$$\"3u$\\yX@ iKE*F-$\"3%fl=D`Bsw$F-7$$\"3!*y$RT8'=u')F-$\"3o:`*G)3zv\\F-7$$\"3)**Qq Pxng%zF-$\"3e7-xlUCrgF-7$$\"3pKc(*Rfc5rF-$\"36=9:4yMJqF-7$$\"3AbWQLke[ gF-$\"3/fkK&eGL'zF-7$$\"3!)>*3=U)o!*\\F-$\"3oPic4Mil')F-7$$\"3IE&>b[\" ysPF-$\"3)Go(eQ***4E*F-7$$\"3-7DD&el'3EF-$\"3G]=!\\p[Pl*F-7$$\"3#>#*3$ znc\"H\"F-$\"3j$eQD*>C;**F-7$$\"3s=GdRQQdIFdt$\"3Ol5\">E`*****F-7$$!35 :Vc\\Ks(G\"F-$\"3cyHhX=u;**F-7$$!37>!Qx]B*F-7$$!3)y*eas0I.]F-$\"3-=%3EhZ$e')F-7$$!3H?2C' R]$4hF-$\"3f^]N%H2o\"zF-7$$!3oJ'yNQ+$*4(F-$\"3e_2]j?sUqF-7$$!3wYp?\"Ry ,!zF-$\"3&R,q\\bQ38'F-7$$!3yW\"pT^PVn)F-$\"39#4Qh(p_v\\F-7$$!3:ZCj^4+D #*F-$\"3;a8\\!eo*fQF-7$$!3--U:o))oa'*F-$\"3hzajVQ=0EF-7$$!3X-tSNb&p!** F-$\"3#\\(3@5t'4O\"F-7$F;$!30BmIq&o?5%F?-F16&F3$\")+++!)F6$\")AR!)\\F6 $\")Vyg>F6-F$6$7$F*7$$!37+++++++V\"Fgp$\"3+++++++!***F-Ff\\lF[]l-F$6$7$7$$! 3=+++++qW?Fgp$!3!)***********f*[F-F`]lF[]l-F$6$7$Fh]l7$$!3%**********> ^>\"Fgp$!31+++++\">B#FgpF[]l-F$6$7$7$$\"33++++Ii_5Fgp$!37++++g8hGFgpF` ^lF[]l-F$6$7$7$$\"3=++++37 " 0 "" {MPLTEXT 1 0 10 " z := 'z';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zGF$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Lets solve the \+ equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Note that we get three answers because the equation is degree 3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " solve( z^3 = 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"\",&#! \"\"\"\"#F#*&^##F#F'F#-%%sqrtG6#\"\"$F#F#,&F%F#*&^#F%F#F+F#F#" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Each of t hese numbers, when cubed, yields 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " z1 := (-1 + I*sqrt(3))/2; \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z1G,&#!\"\"\"\"#\"\"\"*&^##F) F(F)-%%sqrtG6#\"\"$F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 " \+ z1^3;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$),&#!\"\"\"\"#\"\"\"*&^## F)F(F)-%%sqrtG6#\"\"$F)F)F0F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " evalc(z1^3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Lets look at all 12 roots of unity." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " n := 12;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " roots_of_unity := solve( z^n = 1, z);\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%/roots_of_unityG6.^#!\"\"^#\"\"\"F'F),$*$-%%sqrtG6#,$ *$-F-6#,&!\"#F)*&^#\"\"#F)-F-6#\"\"$F)F)F)F4F)#F'F7,$F+#F)F7,$*&-F-6#F 7F))F3#F)\"\"%F)F;,$F?F=,$*$-F-6#,$*$-F-6#,&F4F)*&^#F4F)F8F)F)F)F4F)F; ,$FGF=,$*&F@F))FNFCF)F;,$FSF=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 " display( complexplot( \{roots_of_unity[k] $ k = 1..n\}, x = -1 ..1, \n style = point, color = blue),\n polarplo t(1,scaling = constrained, color = gold ));" }}{PARA 13 "" 1 "" {GLPLOT2D 349 262 262 {PLOTDATA 2 "6'-%'CURVESG6%7.7$$\"\"!F)$\"\"\"F) 7$F*F(7$$!\"\"F)F(7$$!+NSDg')!#5$!+********\\F37$$!+++++]F3$!+SSDg')F3 7$$\"+++++]F3$\"+SSDg')F37$$\"+NSDg')F3$\"+********\\F37$F7F>7$FAN:8\"zFin7$$!3xo t&)f:x5]Fin$!3cSPrOh-a')Fin7$$!3yx!G'QDDFin$!3QaaWL\"oen*Fin7$$!3e6*zm#o8X8Fin$!3\"R\"=>jt64**Fin7$$\" 3%4`u>nl5(f!#@$!2'y&=t@)******!#<7$$\"3S#ox^N#[i8Fin$!39p$y\")GZn!**Fi n7$$\"3W78SY@=YEFin$!33/uE]D`V'*Fin7$$\"3s2bEF*3Jx$Fin$!3cM0PPl'3E*Fin 7$$\"32\"3!3q_JU]Fin$!3([.Z^2&oN')Fin7$$\"3)*>m3t%*[RgFin$!3;-c(3/I-(z Fin7$$\"3G\\6(QM;$*3(Fin$!3AXcFj?x_qFin7$$\"3l*zRI?/U!zFin$!3NnSb(QZc7 'Fin7$$\"3=u&ext1$f')Fin$!3j+a/8/k,]Fin7$$\"3H)*)e:FO4B*Fin$!3cg_tBHvX QFin7$$\"3e=*[$3Nij'*Fin$!3Iy(y0$Q%=d#Fin7$$\"34@*e)=+.2**Fin$!3uuf64] Ug8Fin7$$\"3+Q0F4>&*****Fin$!394rpY)485$!#?7$$\"31&)oF>!Q%3**Fin$\"3U$ eM@&=8]8Fin7$$\"3=7BHoa1u'*Fin$\"3cnrICIGKDFin7$$\"3u$\\yX@iKE*Fin$\"3 %fl=D`Bsw$Fin7$$\"3!*y$RT8'=u')Fin$\"3o:`*G)3zv\\Fin7$$\"3)**QqPxng%zF in$\"3e7-xlUCrgFin7$$\"3pKc(*Rfc5rFin$\"36=9:4yMJqFin7$$\"3AbWQLke[gFi n$\"3/fkK&eGL'zFin7$$\"3!)>*3=U)o!*\\Fin$\"3oPic4Mil')Fin7$$\"3IE&>b[ \"ysPFin$\"3)Go(eQ***4E*Fin7$$\"3-7DD&el'3EFin$\"3G]=!\\p[Pl*Fin7$$\"3 #>#*3$znc\"H\"Fin$\"3j$eQD*>C;**Fin7$$\"3s=GdRQQdIFav$\"3Ol5\">E`***** Fin7$$!35:Vc\\Ks(G\"Fin$\"3cyHhX=u;**Fin7$$!37>!Qx]B*Fin7$$!3)y*eas0I.]Fin$\"3-=%3EhZ $e')Fin7$$!3H?2C'R]$4hFin$\"3f^]N%H2o\"zFin7$$!3oJ'yNQ+$*4(Fin$\"3e_2] j?sUqFin7$$!3wYp?\"Ry,!zFin$\"3&R,q\\bQ38'Fin7$$!3yW\"pT^PVn)Fin$\"39# 4Qh(p_v\\Fin7$$!3:ZCj^4+D#*Fin$\"3;a8\\!eo*fQFin7$$!3--U:o))oa'*Fin$\" 3hzajVQ=0EFin7$$!3X-tSNb&p!**Fin$\"3#\\(3@5t'4O\"Fin7$F.$!30BmIq&o?5%F en-FK6&FM$\")+++!)FP$\")AR!)\\FP$\")Vyg>FP-%(SCALINGG6#%,CONSTRAINEDG- %+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F.F*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "2 0" 1 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }