{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 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 "" -1 256 "" 0 1 110 103 32 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 271 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 1 14 0 0 16 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 276 "" 1 12 0 0 16 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 1 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 6 6 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 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 " " 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 0 0 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 "Bullet \+ Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 } {PSTYLE "Korrektur" 0 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 255 0 255 1 2 1 2 0 0 2 1 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 256 1 }{PSTYLE " " 3 257 1 {CSTYLE "" -1 -1 "" 0 1 220 206 144 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 "" 4 258 1 {CSTYLE "" -1 -1 " " 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } } {SECT 0 {PARA 257 "" 0 "" {TEXT -1 15 "Komplexe Zahlen" }}{PARA 19 "" 0 "" {TEXT -1 26 "Gerhard Bitsch, Oktober 99" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }} {PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and tra ce have been redefined and unprotected\n" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 35 "Komplexe Zahlen als Drehstreckungen" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 320 "In diesem Arbeitsblatt w erden komplexe Zahlen eingef\374hrt. Man muss ber\374cksichtigen, das s die komplexen Zahlen nicht wie die reellen Zahlen aus einer Ordnungs vervollst\344ndigung entstehen, sondern aus einer algebraischen Vervol lst\344ndigung. Es ist deshalb n\366tig, die K\366rpereigenschaften de r reellen Zahlen kurz anzusprechen" }}{PARA 258 "" 0 "" {TEXT -1 4 "Di e " }{TEXT 256 14 "reellen Zahlen" }{TEXT -1 1 " " }{TEXT 258 2 "R " } {TEXT -1 24 "erf\374llen eine Reihe von " }{TEXT 257 8 "Axiomen:" }} {PARA 4 "" 0 "" {TEXT 274 2 "Di" }{TEXT 276 54 "e reellen Zahlen bilde n einen K\366rper bez\374glich + und *" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 32 "Bez\374glich der Addition + bildet " }{TEXT 259 2 "R " } {TEXT -1 50 " eine kommutative Gruppe mit neutralem Element 0:" }} {PARA 15 "" 0 "" {TEXT -1 16 "F\374r alle a,b in " }{TEXT 268 2 "R " } {TEXT -1 85 "gilt: a + b = b + a \+ (Kommutativgesetz)" }}{PARA 15 "" 0 "" {TEXT -1 18 "F\374r a lle a,b,c in " }{TEXT 269 2 "R " }{TEXT -1 77 " gilt: a + ( b + c ) \+ = ( a + b ) + c (Assoziativgesetz)" }}{PARA 15 "" 0 "" {TEXT -1 23 "Es gibt eine Zahl 0 in " }{TEXT 270 2 "R," }{TEXT -1 22 " so da\337 f\374r alle a in " }{TEXT 271 2 "R " }{TEXT -1 49 " \+ gilt: a + 0 = a (0 ist neutrales Element)" }}{PARA 15 "" 0 "" {TEXT -1 14 "F\374r alle a in " }{TEXT 273 2 "R " }{TEXT -1 15 " gibt \+ es -a in " }{TEXT 272 2 "R " }{TEXT -1 91 " mit: a + -a = 0 \+ (-a und a sind zueinander invers bez\374glich +)" } }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 39 "Bez\374glich der Multiplikation * bildet " }{TEXT 267 2 "R " }{TEXT -1 59 "ohne die 0 eine kommutati ve Gruppe mit neutralem Element 1:" }}{PARA 15 "" 0 "" {TEXT -1 16 "F \374r alle a,b in " }{TEXT 260 2 "R " }{TEXT -1 85 "gilt: a * b \+ = b * a (Kommutativgesetz)" } }{PARA 15 "" 0 "" {TEXT -1 18 "F\374r alle a,b,c in " }{TEXT 261 2 "R \+ " }{TEXT -1 77 " gilt: a * ( b * c ) = ( a * b ) * c \+ (Assoziativgesetz)" }}{PARA 15 "" 0 "" {TEXT -1 23 "Es gibt eine \+ Zahl 1 in " }{TEXT 262 2 "R," }{TEXT -1 22 " so da\337 f\374r alle a i n " }{TEXT 263 2 "R " }{TEXT -1 49 " gilt: a * 1 = a (1 ist neu trales Element)" }}{PARA 15 "" 0 "" {TEXT -1 14 "F\374r alle a in " } {TEXT 265 2 "R " }{TEXT -1 16 " gibt es 1/a in " }{TEXT 264 2 "R " } {TEXT -1 90 " mit: a * 1/a = 1 (1/a und a si nd zueinander invers bez\374glich *)" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 46 "Produkte d\374rfen \374ber Summen \"verteilt\" werden:" } }{PARA 15 "" 0 "" {TEXT -1 18 "F\374r alle a,b,c in " }{TEXT 266 2 "R \+ " }{TEXT -1 77 " gilt: a * ( b + c) = a*b + a*c \+ (Distributivgesetz)" }}}{PARA 4 "" 0 "" {TEXT 277 68 "Die reellen \+ Zahlen haben eine Ordnung < mit folgenden Eigenschaften:" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 65 "Die Ordnung < ist transitiv, asymmetrisch , linear und vollst\344ndig" }}{PARA 15 "" 0 "" {TEXT -1 112 "Wenn a " 0 "" {MPLTEXT 1 0 18 "z1:=1+I;z2:= 4-2*I;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z1G^$\"\"\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z2G^$\"\"%!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "z1*z2;z1+z2;z1-z2;z1/z2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$\"\"'\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$\"\"& !\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$!\"$\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$#\"\"\"\"#5#\"\"$F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Realteil und Imagin\344rteil erh\344lt man so:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Re(z2);Im(z2);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"#" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Winkel und L\344nge f\374r die D arstellung mit Polarkoordinaten:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "argument(z1);abs(z1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrt G6#\"\"#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 262 "Die Funktion p olar (muss zuerst mit readlib eingelesen werden) liefert komplexe Zahl en in Polarkoordinaten. Man kann sie auch verwenden, um komplexe Zahle n mit Polarkoordinaten zu definieren. Mit evalc kann man eine komplexe Zahl in kartesischer Form darstellen." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "polar(z1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&pola rG6$*$-%%sqrtG6#\"\"#\"\"\",$%#PiG#F+\"\"%" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "z3:=polar(1,Pi/6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z3G-%&polarG6$\"\"\",$%#PiG#F(\"\"'" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 24 "Re(z3);Im(z3);evalc(z3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$-%%sqrtG6#\"\"$\"\"\"#F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$-%% sqrtG6#\"\"$\"\"\"#F)\"\"#^#F*F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Die Funktion " }{TEXT 278 6 "signum" }{TEXT -1 11 " liefert " } {XPPEDIT 18 0 "x/abs(x);" "6#*&%\"xG\"\"\"-%$absG6#F$!\"\"" }{TEXT -1 7 " f\374r " }{XPPEDIT 18 0 "x <> 0;" "6#0%\"xG\"\"!" }{TEXT -1 103 ". F\374r komplexe Zahlen z ist das eine komplexe Zahl, die den gleich en Winkel wie z hat und die L\344nge 1. " }{TEXT 279 4 "csgn" }{TEXT -1 219 "(x) liefert 1, wenn Re(x)>0 ist oder Re(x)=0 und Im(x)>0 ist. Die Funktion liefert -1, wenn Re(x)<0 oder Re(x)=0 und Im(x)<0 ist. \+ (Sie unterscheidet also, ob x in der linken oder rechten H\344lfte der Zahlenebene liegt. " }{TEXT 280 9 "conjugate" }{TEXT -1 47 "(x) liefe rt die zu x konjugierte komplexe Zahl." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "signum(z1);abs(signum(z1));csgn(z1);csgn(-z1);conjuga te(z1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&^$#\"\"\"\"\"#F%F&-%%sqrt G6#F'F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$\"\"\"!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Maple gibt beim L\366s en von Gleichungen stets auch komplexe L\366sungen aus." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve( x^2=-1,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$^#\"\"\"^#!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve(x^3=1,\{x\});" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%<#/%\"xG\"\"\"<#/F%,&#!\"\"\"\"#F&*&^# #F&F,F&-%%sqrtG6#\"\"$F&F&<#/F%,&F*F&*&^#F*F&F0F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Will man komplexe Werte unterdr\374cken, kann m an so vorgehen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "select(t ype,[solve(x^3=1,x)],realcons);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7# \"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "etwas eleganter:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 216 "reell_solve:=proc(eqn,var) \n local ls;\n ls := [solve(eqn,var)];\n if type(ls[1],set) \n then\n \+ op(map(zzzz->\{op(var)=zzzz\},select(type,map(x->rhs(x[1]),ls),realc ons)));\n else\n op(select(type,ls,realcons));\n fi; \nend:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(x^3+2*x^2-x+1=0,x);" } }{PARA 12 "" 1 "" {XPPMATH 20 "6%,(*$),&\"$W#\"\"\"*&\"#OF(-%%sqrtG6# \"#HF(F(#F(\"\"$F(#!\"\"\"\"'*&#\"#9F0F(*&F(F(*$)F&#F(F0F(F2F(F2#\"\"# F0F2,*F$#F(\"#7*&#\"\"(F0F(*$)F&#F(F0F(F2F(#F*&FAF(*$)F&#F(F0F(F2F(#F " 0 "" {MPLTEXT 1 0 31 "re ell_solve(x^3+2*x^2-x+1=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$) ,&\"$W#\"\"\"*&\"#OF(-%%sqrtG6#\"#HF(F(#F(\"\"$F(#!\"\"\"\"'*&#\"#9F0F (*&F(F(*$)F&#F(F0F(F2F(F2#\"\"#F0F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "5 23 0 0 " 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }