{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Purple Emphasis" -1 261 " Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 283 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 283 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 283 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "T imes" 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 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 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 "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 2 2 2 2 2 1 1 1 1 } 3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "" 256 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 34 "An introduction to complex number s" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canad a" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "Solving quadratic equations and the imaginary unit" }}{PARA 0 "" 0 "" {TEXT -1 23 "The quadratic equation " }{XPPEDIT 18 0 "x^2-4*x-5 = 0;" "6#/,(*$%\"xG\"\"#\"\"\"*&\"\"%F(F&F(!\"\"\"\"&F+\" \"!" }{TEXT -1 29 " can be solved by factoring. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2-4*x-5 = 0" "6#/,(*$%\"xG\"\"#\"\"\" *&\"\"%F(F&F(!\"\"\"\"&F+\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "exactly when " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "(x+1)*(x-5)=0" "6#/*&,&%\"xG\"\"\"F'F'F',&F&F'\"\"&!\" \"F'\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=-1" "6#/%\"xG,$\"\" \"!\"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "x=5" "6#/%\"xG\"\"&" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "factor(x^2-4*x-5=0);\nsolve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%\"xG\"\"\"F'F'F',&F&F'!\"&F'F'\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$!\"\"\"\"&" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 30 "Alternatively, the method of \"" }{TEXT 261 21 "complet ing the square" }{TEXT -1 15 "\" can be used. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2-4*x-5 = 0" "6#/,(*$%\"xG\"\"#\"\"\" *&\"\"%F(F&F(!\"\"\"\"&F+\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "exactly when " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x^2-4*x=5" "6#/,&*$%\"xG\"\"#\"\"\"*&\"\"%F(F&F(!\"\"\" \"&" }{TEXT -1 14 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 2 "A " } {TEXT 261 14 "perfect square" }{TEXT -1 14 " has the form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x+p)^2=x^2+2*p*x+p^2" "6#/ *$,&%\"xG\"\"\"%\"pGF'\"\"#,(*$F&F)F'*(F)F'F(F'F&F'F'*$F(F)F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Comparing the expression " }{XPPEDIT 18 0 "x^2+2*p*x" "6#,&*$%\"xG \"\"#\"\"\"*(F&F'%\"pGF'F%F'F'" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "x ^2-4*x" "6#,&*$%\"xG\"\"#\"\"\"*&\"\"%F'F%F'!\"\"" }{TEXT -1 21 " we s ee that taking " }{XPPEDIT 18 0 "p=-2" "6#/%\"pG,$\"\"#!\"\"" }{TEXT -1 12 ", which is \"" }{TEXT 261 25 "half the coefficient of x" } {TEXT -1 56 "\", enables us to complete the square for the expression \+ " }{XPPEDIT 18 0 "x^2-4*x" "6#,&*$%\"xG\"\"#\"\"\"*&\"\"%F'F%F'!\"\"" }{TEXT -1 12 " by adding " }{XPPEDIT 18 0 "p^2=(-2)^2" "6#/*$%\"pG\" \"#*$,$F&!\"\"F&" }{XPPEDIT 18 0 "``=4" "6#/%!G\"\"%" }{TEXT -1 28 " t o form the perfect square " }{XPPEDIT 18 0 "x^2-4*x+4 = (x-2)^2;" "6#/ ,(*$%\"xG\"\"#\"\"\"*&\"\"%F(F&F(!\"\"F*F(*$,&F&F(F'F+F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 48 "Hence by adding 4 to each side of \+ (i) we obtain " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2 -4*x+4 = 5+4;" "6#/,(*$%\"xG\"\"#\"\"\"*&\"\"%F(F&F(!\"\"F*F(,&\"\"&F( F*F(" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 23 "which is equivale nt to " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x-2)^2=9" "6#/*$,&%\"xG\"\"\"\"\"#!\"\"F(\"\"*" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x-2 =``" "6#/,&%\"xG\"\"\"\"\"#!\"\"%!G" }{TEXT 262 1 "+" }{TEXT -1 4 " 3, " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 1 " " }{TEXT 263 1 "+" }{TEXT -1 3 " 3 " }}{PARA 0 "" 0 "" {TEXT -1 9 "that \+ is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=-1" "6#/%\" xG,$\"\"\"!\"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "x=5" "6#/%\"xG\"\" &" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 11 "as before. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "The foll owing template of Maple commands can be used to show the steps in the \+ solution of a quadratic equation by completing the square." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 732 "x^2 -4*x-5=0;\neq1 := %:\nif rhs(eq1)<>0 then eq2 := lhs(eq1)-rhs(eq1)=0 e lse eq2 := eq1 end if:\neq3 := student[completesquare](eq2,x):\nif pat match(lhs(eq3),_a::algebraic*(x+_p::algebraic)^2+_q::algebraic,'la') t hen\n pp := subs(la,_p): aa := subs(la,_a): qq := subs(la,_q):\n b b := simplify(2*aa*pp): cc := simplify(qq+bb^2/(4*aa)):\n eq4 := x^2 +bb/aa*x+cc/aa=0:\n if eq4<>eq2 and eq4<>eq1 then print(eq4) end if; \n eq5 := x^2+bb/aa*x=-cc/aa:\n if eq5<>eq1 then print(eq5) end if ;\n rr := simplify((bb^2-4*aa*cc)/(4*aa^2));\n print(x^2+bb/aa*x+p p^2=rr);\n print((x+pp)^2=rr);\n ss := simplify(bb^2-4*aa*cc);\n \+ print(x+pp=sqrt(ss)/(2*aa),x+pp=-sqrt(ss)/(2*aa));\n print(x=-pp+sq rt(ss)/(2*aa),x=-pp-sqrt(ss)/(2*aa));\nend if:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$)%\"xG\"\"#\"\"\"F)*&\"\"%F)F'F)!\"\"\"\"&F,\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*$)%\"xG\"\"#\"\"\"F)*&\"\"%F)F 'F)!\"\"\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$)%\"xG\"\"#\"\" \"F)*&\"\"%F)F'F)!\"\"F+F)\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$ ),&%\"xG\"\"\"\"\"#!\"\"F)F(\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/ ,&%\"xG\"\"\"\"\"#!\"\"\"\"$/F$!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $/%\"xG\"\"&/F$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "On the \+ other hand, the quadratic equation" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "x^2-4*x+5 = 0;" "6#/,(*$%\"xG\"\"#\"\"\"*&\"\"%F(F&F (!\"\"\"\"&F(\"\"!" }{TEXT -1 15 " ------- (ii), " }}{PARA 0 "" 0 "" {TEXT -1 13 "in which the " }{XPPEDIT 18 0 "-5" "6#,$\"\"&!\"\"" } {TEXT -1 50 " in the previous equation is replaced by + 5, has " } {TEXT 261 24 "no real number solutions" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Writing the equation (ii) in the form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2-4*x=-5" "6#/,&*$%\"xG\"\"#\"\"\"*&\"\"%F(F&F(!\"\",$\"\"&F+" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 100 "we can complete the sq uare on the left side of the equation by adding 4 to the left side as \+ before. " }}{PARA 0 "" 0 "" {TEXT -1 93 "We must also add 4 to the rig ht to obtain an equation which is equivalent to (ii). This gives" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2-4*x+4=-5+4" "6#/, (*$%\"xG\"\"#\"\"\"*&\"\"%F(F&F(!\"\"F*F(,&\"\"&F+F*F(" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x-2)^2=-1" "6#/*$,&%\"xG\"\"\"\"\"#!\"\"F(,$ F'F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 108 "We cannot proce ed any further using only real numbers, since there is no real number \+ with a square equal to " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 52 "This is consistent with the fa ct that the graph of " }{XPPEDIT 18 0 "y = x^2-4*x+5;" "6#/%\"yG,(*$% \"xG\"\"#\"\"\"*&\"\"%F)F'F)!\"\"\"\"&F)" }{TEXT -1 8 " has no " } {TEXT 264 1 "x" }{TEXT -1 12 " intercepts." }}{PARA 0 "" 0 "" {TEXT -1 55 "Indeed, since this equation can be written in the form " } {XPPEDIT 18 0 "y = (x-2)^2+1;" "6#/%\"yG,&*$,&%\"xG\"\"\"\"\"#!\"\"F*F )F)F)" }{TEXT -1 18 ", we can see that " }{TEXT 266 1 "y" }{TEXT -1 49 " has a minimum value of 1 which is obtained when " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 54 ". Consequently it is impossible \+ find a real value for " }{TEXT 265 1 "x" }{TEXT -1 13 " to give the " }{TEXT 282 1 "y" }{TEXT -1 50 " value 0, which is less than the minimu m value 1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 38 "plot(x^2-4*x+5,x=-0.5..4.5,y=0..7.25);" }}{PARA 13 "" 1 "" {GLPLOT2D 281 286 286 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!1++++++ +]!#;$\"1++++++]s!#:7$$!1nmm\"HU,\"RF*$\"1pu2W!\\pr'F-7$$!1LL3FH'='HF* $\"1S;#3\\rCF'F-7$$!1nm;/OU&*=F*$\"1*)*e![d4%z&F-7$$!1fmm\"H_\">#)!#<$ \"1Fg5Q:_N`F-7$$\"1mm;z%4\\Y#F@$\"1#Q2()>6?!\\F-7$$\"1MLeR-/P7F*$\"1^, r*e'[?XF-7$$\"1***\\il'piAF*$\"1\\?3*H>h9%F-7$$\"1MLe*)>VBLF*$\"1D011# z5y$F-7$$\"1++DJbw!Q%F*$\"1\")y*Q&[gRMF-7$$\"1ommTIOoaF*$\"1#*z)os%o6J F-7$$\"1NL$3_>jU'F*$\"1-^]BJhDF-7$$\" 1,++v=h(e)F*$\"15k:FgU-BF-7$$\"1+++v$[6j*F*$\"1^E@S38v?F-7$$\"1L$e*[z( y0\"F-$\"1d*e!fRf()=F-7$$\"1nm;a/cq6F-$\"11A>g*pzo\"F-7$$\"1nmm;t,m7F- $\"1+KVz0tQ:F-7$$\"1+]iSj0x8F-$\"1;bv-)e!)Q\"F-7$$\"1nmm\"pW`Z\"F-$\"1 XBc#>j_F\"F-7$$\"1+]i!f#=$e\"F-$\"1'3eFvOP<\"F-7$$\"1+](=xpeo\"F-$\"1k =A+yn)4\"F-7$$\"1nm\"H28Iz\"F-$\"1,M)zeVG/\"F-7$$\"1n;zpSS\"*=F-$\"18y 0wIz65F-7$$\"1LL3_?`(*>F-$\"1>pw!41++\"F-7$$\"1M$e*)>px5#F-$\"1@TC+Uh6 5F-7$$\"1+]Pf4t.AF-$\"1ey!QI1:/\"F-7$$\"1MLe*GstI#F-$\"1,PSCxZ%4\"F-7$ $\"1+++DRW9CF-$\"11%ppwj<<\"F-7$$\"1++DJE>>DF-$\"1)HW$))4cp7F-7$$\"1+] i!RU0i#F-$\"1!fi&eG2&Q\"F-7$$\"1++v=S2LFF-$\"1vi'p^(RP:F-7$$\"1mmm\"p) =MGF-$\"1XXIt2(ep\"F-7$$\"1++](=]@%HF-$\"1'G1e(pk()=F-7$$\"1L$e*[$z*RI F-$\"1)zfi/d:3#F-7$$\"1,+]iC$p9$F-$\"1;j`tSX:BF-7$$\"1m;H2qcZKF-$\"1&o #oPMUcDF-7$$\"1+]7.\"fFN$F-$\"1bv3\">d*HGF-7$$\"1mm;/OgbMF-$\"1*)HY_=y =JF-7$$\"1+]ilAFjNF-$\"1?BZw,#QW$F-7$$\"1MLL$)*ppm$F-$\"1OVLD*)yyPF-7$ $\"1ML3xe,tPF-$\"1$e*Q+`eVTF-7$$\"1n;HdO=yQF-$\"1/v^]QdFXF-7$$\"1,++D> #[(RF-$\"122YN;#***[F-7$$\"1nmT&G!e&3%F-$\"1m'=p7X'\\`F-7$$\"1MLL$)Qk% =%F-$\"1\\d)p*)oEx&F-7$$\"1+]iSjE!H%F-$\"1H)**4\"*>`C'F-7$$\"1,]P40O\" R%F-$\"1j#)z&30'=nF-7$$\"1+++++++XF-F+-%'COLOURG6&%$RGBG$\"#5!\"\"\"\" !F^[l-%+AXESLABELSG6$Q\"x6\"Q\"yFc[l-%%VIEWG6$;$!\"&F][l$\"#XF][l;F^[l $\"$D(!\"#" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C urve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 26 "Returning to the equation " }{XPPEDIT 18 0 "(x-2)^2 \+ = -1;" "6#/*$,&%\"xG\"\"\"\"\"#!\"\"F(,$F'F)" }{TEXT -1 23 ", we now i ntroduce the " }{TEXT 261 14 "imaginary unit" }{TEXT -1 12 " denoted b y " }{TEXT 267 1 "i" }{TEXT -1 4 " or " }{TEXT 268 1 "j" }{TEXT -1 140 " in mathematical notation, which is a new number (that is, it is \+ not a real number) with precisely the property we are looking for, nam ely " }{XPPEDIT 18 0 "j^2 = -1;" "6#/*$%\"jG\"\"#,$\"\"\"!\"\"" } {TEXT -1 4 " or " }{XPPEDIT 18 0 "i^2 = -1;" "6#/*$%\"iG\"\"#,$\"\"\"! \"\"" }{TEXT -1 13 ". The symbol " }{TEXT 269 1 "i" }{TEXT -1 44 " is \+ preferred by pure mathematicians, while " }{TEXT 270 1 "j" }{TEXT -1 51 " is commonly used in applied mathematics. I'll use " }{TEXT 271 1 "i" }{TEXT -1 66 " for the imaginary unit in this worksheet. Maple us es the symbol " }{TEXT 0 1 "I" }{TEXT -1 49 ", but this can be changed , if desired, using the " }{TEXT 0 9 "interface" }{TEXT -1 9 " command ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "I^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "The general principal we adopt \+ when working with the imaginary unit " }{TEXT 272 1 "i" }{TEXT -1 40 " \"mixed in\" with real numbers is that we" }}{PARA 256 "" 0 "" {TEXT -1 2 " \"" }{TEXT 261 106 "treat the imaginary unit i just like an (un known) real number, but, whenever it suites us, we can replace " } {XPPEDIT 18 0 "i^2;" "6#*$%\"iG\"\"#" }{TEXT 261 6 " by -1" }{TEXT -1 2 ".\"" }}{PARA 0 "" 0 "" {TEXT -1 24 "Likewise we can replace " } {XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "i^2;" "6#*$%\"iG\"\"#" }{TEXT -1 69 ", and this enables us to conti nue with the solution of the equation " }{XPPEDIT 18 0 "x^2-4*x+5 = 0 ;" "6#/,(*$%\"xG\"\"#\"\"\"*&\"\"%F(F&F(!\"\"\"\"&F(\"\"!" }{TEXT -1 52 " from the step where we had the equivalent equation " }{XPPEDIT 18 0 "(x-2)^2 = -1;" "6#/*$,&%\"xG\"\"\"\"\"#!\"\"F(,$F'F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "Thus we obtain" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x-2)^2 = i^2;" "6#/*$,&%\"xG\"\"\" \"\"#!\"\"F(*$%\"iGF(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "a nd so" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x-2 = i;" "6 #/,&%\"xG\"\"\"\"\"#!\"\"%\"iG" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "x -2 = -i;" "6#/,&%\"xG\"\"\"\"\"#!\"\",$%\"iGF(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 11 "This gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = 2+i;" "6#/%\"xG,&\"\"#\"\"\"%\"iGF'" } {TEXT -1 4 " or " }{XPPEDIT 18 0 "x = 2-i;" "6#/%\"xG,&\"\"#\"\"\"%\"i G!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 44 "This is the re sult given by Maple using the " }{TEXT 0 5 "solve" }{TEXT -1 9 " comma nd." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve(x^2-4*x+5=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$^$\"\"#\"\"\"^$F$!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 106 "We can also get Maple to show the steps \+ of the solution by using the same template of commands as before. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 732 "x^2-4*x+5=0;\neq1 := %:\nif rhs(eq1)<>0 then eq2 := lhs(eq1)-rhs( eq1)=0 else eq2 := eq1 end if:\neq3 := student[completesquare](eq2,x): \nif patmatch(lhs(eq3),_a::algebraic*(x+_p::algebraic)^2+_q::algebraic ,'la') then\n pp := subs(la,_p): aa := subs(la,_a): qq := subs(la,_q ):\n bb := simplify(2*aa*pp): cc := simplify(qq+bb^2/(4*aa)):\n eq 4 := x^2+bb/aa*x+cc/aa=0:\n if eq4<>eq2 and eq4<>eq1 then print(eq4) end if;\n eq5 := x^2+bb/aa*x=-cc/aa:\n if eq5<>eq1 then print(eq5 ) end if;\n rr := simplify((bb^2-4*aa*cc)/(4*aa^2));\n print(x^2+b b/aa*x+pp^2=rr);\n print((x+pp)^2=rr);\n ss := simplify(bb^2-4*aa* cc);\n print(x+pp=sqrt(ss)/(2*aa),x+pp=-sqrt(ss)/(2*aa));\n print( x=-pp+sqrt(ss)/(2*aa),x=-pp-sqrt(ss)/(2*aa));\nend if:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$)%\"xG\"\"#\"\"\"F)*&\"\"%F)F'F)!\"\"\"\"&F) \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*$)%\"xG\"\"#\"\"\"F)*&\" \"%F)F'F)!\"\"!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$)%\"xG\"\"# \"\"\"F)*&\"\"%F)F'F)!\"\"F+F)F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/* $),&%\"xG\"\"\"\"\"#!\"\"F)F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/,& %\"xG\"\"\"\"\"#!\"\"^#F&/F$^#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/% \"xG^$\"\"#\"\"\"/F$^$F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "If we substitute " }{XPPEDIT 18 0 "x = 2 + i" "6#/%\"xG,&\"\"#\"\" \"%\"iGF'" }{TEXT -1 21 " into the expression " }{XPPEDIT 18 0 "x^2-4* x+5;" "6#,(*$%\"xG\"\"#\"\"\"*&\"\"%F'F%F'!\"\"\"\"&F'" }{TEXT -1 8 " \+ we get " }{XPPEDIT 18 0 "(2+i)^2-4*(2+i)+5;" "6#,(*$,&\"\"#\"\"\"%\"iG F'F&F'*&\"\"%F',&F&F'F(F'F'!\"\"\"\"&F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 19 "This simplifies to " }{XPPEDIT 18 0 "4+4*i+i^2-8-4*i +5;" "6#,.\"\"%\"\"\"*&F$F%%\"iGF%F%*$F'\"\"#F%\"\")!\"\"*&F$F%F'F%F+ \"\"&F%" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "1+i^2;" "6#,&\"\"\"F$*$%\" iG\"\"#F$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "If we now us e the relation " }{XPPEDIT 18 0 "i^2 = -1;" "6#/*$%\"iG\"\"#,$\"\"\"! \"\"" }{TEXT -1 14 ", we see that " }{XPPEDIT 18 0 "x = 2+i" "6#/%\"xG ,&\"\"#\"\"\"%\"iGF'" }{TEXT -1 31 " is a solution of the equation " } {XPPEDIT 18 0 "x^2-4*x+5 = 0;" "6#/,(*$%\"xG\"\"#\"\"\"*&\"\"%F(F&F(! \"\"\"\"&F(\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "Simil arly " }{XPPEDIT 18 0 "x = 2 - i" "6#/%\"xG,&\"\"#\"\"\"%\"iG!\"\"" } {TEXT -1 37 " is also a solution of this equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "i := 'i':\ns ubs(x=2+i,x^2-4*x+5);\nexpand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, (*$),&\"\"#\"\"\"%\"iGF(F'F(F(!\"$F(F)!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*$)%\"iG\"\"#F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "subs(x=2+I,x^2-4*x +5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Expressions like " }{XPPEDIT 18 0 "2+i;" "6#,&\"\"# \"\"\"%\"iGF%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "2-i;" "6#,&\"\"#\" \"\"%\"iG!\"\"" }{TEXT -1 12 " are called " }{TEXT 261 15 "complex num bers" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 33 "In general, a num ber of the form " }{XPPEDIT 18 0 "a+b*i;" "6#,&%\"aG\"\"\"*&%\"bGF%%\" iGF%F%" }{TEXT -1 8 ", where " }{TEXT 273 1 "a" }{TEXT -1 5 " and " } {TEXT 274 1 "b" }{TEXT -1 45 " are real numbers is called a complex nu mber." }}{PARA 0 "" 0 "" {TEXT -1 64 "The letters z and w are often us ed to represent complex numbers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "z = a+b*i;" "6#/%\"zG ,&%\"aG\"\"\"*&%\"bGF'%\"iGF'F'" }{TEXT -1 8 ", where " }{TEXT 276 1 " a" }{TEXT -1 5 " and " }{TEXT 275 1 "b" }{TEXT -1 19 " are real number s, " }{TEXT 277 1 "a" }{TEXT -1 5 " and " }{TEXT 278 1 "b" }{TEXT -1 16 " are called the " }{TEXT 261 24 "real and imaginary parts" }{TEXT -1 4 " of " }{TEXT 279 1 "z" }{TEXT -1 14 " respectively." }}{PARA 0 " " 0 "" {TEXT -1 37 "Thus the real and imaginary parts of " }{XPPEDIT 18 0 "2-i;" "6#,&\"\"#\"\"\"%\"iG!\"\"" }{TEXT -1 11 " are 2 and " } {XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 14 " respectively." }} {PARA 0 "" 0 "" {TEXT -1 20 "The Maple functions " }{TEXT 0 2 "Re" } {TEXT -1 5 " and " }{TEXT 0 2 "Im" }{TEXT -1 58 " extract the real and imaginary parts of a complex number." }}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "'Re'(2-I);\nvalue(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#ReG6#^$\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "'Im'(2-I);\nvalue(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%#ImG6#^$\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 54 "If we had suspec ted that the solution of the equation " }{XPPEDIT 18 0 "x^2-4*x+5 = 0; " "6#/,(*$%\"xG\"\"#\"\"\"*&\"\"%F(F&F(!\"\"\"\"&F(\"\"!" }{TEXT -1 124 " could only be expressed in terms of complex numbers, it would ha ve been more appropriate to write the equation in the form " } {XPPEDIT 18 0 "z^2-4*z+5 = 0;" "6#/,(*$%\"zG\"\"#\"\"\"*&\"\"%F(F&F(! \"\"\"\"&F(\"\"!" }{TEXT -1 8 ", using " }{TEXT 280 1 "z" }{TEXT -1 29 " for the unknown rather than " }{TEXT 281 1 "x" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "eq := z^2-4*z+5=0;\nsolve(eq,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#eqG/,(*$)%\"zG\"\"#\"\"\"F+F)!\"%\"\"&F+\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$,&\"\"#\"\"\"%\"IGF%,&F$F%F&!\"\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 62 "Argand diagrams, the modulus and \+ conjugate of a complex number" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 35 "We can picture the complex num bers " }{XPPEDIT 18 0 "2 + i" "6#,&\"\"#\"\"\"%\"iGF%" }{TEXT -1 5 " a nd " }{XPPEDIT 18 0 "2 - i" "6#,&\"\"#\"\"\"%\"iG!\"\"" }{TEXT -1 18 " as points in the " }{TEXT 261 13 "complex plane" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 59 "Set up a rectangular co-ordinate system a nd mark the points" }{XPPEDIT 18 0 " ``(2,1)" "6#-%!G6$\"\"#\"\"\"" } {TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(2,-1)" "6#-%!G6$\"\"#,$\"\"\"!\" \"" }{TEXT -1 107 ", using the real and imaginary parts of each of the two numbers as the horizontal and vertical coordinates." }}{PARA 0 " " 0 "" {TEXT -1 50 "In such a picture we call the horizontal axis the \+ " }{TEXT 261 9 "real axis" }{TEXT -1 27 " and the vertical axis the " }{TEXT 261 14 "imaginary axis" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 394 293 293 {PLOTDATA 2 "60-%'CURVESG6&7#7$$\"\"#\"\"!$\"\"\"F*-%'SYMBOLG6#%'C IRCLEG-%&COLORG6&%$RGBGF*F*F,-%&STYLEG6#%&POINTG-F$6&F&-F.6#%(DIAMONDG F1F5-F$6&F&-F.6#%&CROSSGF1F5-F$6&7#7$F($!\"\"F*F--F26&F4F*$\"\"'FHF*F5 -F$6&FEF;FIF5-F$6&FEF@FIF5-%%TEXTG6%7$$\"#BFH$\"$D\"!\"#Q&2~+~i6\"-F26 &F4$F,FYFhnFhn-FR6%7$FU$!#7FHQ&2~-~iFenFfn-FR6%7$$\"#HFH$FYFHQ*Real~ax isFenFfn-FR6%7$$!\"'FH$\"$X#FYQ*Imag~axisFenFfn-%(SCALINGG6#%,CONSTRAI NEDG-%*AXESTICKSG6$7%/FH%#-1G/F,%\"1G/F)%\"2G7'/FY%%-2~iG/FH%#-iG/F*% \"0G/F,%\"iG/F)%$2~iG-%+AXESLABELSG6%Q!FenFjq-%%FONTG6#%(DEFAULTG-%%VI EWG6$;F\\o$\"\"$F*;$!#BFHF[p" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" }}{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "In ge neral, the complex number " }{XPPEDIT 18 0 "a+b*i;" "6#,&%\"aG\"\"\"*& %\"bGF%%\"iGF%F%" }{TEXT -1 67 " is represented by the point with stan dard rectangular co-ordinates" }{XPPEDIT 18 0 "``(a,b)" "6#-%!G6$%\"aG %\"bG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The complex number " }{XPPEDIT 18 0 "i = 0+i;" "6#/% \"iG,&\"\"!\"\"\"F$F'" }{TEXT -1 44 " is on the vertical axis, 1 unit up from 0." }}{PARA 0 "" 0 "" {TEXT 261 22 "Pure imaginary numbers" } {TEXT -1 15 " have the form " }{XPPEDIT 18 0 "0+b*i;" "6#,&\"\"!\"\"\" *&%\"bGF%%\"iGF%F%" }{TEXT -1 64 " and sit on the vertical axis, while real numbers have the form " }{XPPEDIT 18 0 "a+0*i;" "6#,&%\"aG\"\"\" *&\"\"!F%%\"iGF%F%" }{TEXT -1 32 " and sit on the horizontal axis." }} {PARA 0 "" 0 "" {TEXT -1 107 "In this way we can visualize the set of \+ real numbers as forming a subset of the set of all complex numbers." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "A diagra m used to illustrate complex numbers in the complex plane is called an " }{TEXT 261 14 "Argand Diagram" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "It is sometimes more appr opriate to associate a complex number with the " }{TEXT 261 15 "positi on vector" }{TEXT -1 148 " of the the corresponding point in the compl ex plane, that is, the directed line segment which joins the origin to the point for the complex number." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 400 339 339 {PLOTDATA 2 "6 6-%'CURVESG6&7#7$$\"\"#\"\"!$\"\"\"F*-%'SYMBOLG6#%'CIRCLEG-%&COLORG6&% $RGBGF*F*F,-%&STYLEG6#%&POINTG-F$6&F&-F.6#%(DIAMONDGF1F5-F$6&F&-F.6#%& CROSSGF1F5-F$6&7#7$F($!\"\"F*F--F26&F4F*$\"\"'FHF*F5-F$6&FEF;FIF5-F$6& FEF@FIF5-F$6$7$7$$F*F*FUF'F1-F$6$7$7$$\"3%**************>\"!#<$\"3Q+++ +++qm!#=7$$\"3'************HL\"FfnFgnF1-F$6$7$7$$\"3+++++++]7Ffn$\"3]* ************p&FinFjnF1-F$6$7$FTFFFI-F$6$7$7$FZ$!3Q++++++qmFin7$F[oF\\p FI-F$6$7$7$Fao$!3]*************p&FinF^pFI-%%TEXTG6%7$$\"#BFH$\"$D\"!\" #Q&2~+~i6\"-F26&F4$F,F]qFbqFbq-Ffp6%7$Fip$!#7FHQ&2~-~iF_qF`q-Ffp6%7$$ \"#HFH$F]qFHQ*Real~axisF_qF`q-Ffp6%7$$!\"'FH$\"$X#F]qQ*Imag~axisF_qF`q -%(SCALINGG6#%,CONSTRAINEDG-%*AXESTICKSG6$7%/FH%#-1G/F,%\"1G/F)%\"2G7' /F]q%%-2~iG/FH%#-iG/F*%\"0G/F,%\"iG/F)%$2~iG-%+AXESLABELSG6%Q!F_qFdt-% %FONTG6#%(DEFAULTG-%%VIEWG6$;Ffq$\"\"$F*;$!#BFHFer" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" " Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" }} {TEXT -1 1 " " }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 261 7 "modulus" }{TEXT -1 2 ", " }{TEXT 261 9 "magnitude" }{TEXT -1 4 " or " }{TEXT 261 14 "absolute value" } {TEXT -1 21 " of a complex number " }{XPPEDIT 18 0 "z = a+b*i;" "6#/% \"zG,&%\"aG\"\"\"*&%\"bGF'%\"iGF'F'" }{TEXT -1 6 " is " }{XPPEDIT 18 0 "abs(z) = sqrt(a^2+b^2);" "6#/-%$absG6#%\"zG-%%sqrtG6#,&*$%\"aG\" \"#\"\"\"*$%\"bGF.F/" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 56 "G eometrically, it is the distance of the complex number " }{XPPEDIT 18 0 "z = a+b*i" "6#/%\"zG,&%\"aG\"\"\"*&%\"bGF'%\"iGF'F'" }{TEXT -1 114 " from 0 in the complex plane, or equivalently, it is the length of th e vector corresponding to the complex number." }}{PARA 0 "" 0 "" {TEXT -1 13 "For example, " }{XPPEDIT 18 0 "abs(2+i) = sqrt(2^2+1^2); " "6#/-%$absG6#,&\"\"#\"\"\"%\"iGF)-%%sqrtG6#,&*$F(F(F)*$F)F(F)" } {TEXT -1 3 " = " }{XPPEDIT 18 0 "sqrt(5);" "6#-%%sqrtG6#\"\"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 41 "The Maple command which comput es this is " }{TEXT 0 3 "abs" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "abs(2+I);\nevalf(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$\"\"&#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+xz1OA!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 261 9 "conjugate" }{TEXT -1 23 " of the compl ex number " }{XPPEDIT 18 0 "z = a+b*i;" "6#/%\"zG,&%\"aG\"\"\"*&%\"bGF '%\"iGF'F'" }{TEXT -1 6 " is " }{XPPEDIT 18 0 "conjugate(z) = a-b*i; " "6#/-%*conjugateG6#%\"zG,&%\"aG\"\"\"*&%\"bGF*%\"iGF*!\"\"" }{TEXT -1 91 ". Thus to form the conjugate of a complex number, we change the sign of the imaginary part." }}{PARA 0 "" 0 "" {TEXT -1 146 "The conj ugate of a complex number can be obtained geometrically by reflecting \+ it in the real axis. Thus in the diagrams above the complex numbers " }{XPPEDIT 18 0 "2+i;" "6#,&\"\"#\"\"\"%\"iGF%" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "2-i;" "6#,&\"\"#\"\"\"%\"iG!\"\"" }{TEXT -1 34 " are th e conjugates of each other." }}{PARA 0 "" 0 "" {TEXT -1 69 "The Maple \+ command which computes the conugate of a complex number is " }{TEXT 0 9 "conjugate" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "conjugate(2+I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$\"\"#!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 236 "If you use the unavaluated form of the command by enclosing the name of the command in quotes, Maple simply \+ puts a bar over the number or expression to be conjugated. The value o f the conjugate can then be obtained by using the command " }{TEXT 0 5 "value" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "'conjugate'(sqrt(5)-4*I);\nvalue(%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*conjugateG6#,&*$\"\"&#\"\"\"\" \"#F*^#!\"%F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$\"\"&#\"\"\"\"\"# F'^#\"\"%F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 425 380 380 {PLOTDATA 2 "6D-%'CU RVESG6&7#7$$\"\"#\"\"!$\"\"\"F*-%'SYMBOLG6#%'CIRCLEG-%&COLORG6&%$RGBGF *F*F,-%&STYLEG6#%&POINTG-F$6&F&-F.6#%(DIAMONDGF1F5-F$6&F&-F.6#%&CROSSG F1F5-F$6&7#7$F($!\"\"F*F--F26&F4F*$\"\"'FHF*F5-F$6&FEF;FIF5-F$6&FEF@FI F5-F$6&7#7$$!\"#F*F+F--F26&F4F,F*F*F5-F$6&FSF;FWF5-F$6&FSF@FWF5-F$6&7# 7$FUFGF--F26&F4$\"\"*FH$\"\")FHF*F5-F$6&FinF;F[oF5-F$6&FinF@F[oF5-F$6$ 7$7$$F*F*FioF'F1-F$6$7$7$$\"3%**************>\"!#<$\"3Q++++++qm!#=7$$ \"3'************HL\"F`pFapF1-F$6$7$7$$\"3+++++++]7F`p$\"3]************ *p&FcpFdpF1-F$6$7$FhoFFFI-F$6$7$7$F^p$!3Q++++++qmFcp7$FepFfqFI-F$6$7$7 $F[q$!3]*************p&FcpFhqFI-F$6$7$FhoFTFW-F$6$7$7$$!3%************ **>\"F`pFap7$$!3'************HL\"F`pFapFW-F$6$7$7$$!3+++++++]7F`pF]qFh rFW-F$6$7$FhoFjnF[o-F$6$7$7$FfrFfq7$FirFfqF[o-F$6$7$7$F_sF]rFhsF[o-%%T EXTG6%7$$\"#BFH$\"$D\"FVQ&2~+~i6\"-F26&F4$F,FVFitFit-F^t6%7$Fat$!#7FHQ &2~-~iFftFgt-F^t6%7$$!#BFHFctQ'-2~+~iFftFgt-F^t6%7$FcuF]uQ'-2~-~iFftFg t-F^t6%7$$\"#HFH$FVFHQ*Real~axisFftFgt-F^t6%7$$!\"'FH$\"$X#FVQ*Imag~ax isFftFgt-%(SCALINGG6#%,CONSTRAINEDG-%*AXESTICKSG6$7%/FH%#-1G/F,%\"1G/F )%\"2G7'/FV%%-2~iG/FH%#-iG/F*%\"0G/F,%\"iG/F)%$2~iG-%+AXESLABELSG6%Q!F ftFex-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!\"$F*$\"\"$F*;FcuFfv" 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" "Curve 27" "Curve 28" "C urve 29" "Curve 30" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 75 "Th e absolute value and conjugation operations are related by the equatio n: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z*conjugate(z) = abs(z)^2" "6#/*&%\"zG\"\"\"-%*conjugateG6#F%F&*$-%$absG6#F%\"\"#" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "z \+ = a+b*i;" "6#/%\"zG,&%\"aG\"\"\"*&%\"bGF'%\"iGF'F'" }{TEXT -1 29 " whe re a and b are real, then" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "z*conjugate(z) = (a+b*i)*(a-b*i);" "6#/*&%\"zG\"\"\"-%* conjugateG6#F%F&*&,&%\"aGF&*&%\"bGF&%\"iGF&F&F&,&F,F&*&F.F&F/F&!\"\"F& " }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` ` = a^2-b^2*i^2" "6#/%!G,&*$%\"aG\"\"#\"\"\"*&%\"bGF(%\"iGF(!\"\"" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` \+ = a^2+b^2;" "6#/%!G,&*$%\"aG\"\"#\"\"\"*$%\"bGF(F)" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = abs(z)^2;" "6#/% !G*$-%$absG6#%\"zG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 42 "Note that this means, in particular, that " }{XPPEDIT 18 0 "z*conj ugate(z);" "6#*&%\"zG\"\"\"-%*conjugateG6#F$F%" }{TEXT -1 136 " is a r eal number. This observation provides the method of performing divisio n of complex numbers, as is shown in the following section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "Arithmetic operations on comp lex numbers" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 188 "As suggested in an earlier section, we can add, \+ subtract and multiply complex numbers by the general principle of trea ting the imaginary unit \"like a real number\", and using the relation " }{XPPEDIT 18 0 "i^2 = -1;" "6#/*$%\"iG\"\"#,$\"\"\"!\"\"" }{TEXT -1 19 " where appropriate." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Thus if " }{XPPEDIT 18 0 "z[1] = 3+4*i;" "6#/&%\"z G6#\"\"\",&\"\"$F'*&\"\"%F'%\"iGF'F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "z[2] = 2-i;" "6#/&%\"zG6#\"\"#,&F'\"\"\"%\"iG!\"\"" }{TEXT -1 17 ", we can compute " }{XPPEDIT 18 0 "z[1]+z[2],z[1]-z[2];" "6$,&&%\" zG6#\"\"\"F'&F%6#\"\"#F',&&F%6#F'F'&F%6#F*!\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "z[1]*z[2];" "6#*&&%\"zG6#\"\"\"F'&F%6#\"\"#F'" }{TEXT -1 13 " as follows: " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "z[1]+z[2] = ``(3+4*i)+``(2-i);" "6#/,&&%\"zG6#\"\"\"F(&F&6#\"\"#F(, &-%!G6#,&\"\"$F(*&\"\"%F(%\"iGF(F(F(-F.6#,&F+F(F4!\"\"F(" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "5+3*i;" "6#,&\"\"&\"\"\"*&\"\"$F%%\"iGF%F%" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "z[1]- z[2] = ``(3+4*i)-``(2-i);" "6#/,&&%\"zG6#\"\"\"F(&F&6#\"\"#!\"\",&-%!G 6#,&\"\"$F(*&\"\"%F(%\"iGF(F(F(-F/6#,&F+F(F5F,F," }{TEXT -1 3 " = " } {XPPEDIT 18 0 "1+5*i;" "6#,&\"\"\"F$*&\"\"&F$%\"iGF$F$" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "z[1]*z[2] = ``(3+4 *i)*` .`*``(2-i);" "6#/*&&%\"zG6#\"\"\"F(&F&6#\"\"#F(*(-%!G6#,&\"\"$F( *&\"\"%F(%\"iGF(F(F(%#~.GF(-F.6#,&F+F(F4!\"\"F(" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "6-3*i+8*i-4*i^2 = 6+5*i+4;" "6#/,*\"\"'\"\"\"*&\"\"$F&% \"iGF&!\"\"*&\"\")F&F)F&F&*&\"\"%F&*$F)\"\"#F&F*,(F%F&*&\"\"&F&F)F&F&F .F&" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "10+5*i;" "6#,&\"#5\"\"\"*&\"\"& F%%\"iGF%F%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "z1 := 3 + 4*I;\nz2 := 2 - I;\nz1 + \+ z2;\nz1 - z2;\nz1 * z2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z1G^$\" \"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z2G^$\"\"#!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$\"\"&\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$\"\"\"\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$\"#5 \"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "We now see how to perf orm division of complex numbers." }}{PARA 0 "" 0 "" {TEXT -1 6 "Since \+ " }{XPPEDIT 18 0 "``(3+4*i)*` .`*``(2-i) = 10+5*i;" "6#/*(-%!G6#,&\"\" $\"\"\"*&\"\"%F*%\"iGF*F*F*%#~.GF*-F&6#,&\"\"#F*F-!\"\"F*,&\"#5F**&\" \"&F*F-F*F*" }{TEXT -1 54 " from the example above, we would expect to find that " }{XPPEDIT 18 0 "(10+5*i)/(3+4*i) = 2-i;" "6#/*&,&\"#5\"\" \"*&\"\"&F'%\"iGF'F'F',&\"\"$F'*&\"\"%F'F*F'F'!\"\",&\"\"#F'F*F/" } {TEXT -1 7 ".\nThis " }{TEXT 261 8 "division" }{TEXT -1 16 " is achiev ed by " }{TEXT 261 80 "multiplying the numerator and denominator of by the conjugate of the denominator" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "(10+5*i)/(3+4*i) = ``(10+5*i)*` .`*``( 3-4*i)/(``(3+4*i)*` .`*``(3-4*i));" "6#/*&,&\"#5\"\"\"*&\"\"&F'%\"iGF' F'F',&\"\"$F'*&\"\"%F'F*F'F'!\"\"**-%!G6#,&F&F'*&F)F'F*F'F'F'%#~.GF'-F 26#,&F,F'*&F.F'F*F'F/F'*(-F26#,&F,F'*&F.F'F*F'F'F'F6F'-F26#,&F,F'*&F.F 'F*F'F/F'F/" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 2 "= " } {XPPEDIT 18 0 "(30-20*i^2+15*i-40*i)/25;" "6#*&,*\"#I\"\"\"*&\"#?F&*$% \"iG\"\"#F&!\"\"*&\"#:F&F*F&F&*&\"#SF&F*F&F,F&\"#DF," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "(50-25*i)/25;" "6 #*&,&\"#]\"\"\"*&\"#DF&%\"iGF&!\"\"F&F(F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 "= " }{XPPEDIT 18 0 "2-i;" "6#,&\"\"#\"\"\"%\"i G!\"\"" }{TEXT -1 3 " , " }}{PARA 257 "" 0 "" {TEXT -1 13 "as expected . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Sim ilarly, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "(10+5*i)/(2-i) = ``(10+5*i)*` .`*``(2+i)/(`` (2-i)*` .`*``(2+i));" "6#/*&,&\"#5\"\"\"*&\"\"&F'%\"iGF'F'F',&\"\"#F'F *!\"\"F-**-%!G6#,&F&F'*&F)F'F*F'F'F'%#~.GF'-F06#,&F,F'F*F'F'*(-F06#,&F ,F'F*F-F'F4F'-F06#,&F,F'F*F'F'F-" }{TEXT -1 2 " " }}{PARA 256 "" 0 " " {TEXT -1 2 "= " }{XPPEDIT 18 0 "(20+5*i^2+10*i+10*i)/5;" "6#*&,*\"#? \"\"\"*&\"\"&F&*$%\"iG\"\"#F&F&*&\"#5F&F*F&F&*&F-F&F*F&F&F&F(!\"\"" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "(1 5+20*i)/5;" "6#*&,&\"#:\"\"\"*&\"#?F&%\"iGF&F&F&\"\"&!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 "= " }{XPPEDIT 18 0 "3+4*i;" "6#, &\"\"$\"\"\"*&\"\"%F%%\"iGF%F%" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "(10+5*I)/(3+4*I);\n(10+5*I)/(2-I);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#^$\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ^$\"\"$\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "The reciprocal, or multiplicative inverse, " }{XPPEDIT 18 0 "z^(-1)" "6#)%\"zG,$\"\" \"!\"\"" }{TEXT -1 21 " of a complex number " }{TEXT 287 1 "z" }{TEXT -1 29 " can be obtained by dividing " }{TEXT 288 1 "z" }{TEXT -1 9 " i nto 1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Thus, given " }{XPPEDIT 18 0 "z=2+3*i" "6#/%\"zG,&\"\"#\"\"\"*&\" \"$F'%\"iGF'F'" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "z^(-1)=1/z" "6#/)%\"zG,$\"\"\"!\"\"*&F'F'F%F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/(2+3* i)" "6#/%!G*&\"\"\"F&,&\"\"#F&*&\"\"$F&%\"iGF&F&!\"\"" }{XPPEDIT 18 0 "``=(2-3*i)/((2+3*i)*`.`*(2-3*i))" "6#/%!G*&,&\"\"#\"\"\"*&\"\"$F(%\"i GF(!\"\"F(*(,&F'F(*&F*F(F+F(F(F(%\".GF(,&F'F(*&F*F(F+F(F,F(F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(2-3*i)/13" "6#/%!G*&,&\"\"#\"\"\"*&\"\"$F(% \"iGF(!\"\"F(\"#8F," }{XPPEDIT 18 0 "`` = 2/13-3/13;" "6#/%!G,&*&\"\"# \"\"\"\"#8!\"\"F(*&\"\"$F(F)F*F*" }{TEXT -1 1 " " }{TEXT 286 1 "i" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "(2+3*I)^(-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #^$#\"\"#\"#8#!\"$F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "A note on programming complex arithmetic" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 122 "We can see how complex multiplication and division are performed for general complex numbers by using the Maple procedure " }{TEXT 0 5 "evalc" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "unasign('a','b','c','d'):\n( a + b*I)*(c+d*I);\nevalc(%);\n(a + b*I)/(c+d*I);\nevalc(%);\nsimplify( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"aG\"\"\"*&%\"bGF&^#F&F&F &F&,&%\"cGF&*&%\"dGF&F)F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&% \"aG\"\"\"%\"cGF&F&*&%\"bGF&%\"dGF&!\"\"*&,&*&F%F&F*F&F&*&F)F&F'F&F&F& ^#F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"aG\"\"\"*&%\"bGF&^# F&F&F&F&,&%\"cGF&*&%\"dGF&F)F&F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(%\"aG\"\"\"%\"cGF&,&*$)F'\"\"#F&F&*$)%\"dGF+F&F&!\"\"F&*(%\"bGF &F.F&F(F/F&*&,&*(F1F&F'F&F(F/F&*(F%F&F.F&F(F/F/F&^#F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,**&%\"aG\"\"\"%\"cGF(!\"\"*&%\"bGF(%\"dGF( F**(^#F*F(F,F(F)F(F(*(F'F(F-F(^#F(F(F(F(,&*$)F)\"\"#F(F(*$)F-F5F(F(F*F *" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 167 "Si nce a complex numbers is determined by a pair of real numbers, we can \+ define Maple procedures to perform the operations of multiplication an d division on such pairs." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 293 "mult := proc(u::[algebraic,algebra ic],v::[algebraic,algebraic])\n return [u[1]*v[1]-u[2]*v[2],u[1]*v[2 ]+u[2]*v[1]]\nend proc:\ndiv := proc(u::[algebraic,algebraic],v::[alge braic,algebraic])\n local s;\n s := v[1]^2+v[2]^2;\n return [(u[ 1]*v[1]+u[2]*v[2])/s,(u[2]*v[1]-u[1]*v[2])/s]\nend proc:" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "'mult '([a,b],[c,d]);\nvalue(%);\n'div'([a,b],[c,d]);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%multG6$7$%\"aG%\"bG7$%\"cG%\"dG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,&*&%\"aG\"\"\"%\"cGF'F'*&%\"bGF'%\"dGF'! \"\",&*&F&F'F+F'F'*&F*F'F(F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$d ivG6$7$%\"aG%\"bG7$%\"cG%\"dG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$*&, &*&%\"aG\"\"\"%\"cGF(F(*&%\"bGF(%\"dGF(F(F(,&*$)F)\"\"#F(F(*$)F,F0F(F( !\"\"*&,&*&F+F(F)F(F(*&F'F(F,F(F3F(F-F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The procedures " }{TEXT 0 4 "mult " }{TEXT -1 5 " and " }{TEXT 0 3 "div" }{TEXT -1 155 " can be used to \+ perform multiplication and division complex with the aid of the follow ing two procedures which perform conversions between coordinate form \+ " }{TEXT 283 1 "[" }{TEXT 284 1 "a" }{TEXT 283 2 ", " }{TEXT 285 1 "b " }{TEXT 283 1 "]" }{TEXT -1 20 " and the usual form " }{TEXT 283 7 "a + b*I" }{TEXT -1 21 " for complex numbers." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "getcoords := z - > [Re(z),Im(z)];\nformcomplex := proc(u::[algebraic,algebraic])\n u[ 1]+u[2]*I\nend proc;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*getcoordsGf *6#%\"zG6\"6$%)operatorG%&arrowGF(7$-%#ReG6#9$-%#ImGF/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,formcomplexGf*6#'%\"uG7$%*algebraicGF*6\" F+F+,&&9$6#\"\"\"F0*&&F.6#\"\"#F0^#F0F0F0F+F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The multiplication and d ivision examples of the previous section can now be performed using th e procedures described above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "z1 := 3 + 4*I;\nz2 := 2 - I; \nu := getcoords(z1);\nv := getcoords(z2);\nmult(u,v);\nformcomplex(%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z1G^$\"\"$\"\"%" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#z2G^$\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG7$\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG7$\" \"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"#5\"\"&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#^$\"#5\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "p := 10 + 5*I;\nz2 := 2 - I; \nu := getcoords(p);\nv := getcoords(z2);\ndiv(u,v);\nformcomplex(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG^$\"#5\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z2G^$\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG7$\"#5\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG7$\" \"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"$\"\"%" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#^$\"\"$\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 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" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{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 0 496 "p1 := plot([[[2,1]]$3],sty le=point,symbol=[circle,diamond,cross],\n color=COLOR(RGB,0,0,1)):\n p2 := plot([[[2,-1]]$3],style=point,symbol=[circle,diamond,cross],\n \+ color=COLOR(RGB,0,.6,0)):\nt1 := plots[textplot]([[2.3,1.25,`2 + i`], [2.3,-1.2,`2 - i`],[2.9,-0.2,`Real axis`],[-0.6,2.45,`Imag axis`]],\n \+ color=COLOR(RGB,.01,.01,.01)):\nplots[display]([p1,p2,t1],xtickmarks =[-1=`-1`,1=`1`,2=`2`],\n ytickmarks=[-2=`-2 i`,-1=`-i`,0=`0`,1=`i`,2= `2 i`],\n view=[-1.2..3,-2.3..2.45],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 0 755 "p1 := plot([[[2,1]]$3],style=point,symbol=[circle,diamond,cro ss],\n color=COLOR(RGB,0,0,1)):\np2 := plot([[[2,-1]]$3],style=point ,symbol=[circle,diamond,cross],\n color=COLOR(RGB,0,.6,0)):\np3 := p lot([[[0,0],[2,1]],[[1.2,0.667],[1.333,0.667]],\n [[1.25,0.57],[1 .333,0.667]]],color=COLOR(RGB,0,0,1)):\np4 := plot([[[0,0],[2,-1]],[[1 .2,-0.667],[1.333,-0.667]],\n [[1.25,-0.57],[1.333,-0.667]]],colo r=COLOR(RGB,0,0.6,0)):\nt1 := plots[textplot]([[2.3,1.25,`2 + i`],[2.3 ,-1.2,`2 - i`],\n [2.9,-0.2,`Real axis`],[-0.6,2.45,`Imag axis`]],\n color=COLOR(RGB,.01,.01,.01)):\nplots[display]([p1,p2,p3,p4,t1], xtickmarks=[-1=`-1`,1=`1`,2=`2`],\n ytickmarks=[-2=`-2 i`,-1=`-i` ,0=`0`,1=`i`,2=`2 i`],\n view=[-1.2..3,-2.3..2.45],scaling=constr ained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 28 "Code for d rawing 3rd picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1250 "p1 := plot([[[2,1]]$3],style=point,symbol= [circle,diamond,cross],\n color=COLOR(RGB,0,0,1)):\np2 := plot([[[2, -1]]$3],style=point,symbol=[circle,diamond,cross],\n color=COLOR(RGB ,0,.6,0)):\np3 := plot([[[-2,1]]$3],style=point,symbol=[circle,diamond ,cross],\n color=COLOR(RGB,1,0,0)):\np4 := plot([[[-2,-1]]$3],style= point,symbol=[circle,diamond,cross],\n color=COLOR(RGB,.9,.8,0)):\np 5 := plot([[[0,0],[2,1]],[[1.2,0.667],[1.333,0.667]],\n [[1.25,0. 57],[1.333,0.667]]],color=COLOR(RGB,0,0,1)):\np6 := plot([[[0,0],[2,-1 ]],[[1.2,-0.667],[1.333,-0.667]],\n [[1.25,-0.57],[1.333,-0.667]] ],color=COLOR(RGB,0,.6,0)):\np7 := plot([[[0,0],[-2,1]],[[-1.2,0.667], [-1.333,0.667]],\n [[-1.25,0.57],[-1.333,0.667]]],color=COLOR(RGB ,1,0,0)):\np8 := plot([[[0,0],[-2,-1]],[[-1.2,-0.667],[-1.333,-0.667]] ,\n [[-1.25,-0.57],[-1.333,-0.667]]],color=COLOR(RGB,.9,.8,0)):\n t1 := plots[textplot]([[2.3,1.25,`2 + i`],[2.3,-1.2,`2 - i`],\n [-2. 3,1.25,`-2 + i`],[-2.3,-1.2,`-2 - i`],\n [2.9,-0.2,`Real axis`],[-0. 6,2.45,`Imag axis`]],\n color=COLOR(RGB,.01,.01,.01)):\nplots[dis play]([p1,p2,p3,p4,p5,p6,p7,p8,t1],xtickmarks=[-1=`-1`,1=`1`,2=`2`],\n ytickmarks=[-2=`-2 i`,-1=`-i`,0=`0`,1=`i`,2=`2 i`],\n view= [-3..3,-2.3..2.45],scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }