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" }} {PARA 0 "" 0 "" {TEXT -1 36 "For example, for the complex number " } {XPPEDIT 18 0 "z = 4 + 3*i " "6#/%\"zG,&\"\"%\"\"\"*&\"\"$F'%\"iGF'F' " }{TEXT -1 59 ", the acute angle which its position vector makes with the " }{TEXT 272 1 "x" }{TEXT -1 10 " axis is " }{XPPEDIT 18 0 "arct an(3/4);" "6#-%'arctanG6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 1 " " } {TEXT 259 1 "~" }{TEXT -1 20 " 0.6435 radians or " }{XPPEDIT 18 0 "36 .87^o" "6#)-%&FloatG6$\"%(o$!\"#%\"oG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 42 "Alternatively, we could quote the angle 0" } {XPPEDIT 18 0 ".6435-2*Pi;" "6#,&-%&FloatG6$\"%Nk!\"%\"\"\"*&\"\"#F)%# PiGF)!\"\"" }{TEXT -1 1 " " }{TEXT 260 1 "~" }{TEXT -1 13 " -5.6397 or " }{XPPEDIT 18 0 "- 323.136^o" "6#,$)-%&FloatG6$\"'OJK!\"$%\"oG!\"\" " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 15 "The expression " } {XPPEDIT 18 0 "arctan(3/4)+2*n*Pi;" "6#,&-%'arctanG6#*&\"\"$\"\"\"\"\" %!\"\"F)*(\"\"#F)%\"nGF)%#PiGF)F)" }{TEXT -1 8 ", where " }{TEXT 273 1 "n" }{TEXT -1 107 " is any integer, gives all possible ways of speci fying the angle which the vector makes with the real axis." }}{PARA 0 "" 0 "" {TEXT -1 32 "Any of these angles provides an " }{TEXT 269 8 "a rgument" }{TEXT -1 24 " for the complex number " }{XPPEDIT 18 0 "z = 4 + 3*i" "6#/%\"zG,&\"\"%\"\"\"*&\"\"$F'%\"iGF'F'" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 437 312 312 {PLOTDATA 2 "60-%'CURVESG6&7#7$$\"\"%\"\"!$\"\"$ F*-%'SYMBOLG6#%'CIRCLEG-%'COLOURG6&%$RGBG$F*F*F5$\"*++++\"!\")-%&STYLE G6#%&POINTG-F$6&F&-F.6#%(DIAMONDGF1F9-F$6&F&-F.6#%&CROSSGF1F9-F$6$7$7$ F5F5F'-%&COLORG6&F4F5F5$\"\"\"F*-F$6$7,7$$\"3a**************p!#=F57$$ \"3[p1`BBU!)pFV$\"32Dn>UXnJ_!#>7$$\"3'Q[^2g6;$pFV$\"3g@caMd%4w*Ffn7$$ \"34(y@%o@mToFV$\"3_s?8+_U![\"FV7$$\"3(o/c_K4Tr'FV$\"3I(>:(Q[3!)>FV7$$ \"3Y&)zD-C)4b'FV$\"3M]-)3>/nY#FV7$$\"3!3rFElg#ojFV$\"3Ub&\\\"3B21HFV7$ $\"3)\\N;1jj![hFV$\"3?Kt+cj$oM$FV7$$\"3AsVtf.J))eFV$\"3yi>p)>3_y$FV7$$ \"3?%zr********f&FV$\"3)pgP++++?%FV-F26&F4F*F*F*-%%TEXTG6%7$$\"#X!\"\" $\"#GFgqQ,z~=~4~+~3~i6\"-FL6&F4$FO!\"#F^rF^r-Fbq6%7$$\"#>Fgq$\"$&=F_rQ *|gr~z~|gr~=~5F[rF\\r-Fbq6%7$$\"#t\"!#<$\" \"\"F)-%&COLORG6&%$RGBGF/F)F)-F$6$7$F'7$$!3*************>t\"F-F.-F16&F 3F)$\"\"'!\"\"F)-F$6$7$F'7$F8$F>F)-F16&F3$\"\"*F>$\"\")F>F)-F$6$7$F'7$ F+FC-F16&F3F)F)F/-F$6&7#F*-%'SYMBOLG6#%'CIRCLEGF0-%&STYLEG6#%&POINTG-F $6&FR-FT6#%(DIAMONDGF0FW-F$6&FR-FT6#%&CROSSGF0FW-F$6&7#F7FSF:FW-F$6&Fa oFgnF:FW-F$6&FaoF\\oF:FW-F$6&7#FBFSFDFW-F$6&FhoFgnFDFW-F$6&FhoF\\oFDFW -F$6&7#FMFSFNFW-F$6&F_pFgnFNFW-F$6&F_pF\\oFNFW-%%TEXTG6&7$$\"#$\"#7 F>Q(/~3~+~i6\"-F16&F3$F/!\"#F`qF`q-%%FONTG6$%*HELVETICAG\"#5-Fep6&7$$F -F>FjpQ*-~/~3~+~iF]qF^qFbq-Fep6&7$Fjq$!#7F>Q*-~/~3~-~iF]qF^qFbq-Fep6&7 $FhpF_rQ(/~3~-~iF]qF^qFbq-Fep6&7$$\"#EF>$F-FaqQ*Real~axisF]qF^qFbq-Fep 6&7$$!\"%F>$\"#8F>Q*Imag~axisF]qF^qFbq-Fep6&7$$\"$\\\"Faq$\"$=\"FaqQ\" vF]qF^qFbq-Fep6&7$$!$&=FaqFjsF\\tF^qFbq-Fep6&7$$!$$=Faq$!$A\"FaqF\\tF^ qFbq-Fep6&7$$\"$^\"FaqFgtF\\tF^qFbq-%(SCALINGG6#%,CONSTRAINEDG-%*AXEST ICKSG6$7&/Faq%#-2G/F>%#-1G/F/%\"1G/\"\"#%\"2G7%/F>%#-iG/F)%\"0G/F/%\"i G-%+AXESLABELSG6%Q!F]qFiv-Fcq6#%(DEFAULTG-%%VIEWG6$F\\wF\\w" 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" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "C urve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22 " "Curve 23" "Curve 24" "Curve 25" "Curve 26" }}{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 " If the r eal part " }{TEXT 267 1 "a" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "z = a+b *i;" "6#/%\"zG,&%\"aG\"\"\"*&%\"bGF'%\"iGF'F'" }{TEXT -1 14 " is 0 so \+ that " }{TEXT 284 1 "z" }{TEXT -1 50 " is a pure imaginary number, the n its argument is:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "arg(z);" "6#-%$argG6#%\"zG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "arg(b *i) = PIECEWISE([Pi/2, 0 < b],[-Pi/2, b < 0]);" "6#/-%$argG6#*&%\"bG\" \"\"%\"iGF)-%*PIECEWISEG6$7$*&%#PiGF)\"\"#!\"\"2\"\"!F(7$,$*&F0F)F1F2F 22F(F4" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Finally, if " }{XPPEDIT 18 0 "b = 0" "6#/%\"bG\"\"! " }{TEXT -1 5 " and " }{TEXT 268 1 "a" }{TEXT -1 23 " is negative, so \+ that " }{XPPEDIT 18 0 "z = a+b*i;" "6#/%\"zG,&%\"aG\"\"\"*&%\"bGF'%\" iGF'F'" }{TEXT -1 36 " is on the negative real axis, then " }{XPPEDIT 18 0 "arg(z) = Pi;" "6#/-%$argG6#%\"zG%#PiG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Putting this to gether we have:" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 " arg(a+b*i) = PIECEWISE([arctan(b/a), 0 < a],[arctan(b/a)+Pi, a < 0 and 0 < b],[arctan(b/a)-Pi, a < 0 and b < 0],[Pi/2, a = 0 and 0 < b],[-Pi /2, a = 0 and b < 0],[Pi, a < 0 and b = 0]);" "6#/-%$argG6#,&%\"aG\"\" \"*&%\"bGF)%\"iGF)F)-%*PIECEWISEG6(7$-%'arctanG6#*&F+F)F(!\"\"2\"\"!F( 7$,&-F26#*&F+F)F(F5F)%#PiGF)32F(F72F7F+7$,&-F26#*&F+F)F(F5F)F=F532F(F7 2F+F77$*&F=F)\"\"#F53/F(F72F7F+7$,$*&F=F)FKF5F53/F(F72F+F77$F=32F(F7/F +F7" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 29 "The built-in Maple procedure " }{TEXT 0 8 "argument" } {TEXT -1 45 " calculates the argument of a complex number." }}{PARA 0 "" 0 "" {TEXT -1 25 "Similarly, the procedure " }{TEXT 0 3 "arg" } {TEXT -1 108 " defined below calculates the argument according to the \+ formula given above (with one small simplification)." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 374 "arg := pr oc(z::complexcons)\n local theta, a, b, aa, bb;\n a := Re(z);\n \+ b := Im(z);\n aa := evalf(a);\n bb := evalf(b);\n if aa<>0 then \+ theta := arctan(b/a) end if;\n if aa>0 then return theta\n elif aa <0 then\n if bb>=0 then return theta+Pi else return theta-Pi end \+ if\n else # a=0\n if bb>0 then return Pi/2 else return -Pi/2 en d if;\n end if;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 55 "The following calculations compare the tw o procedures. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "interface(imaginaryunit=i);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "'arg'(sqrt(3 )+i);\nvalue(%);\n'argument'(sqrt(3)+i);\nvalue(%);\n%*180/Pi*`degrees `;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$argG6#,&*$\"\"$#\"\"\"\"\"#F* ^#F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"'!\"\"%#PiG\"\"\"F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%)argumentG6#,&*$\"\"$#\"\"\"\"\" #F*^#F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"'!\"\"%#PiG\"\"\" F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"#I\"\"\"%(degreesGF&F&" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "'arg'(sqrt(3)-i);\nvalue(%);\n'argument'(sqrt(3)-i);\nvalue(%); \n%*180/Pi*`degrees`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$argG6#,&*$ \"\"$#\"\"\"\"\"#F*^#!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\" \"'!\"\"%#PiG\"\"\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%)argumentG6 #,&*$\"\"$#\"\"\"\"\"#F*^#!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, $*&\"\"'!\"\"%#PiG\"\"\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"#I \"\"\"%(degreesGF&!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "'arg'(-sqrt(3)+i);\nvalue(%);\n'arg ument'(-sqrt(3)+i);\nvalue(%);\n%*180/Pi*`degrees`;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$argG6#,&*$\"\"$#\"\"\"\"\"#!\"\"^#F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"&\"\"\"\"\"'!\"\"%#PiGF&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%)argumentG6#,&*$\"\"$#\"\"\"\"\"#!\"\"^#F* F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"&\"\"\"\"\"'!\"\"%#PiGF& F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"$]\"\"\"\"%(degreesGF&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "'arg'(-sqrt(3)-i);\nvalue(%);\n'argument'(-sqrt(3)-i);\nvalue( %);\n%*180/Pi*`degrees`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$argG6#, &*$\"\"$#\"\"\"\"\"#!\"\"^#F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$* (\"\"&\"\"\"\"\"'!\"\"%#PiGF&F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%) argumentG6#,&*$\"\"$#\"\"\"\"\"#!\"\"^#F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"&\"\"\"\"\"'!\"\"%#PiGF&F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"$]\"\"\"\"%(degreesGF&!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "'arg'(-sqrt( 3)*i);\nvalue(%);\n'argument'(-sqrt(3)*i);\nvalue(%);\n%*180/Pi*`degre es`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$argG6#*&^#!\"\"\"\"\"\"\"$# F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#!\"\"%#PiG\"\"\"F& " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%)argumentG6#*&^#!\"\"\"\"\"\"\"$ #F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#!\"\"%#PiG\"\"\"F &" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"#!*\"\"\"%(degreesGF&!\"\" " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "'arg'(-4+3*i);\nvalue(%);\n'argument'(-4+3*i);\nvalue (%);\nevalf[5](evalf[10](%*180/Pi))*`degrees`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$argG6#^$!\"%\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,&-%'arctanG6##\"\"$\"\"%!\"\"%#PiG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%)argumentG6#^$!\"%\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%'arctanG6##\"\"$\"\"%!\"\"%#PiG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&$\"&8V\"!\"#\"\"\"%(degreesGF(F(" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "'arg'(-4-3 *i);\nvalue(%);\n'argument'(-4-3*i);\nvalue(%);\nevalf[4](evalf[10](%* 180/Pi))*`degrees`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$argG6#^$!\"% !\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%'arctanG6##\"\"$\"\"%\"\" \"%#PiG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%)argumentG6#^$!\"%! \"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%'arctanG6##\"\"$\"\"%\"\"\" %#PiG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&$\"%J9!\"\"\"\"\"%(d egreesGF(F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Maple's built-in procedure " }{TEXT 0 8 "argument" }{TEXT -1 43 " can perform symbolic manipulations, while " }{TEXT 0 3 "arg" } {TEXT -1 8 " cannot." }}{PARA 0 "" 0 "" {TEXT -1 23 "For any complex n umber " }{TEXT 285 1 "z" }{TEXT -1 39 ", is a non-negative real number , so . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "z := 'z':\nargument(abs(z));\narg(abs(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 8 "" 1 "" {TEXT -1 102 "Error , invalid input: arg expects its 1st argument, z, to be of type comple xcons, but received abs(z)\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "argument(-abs(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#PiG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 34 "The polar form of a complex number" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 17 "A complex n umber " }{TEXT 282 1 "z" }{TEXT -1 42 " is determined completely by i ts modulus " }{XPPEDIT 18 0 "r = abs(z);" "6#/%\"rG-%$absG6#%\"zG" } {TEXT -1 19 ", and its argument " }{XPPEDIT 18 0 "theta = arg(z);" "6# /%&thetaG-%$argG6#%\"zG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "The pair of numbers " }{TEXT 281 1 "r" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 29 ", often written in t he form " }{XPPEDIT 18 0 "r*`/_`*theta;" "6#*(%\"rG\"\"\"%#/_GF%%&the taGF%" }{TEXT -1 18 " , constitute the " }{TEXT 269 10 "polar form" } {TEXT -1 24 " for the complex number " }{TEXT 283 1 "z" }{TEXT -1 1 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 432 257 257 {PLOTDATA 2 "68-%'CURVESG6%7$7$$\"\"!F)F(7$$\"\"%F)$\"\"$F)-%'COLOURG6&%$RGBG$\" *++++\"!\")F(F(-%*THICKNESSG6#F.-F$6%7$F'7$F+F(-F06&F2F(F3F(F6-F$6%7$F s6b)pFin$\"34+++#f*R,X!#>7$$\"3Q+++io5UpFin$ \"3Y*****HglT)*)F_o7$$\"3=+++&4l*poFin$\"3')*****z:uHM\"Fin7$$\"3%)*** **>0&QpnFin$\"3))*****\\]s>y\"Fin7$$\"3X*****f3$ySmFin$\"34+++iVf8AFin 7$$\"3')*****fa\"p%['Fin$\"3))*****p-`gj#Fin7$$\"3<+++slv,jFin$\"3=+++ I(*fZIFin7$$\"3%******fUNF4'Fin$\"3;+++p3`YMFin7$$\"3_+++`L\\eeFin$\"3 $******f1&>JQFin7$$\"3a+++++++cFin$\"3%)*************>%Fin-F06&F2F)F)F )-%%TEXTG6&7$$\"#[!\"\"$\"#KFerQ,z~=~x~+~i~y6\"F]r-%%FONTG6$%*HELVETIC AG\"#5-F`r6&7$$\"#bFer$\"#HFerQ9=~r~(~cos~~~~+~i~sin~~~)FirF]rFjr-F`r6 &7$$\"# " 0 "" {MPLTEXT 1 0 27 "interface(imaginaryunit=i);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "convert(sqrt (3)+i,polar);\nevalc(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&polarG6 $\"\"#,$*&\"\"'!\"\"%#PiG\"\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, &*$\"\"$#\"\"\"\"\"#F'^#F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "convert(-5,polar);\nevalc(%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&polarG6$\"\"&%#PiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "convert(3+4*i,polar);\nevalc (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&polarG6$\"\"&-%'arctanG6## \"\"%\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$\"\"$\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "po lar(r,theta);\nevalc(%);\nfactor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%&polarG6$%\"rG%&thetaG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"r G\"\"\"-%$cosG6#%&thetaGF&F&*(F%F&-%$sinGF)F&^#F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"rG\"\"\",&-%$cosG6#%&thetaGF%*&-%$sinGF)F%^#F %F%F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 53 "Examples o f converting complex numbers to polar form " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "inte rface(imaginaryunit=i);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 292 8 "Question" }{TEXT -1 10 ": Convert " }{XPPEDIT 18 0 "3+4*i" "6#,&\"\"$\"\"\"*&\"\"%F%%\"iGF%F% " }{TEXT -1 16 " to polar form. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 293 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 676 "x1 := \+ 3: y1 := 4: \np1 := plot([[0,0],[x1,y1]],color=red,thickness=2):\np2 : = plot([[0,0],[x1,0]],color=green,thickness=2):\np3 := plot([[x1,0],[x 1,y1]],color=blue,thickness=2):\np4 := plot([.7*cos(t),.7*sin(t),t=0.. arctan(y1/x1)],\n adaptive=false,numpoints=10, color=black):\nt1 := plots[textplot]([[4,2.8,`z = 3 + 4 i`],[1.9,1.85, `| z | = 5`],\n [1.8,0.4,`arg(z) = arctan(4/3)`],[4.8,-.2,`Real a xis`],\n [-0.6,4.5,`Imag axis`]],color=COLOR(RGB,.01,.01,.01)):\n plots[display]([p1,p2,p3,p4,t1],xtickmarks=[1=`1`,2=`2`,3=`3`,4=`4`], \n ytickmarks=[0=`0`,1=` i`,2=`2 i`,3=`3 i`,4=`4 i`],\n \+ scaling=constrained,font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 359 312 312 {PLOTDATA 2 "60-%'CURVESG6%7$7$$\"\"!F)F(7$$ \"\"$F)$\"\"%F)-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNESSG6#\"\" #-F$6%7$F'7$F+F(-F06&F2F(F3F(F6-F$6%7$F=F*-F06&F2F(F(F3F6-F$6$7,7$$\"3 a**************p!#=F(7$$\"31lE*p%oOfpFK$\"3`mEe]]NJv!#>7$$\"3?()*R:$*Q #eoFK$\"38[(4A=F;S\"FK7$$\"3Ed^.0casmFK$\"3)))HBJ=Cf6#FK7$$\"3t9FgRPr5 kFK$\"3Bw0=4H=6GFK7$$\"3'=ru!4!z`O=q_FK$\"3#>rhjixqg%FK7$$\"3Gn+5z*>(eZFK $\"3F\\q$\"$&=FdqQ*|gr~z~|gr~=~5F^qF_q-Fgp6%7$$\"#=F\\q$F.F\\qQ5ar g(z)~=~arctan(4/3)F^qF_q-Fgp6%7$$\"#[F\\q$FdqF\\qQ*Real~axisF^qF_q-Fgp 6%7$$!\"'F\\q$\"#XF\\qQ*Imag~axisF^qF_q-%(SCALINGG6#%,CONSTRAINEDG-%+A XESLABELSG6%Q!F^qFjs-%%FONTG6#%(DEFAULTG-%*AXESTICKSG6$7&/Fcq%\"1G/F9% \"2G/F,%\"3G/F.%\"4G7'/F)%\"0G/Fcq%#~iG/F9%$2~iG/F,%$3~iG/F.%$4~iG-F\\ t6$%*HELVETICAG\"\"*-%%VIEWG6$F^tF^t" 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" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(z)=sqrt(3^2+4^2)" "6#/-%$absG6#% \"zG-%%sqrtG6#,&*$\"\"$\"\"#\"\"\"*$\"\"%F.F/" }{XPPEDIT 18 0 "``=5" " 6#/%!G\"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arg(z)=arctan(4/3)" "6#/-%$ar gG6#%\"zG-%'arctanG6#*&\"\"%\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 294 1 "~" }{TEXT -1 16 " 53.13 degrees. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z = 3+4*i" "6#/% \"zG,&\"\"$\"\"\"*&\"\"%F'%\"iGF'F'" }{TEXT -1 1 " " }{TEXT 295 1 "~" }{TEXT -1 22 " 5 cis 53.13 degrees. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "z := 3+4*i;\n'abs(z)'=abs( z);\n'arg(z)'=argument(z);\n``=evalf[4](evalf[10](rhs(%)*180/Pi))*`deg rees`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG^$\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$absG6#%\"zG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$argG6#%\"zG-%'arctanG6##\"\"%\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&$\"%8`!\"#\"\"\"%(degreesGF*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "z \+ := 3+4*i;\nconvert(%,polar);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG ^$\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&polarG6$\"\"&-%'arc tanG6##\"\"%\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "E xample 2 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 290 8 "Question" }{TEXT -1 10 ": Convert " }{XPPEDIT 18 0 "2-2*i;" "6# ,&\"\"#\"\"\"*&F$F%%\"iGF%!\"\"" }{TEXT -1 17 " to polar form. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 291 8 "Solution " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 643 "x1 := 2: y1 := -2: \np1 := plot([[0,0],[x1,y 1]],color=red,thickness=2):\np2 := plot([[0,0],[x1,0]],color=green,thi ckness=2):\np3 := plot([[x1,0],[x1,y1]],color=blue,thickness=2):\np4 : = plot([.4*cos(t),.4*sin(t),t=0..arctan(y1/x1)],\n \+ adaptive=false,numpoints=10,color=black):\nt1 := plots[textplot]( [[.9,-1.3,`z = 2 - 2 i`],#[.5,-.5,`| z | = 2^(1/2)`],\n [.95,-.2, `arg(z) = arctan(-1)`],[2.5,-.14,`Real axis`],\n [-0.35,.5,`Imag \+ axis`]],color=black,font=[HELVETICA,9]):\nplots[display]([p1,p2,p3,p4, t1],xtickmarks=[1=`1`,2=`2`],\n ytickmarks=[0=`0`,-1=`-i`,-2=`-2 \+ i`],scaling=constrained,\n font=[HELVETICA,9]);" }}{PARA 13 "" 1 " " {GLPLOT2D 353 340 340 {PLOTDATA 2 "6/-%'CURVESG6%7$7$$\"\"!F)F(7$$\" \"#F)$!\"#F)-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNESSG6#F,-F$6% 7$F'7$F+F(-F06&F2F(F3F(F6-F$6%7$Fjvoe(eDFJ7$$\"3-]Hoew*yE$ FJ$!3,aJJlcp1BFJ7$$\"3!e))4$QXngMFJ$!3[rB0hT#f+#FJ7$$\"3MdJfHl%oi$FJ$! 3!yG#3?X+(o\"FJ7$$\"3_jR.$fCJw$FJ$!3Ql+C**z0c8FJ7$$\"3!e)o\"R,xD'QFJ$! 38s6mZ/ZR5FJ7$$\"3XJuTXO`PRFJ$!3jEO**Q6^Tq!#>7$$\"3C,$*=T#)[%)RFJ$!3qt Ksp>G>NFjo7$$\"3A+++++++SFJF(-F06&F2F)F)F)-%%TEXTG6&7$$\"\"*!\"\"$!#8F [qQ,z~=~2~-~2~i6\"Fcp-%%FONTG6$%*HELVETICAGFjp-Ffp6&7$$\"#&*F.$F.F[qQ4 arg(z)~=~arctan(-1)F_qFcpF`q-Ffp6&7$$\"#DF[q$!#9F.Q*Real~axisF_qFcpF`q -Ffp6&7$$!#NF.$\"\"&F[qQ*Imag~axisF_qFcpF`q-%(SCALINGG6#%,CONSTRAINEDG -%+AXESLABELSG6%Q!F_qFbs-Faq6#%(DEFAULTG-%*AXESTICKSG6$7$/\"\"\"%\"1G/ F,%\"2G7%/F)%\"0G/F[q%#-iG/F.%%-2~iGF`q-%%VIEWG6$FesFes" 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" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(z) = sqrt(2^2+(-2)^ 2);" "6#/-%$absG6#%\"zG-%%sqrtG6#,&*$\"\"#F-\"\"\"*$,$F-!\"\"F-F." } {XPPEDIT 18 0 "`` = sqrt(8);" "6#/%!G-%%sqrtG6#\"\")" }{XPPEDIT 18 0 " ``=2*sqrt(2)" "6#/%!G*&\"\"#\"\"\"-%%sqrtG6#F&F'" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "arg(z) = arctan((-2)/2);" "6#/-%$argG6#%\"zG-%'arctanG6 #*&,$\"\"#!\"\"\"\"\"F-F." }{XPPEDIT 18 0 "``=arctan(-1)" "6#/%!G-%'ar ctanG6#,$\"\"\"!\"\"" }{XPPEDIT 18 0 " ``=-Pi/4" "6#/%!G,$*&%#PiG\"\" \"\"\"%!\"\"F*" }{XPPEDIT 18 0 "`` = 45;" "6#/%!G\"#X" }{TEXT -1 10 " \+ degrees. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z = 2-2*i;" "6#/%\"zG,&\"\"#\"\"\"*&F&F'%\"iGF'! \"\"" }{XPPEDIT 18 0 "`` = 2*sqrt(2)*cis(-Pi/4);" "6#/%!G*(\"\"#\"\"\" -%%sqrtG6#F&F'-%$cisG6#,$*&%#PiGF'\"\"%!\"\"F2F'" }{XPPEDIT 18 0 "`` = 2*sqrt(2)*cis(-45^o);" "6#/%!G*(\"\"#\"\"\"-%%sqrtG6#F&F'-%$cisG6#,$) \"#X%\"oG!\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "z := 2-2*i;\n'abs(z)'=abs(z) ;\n'arg(z)'=argument(z);\n``=rhs(%)*180/Pi*`degrees`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG^$\"\"#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$absG6#%\"zG,$*&\"\"#\"\"\"F*#F+F*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$argG6#%\"zG,$*&\"\"%!\"\"%#PiG\"\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&\"#X\"\"\"%(degreesGF(!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "z \+ := 2-2*i;\nconvert(%,polar);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG ^$\"\"#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&polarG6$,$*&\"\"#\" \"\"F(#F)F(F),$*&\"\"%!\"\"%#PiGF)F." }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 10 "Example 3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 300 8 "Question" }{TEXT -1 10 ": Convert " } {XPPEDIT 18 0 "-2-4*i;" "6#,&\"\"#!\"\"*&\"\"%\"\"\"%\"iGF(F%" }{TEXT -1 16 " to polar form. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 301 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 638 "x1 := -2: y1 := - 4: \np1 := plot([[0,0],[x1,y1]],color=red,thickness=2):\np2 := plot([[ 0,0],[x1,0]],color=green,thickness=2):\np3 := plot([[x1,0],[x1,y1]],co lor=blue,thickness=2):\np4 := plot([.5*cos(t),.5*sin(t),t=0..arctan(y1 /x1)-Pi],\n adaptive=false,numpoints=10,color= black):\nt1 := plots[textplot]([[-.9,-3.3,`z = -2 - 4 i`],[1.4,-.2,`Re al axis`],\n [-0.6,1.2,`Imag axis`]],color=COLOR(RGB,.01,.01,.01) ):\nplots[display]([p1,p2,p3,p4,t1],xtickmarks=[-1=`-1`,-2=`-2`,-3=`-3 `],\n ytickmarks=[0=`0`,-1=`-i`,-2=`-2 i`,-3=`-3 i`,-4=`-4 i`],\n scaling=constrained,font=[HELVETICA,9],view=[-3..1.4,-4.3..1.2]) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 359 312 312 {PLOTDATA 2 "6.-%'CURVESG6 %7$7$$\"\"!F)F(7$$!\"#F)$!\"%F)-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%* THICKNESSG6#\"\"#-F$6%7$F'7$F+F(-F06&F2F(F3F(F6-F$6%7$F=F*-F06&F2F(F(F 3F6-F$6$7,7$$!3uNMTyz1OA!#=$!3OcFaaf8sWFK7$$!3**oq_Z='f7\"FK$!3wutc1;d r[FK7$$!3GwEQ0Fwn5!#>$!3gB')y_(f))*\\FK7$$\"3$*[?c![LE/\"FK$!3w_,5wL3! *[FK7$$\"3%*>&H)fKQV@FK$!3csF!y\"**G&R?;S7$FK$!3#*pD!G@6R !RFK7$$\"3G++v>D%[)QFK$!3ML$R$[tpZJFK7$$\"3gltzK2%Q[%FK$!3&Q*GDBO]7AFK 7$$\"35R\"yx:y.([FK$!3lT$3=*H6J6FK7$$\"3++++++++]FKF(-F06&F2F)F)F)-%%T EXTG6%7$$!\"*!\"\"$!#LF\\qQ-z~=~-2~-~4~i6\"-%&COLORG6&F2$\"\"\"F,FdqFd q-Fgp6%7$$\"#9F\\q$F,F\\qQ*Real~axisF`qFaq-Fgp6%7$$!\"'F\\q$\"#7F\\qQ* Imag~axisF`qFaq-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%Q!F`qF\\s-%% FONTG6#%(DEFAULTG-%*AXESTICKSG6$7%/F\\q%#-1G/F,%#-2G/!\"$%#-3G7'/F)%\" 0G/F\\q%#-iG/F,%%-2~iG/Fjs%%-3~iG/F.%%-4~iG-F^s6$%*HELVETICAG\"\"*-%%V IEWG6$;$FjsF)Fiq;$!#VF\\qFbr" 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" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "abs(z) = sqrt((-2)^2+(-4)^2);" "6#/-%$absG6#%\"zG-%%sqr tG6#,&*$,$\"\"#!\"\"F.\"\"\"*$,$\"\"%F/F.F0" }{XPPEDIT 18 0 "`` = sqrt (20);" "6#/%!G-%%sqrtG6#\"#?" }{XPPEDIT 18 0 "``=2*sqrt(5)" "6#/%!G*& \"\"#\"\"\"-%%sqrtG6#\"\"&F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arg(z) \+ = arctan((-4)/(-2))-Pi;" "6#/-%$argG6#%\"zG,&-%'arctanG6#*&,$\"\"%!\" \"\"\"\",$\"\"#F/F/F0%#PiGF/" }{XPPEDIT 18 0 "``=arctan(2)-Pi" "6#/%!G ,&-%'arctanG6#\"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 1 " " }{TEXT 302 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-116.57" "6#,$-%&FloatG6$\"&d;\"!\"#! \"\"" }{TEXT -1 10 " degrees. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z = -2-4*i;" "6#/%\"zG, &\"\"#!\"\"*&\"\"%\"\"\"%\"iGF*F'" }{TEXT -1 1 " " }{TEXT 303 1 "~" } {TEXT -1 1 " " }{XPPEDIT 18 0 "2*sqrt(5)*cis(-116.57^o)" "6#*(\"\"#\" \"\"-%%sqrtG6#\"\"&F%-%$cisG6#,$)-%&FloatG6$\"&d;\"!\"#%\"oG!\"\"F%" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "z := -2-4*i;\n'abs(z)'=abs(z);\n'arg(z)'=argume nt(z);\n``=evalf[5](evalf[10](rhs(%)*180/Pi))*`degrees`;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"zG^$!\"#!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$absG6#%\"zG,$*&\"\"#\"\"\"\"\"&#F+F*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$argG6#%\"zG,&-%'arctanG6#\"\"#\"\"\"%#PiG!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&$\"&d;\"!\"#\"\"\"%(degreesGF* !\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "z := -2-4*i;\nconvert(%,polar);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG^$!\"#!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%& polarG6$,$*&\"\"#\"\"\"\"\"&#F)F(F),&-%'arctanG6#F(F)%#PiG!\"\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 296 8 "Question" }{TEXT -1 10 ": Convert " }{XPPEDIT 18 0 "-3+5*i;" "6#,&\"\"$!\"\"*&\"\"&\"\" \"%\"iGF(F(" }{TEXT -1 16 " to polar form. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 297 8 "Solution" }{TEXT -1 3 ": " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 639 "x1 := -3: y1 := 5: \np1 := plot([[0,0],[x1,y1]],color=red,thickne ss=2):\np2 := plot([[0,0],[x1,0]],color=green,thickness=2):\np3 := plo t([[x1,0],[x1,y1]],color=blue,thickness=2):\np4 := plot([.5*cos(t),.5* sin(t),t=0..arctan(y1/x1)+Pi],\n adaptive=fals e,numpoints=10,color=black):\nt1 := plots[textplot]([[-1.1,3.3,`z = -3 + 5 i`],[1.4,-.2,`Real axis`],\n [-0.6,5.5,`Imag axis`]],color=C OLOR(RGB,.01,.01,.01)):\nplots[display]([p1,p2,p3,p4,t1],xtickmarks=[- 1=`-1`,-2=`-2`,-3=`-3`],\n ytickmarks=[0=`0`,1=` i`,2=`2 i`,3=`3 \+ i`,4=`4 i`,5=`5 i`],\n scaling=constrained,font=[HELVETICA,9],vie w=[-3.4..1.4,-.3..5.5]);" }}{PARA 13 "" 1 "" {GLPLOT2D 359 312 312 {PLOTDATA 2 "6.-%'CURVESG6%7$7$$\"\"!F)F(7$$!\"$F)$\"\"&F)-%'COLOURG6& %$RGBG$\"*++++\"!\")F(F(-%*THICKNESSG6#\"\"#-F$6%7$F'7$F+F(-F06&F2F(F3 F(F6-F$6%7$F=F*-F06&F2F(F(F3F6-F$6$7,7$$\"3++++++++]!#=F(7$$\"3%RMM4Fi ,&[FK$\"3KKV6Ep([@\"FK7$$\"3'f[H\"*)>_#[%FK$\"3Bl^0tT<:AFK7$$\"3Mkhp\\ >)p#QFK$\"3MV!4%)p&z$\"3!*)Hpngr5$\\FK7$$!3'pl /D*RcMKFbo$\"3'RSBmiE&*)\\FK7$$!3e_'>%Gi$[\\\"FK$\"3gc>^noJrZFK7$$!3I' )o&oxyCd#FK$\"3_o6tGYY(G%FK-F06&F2F)F)F)-%%TEXTG6%7$$!#6!\"\"$\"#LF\\q Q-z~=~-3~+~5~i6\"-%&COLORG6&F2$\"\"\"!\"#FdqFdq-Fgp6%7$$\"#9F\\q$FfqF \\qQ*Real~axisF`qFaq-Fgp6%7$$!\"'F\\q$\"#bF\\qQ*Imag~axisF`qFaq-%(SCAL INGG6#%,CONSTRAINEDG-%+AXESLABELSG6%Q!F`qF]s-%%FONTG6#%(DEFAULTG-%*AXE STICKSG6$7%/F\\q%#-1G/Ffq%#-2G/F,%#-3G7(/F)%\"0G/Feq%#~iG/F9%$2~iG/\" \"$%$3~iG/\"\"%%$4~iG/F.%$5~iG-F_s6$%*HELVETICAG\"\"*-%%VIEWG6$;$!#MF \\qFjq;$F,F\\qFcr" 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" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "abs(z) = sqrt((-3)^2+5^2);" "6#/-%$absG6#%\"zG-%%sqrtG6 #,&*$,$\"\"$!\"\"\"\"#\"\"\"*$\"\"&F0F1" }{XPPEDIT 18 0 "`` = sqrt(34) ;" "6#/%!G-%%sqrtG6#\"#M" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arg(z) = arctan( 5/(-3))+Pi;" "6#/-%$argG6#%\"zG,&-%'arctanG6#*&\"\"&\"\"\",$\"\"$!\"\" F1F.%#PiGF." }{XPPEDIT 18 0 "`` = -arctan(5/3)+Pi;" "6#/%!G,&-%'arctan G6#*&\"\"&\"\"\"\"\"$!\"\"F-%#PiGF+" }{TEXT -1 1 " " }{TEXT 298 1 "~" }{TEXT -1 17 " 120.96 degrees. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z = -3+5*i;" "6#/%\"zG, &\"\"$!\"\"*&\"\"&\"\"\"%\"iGF*F*" }{TEXT -1 1 " " }{TEXT 299 1 "~" } {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(34);" "6#-%%sqrtG6#\"#M" }{TEXT -1 5 " cis " }{XPPEDIT 18 0 "120.96^o" "6#)-%&FloatG6$\"&'47!\"#%\"oG " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "z := -3+5*i;\n'abs(z)'=abs(z);\n'arg(z)'=arg ument(z);\n``=evalf[5](evalf[10](rhs(%)*180/Pi))*`degrees`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG^$!\"$\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$absG6#%\"zG*$\"#M#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$argG6#%\"zG,&-%'arctanG6##\"\"&\"\"$!\"\"%#PiG\"\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&$\"&'47!\"#\"\"\"%(degre esGF*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "z := -3+5*i;\nconvert(%,polar);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG^$!\"$\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% &polarG6$*$\"#M#\"\"\"\"\"#,&-%'arctanG6##\"\"&\"\"$!\"\"%#PiGF)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 60 "Multiplication and division of complex nu mbers in polar form" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "z[1] = r[1]*(cos*th eta[1]+i*sin*theta[1]);" "6#/&%\"zG6#\"\"\"*&&%\"rG6#F'F',&*&%$cosGF'& %&thetaG6#F'F'F'*(%\"iGF'%$sinGF'&F06#F'F'F'F'" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "z[2] = r[2]*(cos*theta[2]+i*sin*theta[2]);" "6#/&%\"zG6 #\"\"#*&&%\"rG6#F'\"\"\",&*&%$cosGF,&%&thetaG6#F'F,F,*(%\"iGF,%$sinGF, &F16#F'F,F,F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 15 "The prod uct of " }{XPPEDIT 18 0 "z[1];" "6#&%\"zG6#\"\"\"" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "z[2];" "6#&%\"zG6#\"\"#" }{TEXT -1 3 " is" }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "z[1]*z[2] = r[1]*r[2]*(c os*theta[1]+i*sin*theta[1])*(cos*theta[2]+i*sin*theta[2]);" "6#/*&&%\" zG6#\"\"\"F(&F&6#\"\"#F(**&%\"rG6#F(F(&F.6#F+F(,&*&%$cosGF(&%&thetaG6# F(F(F(*(%\"iGF(%$sinGF(&F66#F(F(F(F(,&*&F4F(&F66#F+F(F(*(F9F(F:F(&F66# F+F(F(F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 5 " = " } {XPPEDIT 18 0 "r[1]*r[2]*``(cos*theta[1]*cos*theta[2]-sin*theta[1]*sin *theta[2]+i*``(sin*theta[1]*cos*theta[2]+cos*theta[1]*sin*theta[2])); " "6#*(&%\"rG6#\"\"\"F'&F%6#\"\"#F'-%!G6#,(**%$cosGF'&%&thetaG6#F'F'F0 F'&F26#F*F'F'**%$sinGF'&F26#F'F'F7F'&F26#F*F'!\"\"*&%\"iGF'-F,6#,&**F7 F'&F26#F'F'F0F'&F26#F*F'F'**F0F'&F26#F'F'F7F'&F26#F*F'F'F'F'F'" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 "= " }{XPPEDIT 18 0 "r[1 ]*r[2]*``(cos(theta[1]+theta[2])+i*sin(theta[1]+theta[2]));" "6#*(&%\" rG6#\"\"\"F'&F%6#\"\"#F'-%!G6#,&-%$cosG6#,&&%&thetaG6#F'F'&F46#F*F'F'* &%\"iGF'-%$sinG6#,&&F46#F'F'&F46#F*F'F'F'F'" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 5 "Thus " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "z[1]*z[2] =r[1]*`/_`*theta*`.`*r[2]*`/_`*theta[2] " "6 #/*&&%\"zG6#\"\"\"F(&F&6#\"\"#F(*0&%\"rG6#F(F(%#/_GF(%&thetaGF(%\".GF( &F.6#F+F(F0F(&F16#F+F(" }{XPPEDIT 18 0 "`` = r[1]*r[2]*`/_`*(theta[1]+ theta[2]);" "6#/%!G**&%\"rG6#\"\"\"F)&F'6#\"\"#F)%#/_GF),&&%&thetaG6#F )F)&F06#F,F)F)" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z[1]*z[2] =r[1]" "6 #/*&&%\"zG6#\"\"\"F(&F&6#\"\"#F(&%\"rG6#F(" }{TEXT -1 5 " cis " } {XPPEDIT 18 0 "theta[1]*`.`*r[2]" "6#*(&%&thetaG6#\"\"\"F'%\".GF'&%\"r G6#\"\"#F'" }{TEXT -1 5 " cis " }{XPPEDIT 18 0 "theta[2] = r[1]*r[2]*c is(theta[1]+theta[2]);" "6#/&%&thetaG6#\"\"#*(&%\"rG6#\"\"\"F,&F*6#F'F ,-%$cisG6#,&&F%6#F,F,&F%6#F'F,F," }{TEXT -1 2 ", " }}{PARA 257 "" 0 " " {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " polar(r[1],theta[1])*`.`*polar(r[2],theta[2]) = polar(r[1]*r[2],theta[ 1]+theta[2]);" "6#/*(-%&polarG6$&%\"rG6#\"\"\"&%&thetaG6#F+F+%\".GF+-F &6$&F)6#\"\"#&F-6#F4F+-F&6$*&&F)6#F+F+&F)6#F4F+,&&F-6#F+F+&F-6#F4F+" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " }{TEXT 269 84 "to multiply complex numbers in polar form, multiply the moduli and add the arguments" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "polar(2,Pi/6)*polar(3,Pi/3);\nevalc(%);\nconvert(%,polar);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%&polarG6$\"\"#,$%#PiG#\"\"\"\"\"'F +-F%6$\"\"$,$F)#F+F/F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%\"IG\"\"' " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&polarG6$\"\"',$%#PiG#\"\"\"\"\" #" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Map le can also handle the general case." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "a := 'a': b := 'b':\npolar (a,A)*polar(b,B);\nevalc(%);\nfactor(combine(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%&polarG6$%\"aG%\"AG\"\"\"-F%6$%\"bG%\"BGF)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(**%\"aG\"\"\"-%$cosG6#%\"AGF&%\"bGF& -F(6#%\"BGF&F&**F%F&-%$sinGF)F&F+F&-F1F-F&!\"\"*&%\"IGF&,&**F%F&F0F&F+ F&F,F&F&**F%F&F'F&F+F&F2F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*( %\"aG\"\"\"%\"bGF%,&-%$cosG6#,&%\"AGF%%\"BGF%F%*&%\"IGF%-%$sinGF*F%F%F %" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "z[1] = r[1]*(cos*theta[1]+i*sin*theta[1]);" "6#/&%\"zG6#\"\"\"*&&%\"rG6#F' F',&*&%$cosGF'&%&thetaG6#F'F'F'*(%\"iGF'%$sinGF'&F06#F'F'F'F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "z[2] = r[2]*(cos*theta[2]+i*sin*theta[2]) ;" "6#/&%\"zG6#\"\"#*&&%\"rG6#F'\"\"\",&*&%$cosGF,&%&thetaG6#F'F,F,*(% \"iGF,%$sinGF,&F16#F'F,F,F," }{TEXT -1 12 ", as before," }}{PARA 256 " " 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "z[1]/z[2] = r[1]*(cos*theta[1]+ i*sin*theta[1])/(r[2]*(cos*theta[2]+i*sin*theta[2]));" "6#/*&&%\"zG6# \"\"\"F(&F&6#\"\"#!\"\"*(&%\"rG6#F(F(,&*&%$cosGF(&%&thetaG6#F(F(F(*(% \"iGF(%$sinGF(&F56#F(F(F(F(*&&F/6#F+F(,&*&F3F(&F56#F+F(F(*(F8F(F9F(&F5 6#F+F(F(F(F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "z[1]/z[2] = r[1]*(cos*thet a[1]+i*sin*theta[1])*(cos*theta[2]-i*sin*theta[2])/(r[2]*(cos*theta[2] +i*sin*theta[2])*(cos*theta[2]-i*sin*theta[2]));" "6#/*&&%\"zG6#\"\"\" F(&F&6#\"\"#!\"\"**&%\"rG6#F(F(,&*&%$cosGF(&%&thetaG6#F(F(F(*(%\"iGF(% $sinGF(&F56#F(F(F(F(,&*&F3F(&F56#F+F(F(*(F8F(F9F(&F56#F+F(F,F(*(&F/6#F +F(,&*&F3F(&F56#F+F(F(*(F8F(F9F(&F56#F+F(F(F(,&*&F3F(&F56#F+F(F(*(F8F( F9F(&F56#F+F(F,F(F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 6 " \+ = " }{XPPEDIT 18 0 "r[1]/r[2];" "6#*&&%\"rG6#\"\"\"F'&F%6#\"\"#!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(cos*theta[1]*cos*theta[2]+sin*t heta[1]*sin*theta[2]+i*``(sin*theta[1]*cos*theta[2]-cos*theta[1]*sin*t heta[2]));" "6#-%!G6#,(**%$cosG\"\"\"&%&thetaG6#F)F)F(F)&F+6#\"\"#F)F) **%$sinGF)&F+6#F)F)F1F)&F+6#F/F)F)*&%\"iGF)-F$6#,&**F1F)&F+6#F)F)F(F)& F+6#F/F)F)**F(F)&F+6#F)F)F1F)&F+6#F/F)!\"\"F)F)" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 2 "= " }{XPPEDIT 18 0 "r[1]/r[2];" "6#*&&% \"rG6#\"\"\"F'&F%6#\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(cos( theta[1]-theta[2])+i*sin(theta[1]-theta[2]));" "6#-%!G6#,&-%$cosG6#,&& %&thetaG6#\"\"\"F.&F,6#\"\"#!\"\"F.*&%\"iGF.-%$sinG6#,&&F,6#F.F.&F,6#F 1F2F.F." }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 5 "Thus " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z[1]/z[2] = r[1]*`/_` *theta[1]/(r[2]*`/_`*theta[2]);" "6#/*&&%\"zG6#\"\"\"F(&F&6#\"\"#!\"\" **&%\"rG6#F(F(%#/_GF(&%&thetaG6#F(F(*(&F/6#F+F(F1F(&F36#F+F(F," } {XPPEDIT 18 0 "`` = r[1]/r[2];" "6#/%!G*&&%\"rG6#\"\"\"F)&F'6#\"\"#!\" \"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "`/_`*(theta[1]-theta[2]);" "6#*&% #/_G\"\"\",&&%&thetaG6#F%F%&F(6#\"\"#!\"\"F%" }{TEXT -1 3 " , " }} {PARA 257 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z[1]/z[2]=r[1]/r[2]" "6#/*&&%\"zG6#\"\"\"F(&F&6#\"\"#! \"\"*&&%\"rG6#F(F(&F/6#F+F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "cis(theta [1]-theta[2])" "6#-%$cisG6#,&&%&thetaG6#\"\"\"F*&F(6#\"\"#!\"\"" } {TEXT -1 2 " " }}{PARA 257 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "polar(r[1],theta[1])/polar(r[2],theta [2]) = polar(r[1]/r[2],theta[1]-theta[2]);" "6#/*&-%&polarG6$&%\"rG6# \"\"\"&%&thetaG6#F+F+-F&6$&F)6#\"\"#&F-6#F3!\"\"-F&6$*&&F)6#F+F+&F)6#F 3F6,&&F-6#F+F+&F-6#F3F6" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " }{TEXT 269 85 "to divide compl ex numbers in polar form, divide the moduli and subtract the arguments " }{TEXT -1 26 " in the appropriate order." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "polar(6,Pi/2)/pola r(3,Pi/3);\nevalc(%);\nconvert(%,polar);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%&polarG6$\"\"',$%#PiG#\"\"\"\"\"#F+-F%6$\"\"$,$F)#F+F/!\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$-%%sqrtG6#\"\"$\"\"\"F)%\"IGF) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&polarG6$\"\"#,$%#PiG#\"\"\"\"\" '" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Map le can also handle the general case." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "polar(r[1],theta[1])/polar (r[2],theta[2]);\nevalc(%);\nfactor(combine(%));\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*&-%&polarG6$&%\"rG6#\"\"\"&%&thetaGF)F*-F%6$&F(6#\" \"#&F,F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*,&%\"rG6#\"\"\"F(- %$cosG6#&%&thetaGF'F(&F&6#\"\"#F(-F*6#&F-F/F(,&*&)F.F0F()F1F0F(F(*&F6F ()-%$sinGF2F0F(F(!\"\"F(*,F%F(-F;F+F(F.F(F:F(F4FF(F.F(F 1F(F4F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 25 "Code for drawing pictures" }}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 28 "Code for drawing 1st picture" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 580 "p1 : = plot([[[4,3]]$3],style=point,symbol=[circle,diamond,cross],\n colo r=blue):\np2 := plot([[0,0],[4,3]],color=COLOR(RGB,0,0,1)):\np3 := plo t([.7*cos(t),.7*sin(t),t=0..arctan(3/4)],\n ad aptive=false,numpoints=10,color=black):\nt1 := plots[textplot]([[4.5,2 .8,`z = 4 + 3 i`],[1.9,1.85,`| z | = 5`],\n [1.7,0.4,`arg(z) = ar ctan(3/4)`],[4.8,-.2,`Real axis`],\n [-0.6,3.35,`Imag axis`]],col or=COLOR(RGB,.01,.01,.01)):\nplots[display]([p1,p2,p3,t1],xtickmarks=[ 1=`1`,2=`2`,3=`3`,4=`4`],\n ytickmarks=[0=`0`,1=`i`,2=`2 i`,3=`3 \+ i`],scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 28 "Code for drawing 2nd picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1161 "v := 'v': i := 'i' :\np1 := plot([[0,0],[1.732,1]],color=COLOR(RGB,1,0,0)):\np2 := plot([ [0,0],[-1.732,1]],color=COLOR(RGB,0,.6,0)):\np3 := plot([[0,0],[-1.732 ,-1]],color=COLOR(RGB,.9,.8,0)):\np4 := plot([[0,0],[1.732,-1]],color= COLOR(RGB,0,0,1)):\np5 := plot([[[1.732,1]]$3],style=point,\n symbol =[circle,diamond,cross],color=COLOR(RGB,1,0,0)):\np6 := plot([[[-1.732 ,1]]$3],style=point,\n symbol=[circle,diamond,cross],color=COLOR(RGB ,0,.6,0)):\np7 := plot([[[-1.732,-1]]$3],style=point,\n symbol=[circ le,diamond,cross],color=COLOR(RGB,.9,.8,0)):\np8 := plot([[[1.732,-1]] $3],style=point,\n symbol=[circle,diamond,cross],color=COLOR(RGB,0,0 ,1)):\nt1 := plots[textplot]([[1.7,1.2,`/ 3 + i`],[-1.7,1.2,`- / 3 + i `],\n [-1.7,-1.2,`- / 3 - i`],[1.7,-1.2,`/ 3 - i`],[2.6,-.17,`Real axis`],\n [-0.4,1.3,`Imag axis`]],font=[HELVETICA,10],color=COLOR(R GB,.01,.01,.01)):\nt2 := plots[textplot]([[1.49,1.18,`v`],[-1.85,1.18, `v`],\n [-1.83,-1.22,`v`],[1.51,-1.22,`v`]],font=[HELVETICA,10],co lor=COLOR(RGB,.01,.01,.01)):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,p8, t1,t2],\n xtickmarks=[-2=`-2`,-1=`-1`,1=`1`,2=`2`],\n ytickmarks=[ -1=`-i`,0=`0`,1=`i`],scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 28 "Code for drawing 3rd picture" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 816 "p1 : = plot([[0,0],[4,3]],color=red,thickness=3):\np2 := plot([[0,0],[4,0]] ,color=green,thickness=3):\np3 := plot([[4,0],[4,3]],color=blue,thickn ess=3):\np4 := plot([[[4,3]]$3],style=point,symbol=[circle,diamond,cro ss],\n symbolsize=15,color=red):\nhh := evalf(arctan(.75)/10):\np 5 := plot([seq([.7*cos(k*hh),.7*sin(k*hh)],k=0..10)],color=black):\nt1 := plots[textplot]([[4.8,3.2,`z = x + i y`],\n [5.5,2.9,`= r ( cos + i sin )`],[1.7,1.8,`|z| = r`],\n [1.2,.3,`arg(z) =`],[4.6,1 .5,`y = r sin`],[2,-.2,`x = r cos`],\n [4.9,-.17,`Real axis`],[-.6, 3,`Imag axis`]],\n font=[HELVETICA,10],color=black):\nt2 := plots[t extplot]([[5.9,2.9,`q q`],[1.75,.3,`q`],\n [5.15,1.5, `q`],[2.58,-.2,`q`]],font=[SYMBOL,10],color=COLOR(RGB,.01,.01,.01)):\n plots[display]([p1,p2,p3,p4,p5,t1,t2],tickmarks=[0,0]); " }}{PARA 13 " " 1 "" {GLPLOT2D 476 476 476 {PLOTDATA 2 "68-%'CURVESG6%7$7$$\"\"!F)F( 7$$\"\"%F)$\"\"$F)-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNESSG6#F .-F$6%7$F'7$F+F(-F06&F2F(F3F(F6-F$6%7$Fs6b)p Fin$\"34+++#f*R,X!#>7$$\"3Q+++io5UpFin$\"3Y*****HglT)*)F_o7$$\"3=+++&4 l*poFin$\"3')*****z:uHM\"Fin7$$\"3%)*****>0&QpnFin$\"3))*****\\]s>y\"F 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