{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 "" -1 256 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 262 "" 0 1 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 128 0 128 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 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 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 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 "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 268 53 "High School Modul es > Precalculus by Gregory A. Moore" }}{PARA 3 "" 0 "" {TEXT -1 4 " \+ " }{TEXT 267 46 "Powers of Complex Numbers & DeMoivre's Formula" }} {PARA 0 "" 0 "" {TEXT -1 151 "\nThis is a further development of compl ex powers. Also see worksheets on Complex Numbers and Complex Number O perations in the Algebra I & II PowerTool." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 153 "[Directions : Execute the Cod e Resource section first. Although there will be no output immediately , these definitions are used later in this worksheet.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 258 7 "0. Code" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; w ith(plots): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 917 "ComplexPlo t := proc()\n local k,A,Pt,c,cg,shade,r,u,Lines; \n u:= 0;\n c g := COLOR(RGB, .7,.8,.7);\n for k from 1 to nargs do\n shade \+ := evalf(rand()/10^12,2)/4 + .3;\n c := COLOR(RGB, shade,shade+. 2, shade); \n A||k := complexplot( [0,args[k]], linestyle = 2, \+ scaling = constrained,\n color = c);\n \+ Lines||k := plot( [[ 0, Im(args[k])], [Re(args[k]), Im(args[k])],\n \+ [Re(args[k]), 0]], color = cg); \n\n u \+ := max( u, abs(Re(args[k])),abs(Im(args[k])) ); \n od;\n\n r := ev alf(u/40,2);\n for k from 1 to nargs do \n shade := evalf(rand ()/10^12,2)/4 + .1;\n c := COLOR(RGB, shade,.8, shade); \n \+ Pt||k := plottools[disk]( [Re(args[k]), Im(args[k])],r, \n \+ color = c ); od;\n\n display( [seq( A||k, k = 1..narg s), seq( Pt||k, k = 1..nargs),\n seq( Lines||k, k = 1..nar gs)] ); \nend proc:\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " \n" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 259 19 "1. Compl ex Products" }}{PARA 0 "" 0 "" {TEXT -1 66 "\nHere are two complex num bers and their geometric representation.\n" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 30 "z := 2 + I; w := -1 + 3*I; " }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 17 "ComplexPlot(z,w);" }}}{PARA 0 "" 0 "" {TEXT -1 139 "\nNow, here are the same two numbers together with their produ ct. (Maple scales the picture to fit, which results in a 'zoomed out' \+ view.) \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "z*w;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ComplexPlot(z,w, z*w );" }}}{PARA 0 "" 0 "" {TEXT -1 192 "\nApparently the product is both longer and at a different angle than the two original numbers. At first guess, it a ppears both the length and angle increases. Lets look at some more exa mples.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "z := -5 + I; \+ w := 3 + 2*I;\nz*w;\nComplexPlot(z,w, z*w );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "z := -2 + 3*I; w := 3 - 2*I;\nz*w;\nComplexPl ot(z,w, z*w );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "z := 5*I ; w := 4+4*I;\nz*w;\nComplexPlot(z,w, z*w );" }}}{PARA 0 "" 0 "" {TEXT -1 129 "\nActualy there is some logic to this situation. Lets lo ok at the arguments and absolute values of the numbers and their produ ct.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "z1 := 2 + 3*I; \+ \nz2 := 5 + 1*I;\nz*w;\nComplexPlot(z,w, z*w );" }}}{PARA 0 "" 0 "" {TEXT -1 74 "\nThe absolute value is the product is the product of the absolute values.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "a1 := abs(z1); \na2 := evalf(abs(z2));\na3 := abs(z1*z2);\na1 * a2;" }}} {PARA 0 "" 0 "" {TEXT -1 72 "\n\nThe argument of a product is the sum \+ of the arguments of the factors.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "theta||1 := evalf(argument(z1));\ntheta||2 := eval f(argument(z2));\ntheta := argument(z1*z2);\ntheta||1 + theta||2; " }}}{PARA 0 "" 0 "" {TEXT -1 112 "\nHere is an example where its eas y to see that product of absolute values is the absolute value of the \+ product.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "z1 := 3 + 4 *I; abs(z1); \nz2 := 12 + 5*I; abs(z2);\nz1*z2; \nabs(z1*z2);\nComp lexPlot( z1,z2, z1*z2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 270 28 "2. Powers \+ of Complex Numbers" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " \+ " }}{PARA 0 "" 0 "" {TEXT 269 22 " Complex Powers" }}{PARA 0 " " 0 "" {TEXT -1 149 "\nTaking the power of any number, variable, expre ssion, or complex number simply involves repeated multiplication. So t he same principles will apply. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "(3 + 2*x)^3; \n% = expand(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "(3 + 2*I)^3;" }}}{PARA 0 "" 0 "" {TEXT -1 318 "\nNote that 46 = 54 - 8, and -9 = 27 - 36. Can you explain this?\n\nHere is \+ the geometric view of the first four powers of a complex number. The p owers are \"rotating\" to the left, in a counter-clockwise direction. \+ Notice that in some cases the absolute value of the powers is getting \+ bigger or smaller, faster or slower.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "z := 1.2 + .9*I;\nComplexPlot( z, z^2, z^3, z^4, z^5, z^6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "z := .6 + .5*I;\n ComplexPlot( z, z^2, z^3, z^4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "z := 2 + 1*I;\nComplexPlot( z, z^2, z^3, z^4);" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 124 "\nAlso, it's poss ible for the powers to stay on the unit circle. This happens if the or iginal number has absolute value of 1." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "z := sqrt(3)/2 + I/2;\ndisplay( polarplot( 1, color = gray), ComplexPlot( seq(z^k, k = 1..11) ));" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 260 22 "3. De Moivre's Formula" }}{PARA 0 "" 0 "" {TEXT -1 103 "\nJust as \+ with real numbers or variables, powers are simply repeated multiplicat ion of the same factor.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "(3 + 2*I)^10;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "(-1/3 + ( 1/4)*I)^10;" }}}{PARA 0 "" 0 "" {TEXT -1 108 "\nHowever, when a comple x number is in trig form, we can apply this useful formula by the Marq uis De Moivre.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "z:='z': \+ r:='r': theta:='theta':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " demoivre := \nz^n = (r^n)*( cos(n*theta) + I*sin(n*theta) );" }}} {PARA 0 "" 0 "" {TEXT -1 248 "\n\nGiven a complex number in polar form , we can compute the power much faster by raising the modulus (or abso lute value, which is a real number) to the power, and then multiplying the angle by the power and computing the resulting sine and cosine.\n \n" }{TEXT 265 15 " " }{TEXT 266 7 "Example" }{TEXT -1 3 " : " }{XPPEDIT 18 0 "(2*(cos(Pi/7)+I*sin(Pi/7)))^14;" "6#*$*&\"\"# \"\"\",&-%$cosG6#*&%#PiGF&\"\"(!\"\"F&*&%\"IGF&-%$sinG6#*&F,F&F-F.F&F& F&\"#9" }}{PARA 0 "" 0 "" {TEXT -1 204 "\nA problem tailor made for th is formula! We can get Maple to give us the result. However, this woul d be quite difficult to expand and compute without a calculator or com puter because of the unusual angle." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "z := 2*( cos( Pi/7) + I*sin(Pi/7) );\nz^14: % = evalc (%); evalf(%);" }}}{PARA 0 "" 0 "" {TEXT -1 42 "\nLets allow the formu la to work its magic." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "2^1 4; (Pi/7)*14;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "result = \+ 16384* (cos( 2*Pi) + I*sin( 2*Pi));" }}}{PARA 0 "" 0 "" {TEXT -1 119 " \n\n\nWhat if we are not so lucky as to be given a number in this form ? Convert it, and then use the formula, of course!\n\n" }{TEXT 261 15 " " }{TEXT 262 7 "Example" }{TEXT -1 3 " : " }{XPPEDIT 18 0 "(3 - 3*I)^12" "6#*$,&\"\"$\"\"\"*&F%F&%\"IGF&!\"\"\"#7" }}{PARA 0 "" 0 "" {TEXT -1 52 "\nOf course, Maple can do the dirty work for us , ...." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "z := 3 - 3*I;\nz^1 2;" }}}{PARA 0 "" 0 "" {TEXT -1 262 "\n... but if we were stranded on \+ a desert isle and needed to do this without the tools of modern life, \+ it would be quite tedious to multiply it out. In fact, we would need q uite a few papryus scratch husks to write on. We'll convert the form a nd use the formula.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "r : = abs(z);\ntheta := arctan( Im(z)/Re(z));" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 38 "z = r*('cos'(theta) + I*'sin'(theta));" }}}{PARA 0 "" 0 "" {TEXT -1 103 "\nNow that we have expressed z in polar form, we can plug into the formula. The main part is to compute " }{XPPEDIT 18 0 "r^n" "6#)%\"rG%\"nG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "n*theta " "6#*&%\"nG\"\"\"%&thetaGF%" }{TEXT -1 46 ", then evaluate the result using those values." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "subs ( \{z=z, r=r\}, demoivre);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "r^12; theta*12;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "resu lt = 34012224* (cos( -3*Pi) + I*sin( -3*Pi));" }}}{PARA 0 "" 0 "" {TEXT -1 4 "\n \n\n" }{TEXT 263 14 " " }{TEXT 264 7 "Exam ple" }{TEXT -1 3 " : " }{XPPEDIT 18 0 "(8-8*I*sqrt(3))^15;" "6#*$,&\" \")\"\"\"*(F%F&%\"IGF&-%%sqrtG6#\"\"$F&!\"\"\"#:" }}{PARA 0 "" 0 "" {TEXT -1 63 "\nWe can \"cheat\" and get Maple to do the dirty work for us, ...." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "z := 8-8*I*sqrt (3);\nz^15: \n% = evalc(%);" }}}{PARA 0 "" 0 "" {TEXT -1 62 "\n... but if we were stranded ...well you know what to do.....\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "r := abs(z);\ntheta := arctan( Im(z )/Re(z));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "z = r*('cos'(t heta) + I*'sin'(theta));" }}}{PARA 0 "" 0 "" {TEXT -1 75 "\nNow that w e have expressed z in polar form, we can plug into the formula. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "subs( \{z=z, r=r\}, demoivre );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "r^15; theta*15;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "result = 1152921504606846976 * (cos( -5*Pi) + I*sin( -5*Pi));\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT 257 35 "\n \251 20 02 Waterloo Maple Inc" }}}{MARK "0 1" 33 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }