{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 "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" 20 256 "" 0 1 0 0 0 1 0 0 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 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 116 103 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 128 20 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 64 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "Symbol" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 276 "Symbol" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 277 "Symbol" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 " Symbol" 0 1 0 0 0 0 0 0 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 } {CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 287 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 292 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 295 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 296 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 297 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 298 "" 0 1 0 0 7 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 299 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 300 "" 0 1 0 0 236 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 301 "" 1 12 0 0 0 0 0 2 2 0 0 0 0 0 0 1 } {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 "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courie r" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 1 1 }1 1 0 0 0 0 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 Outpu t" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 24 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }2 1 0 0 12 12 1 0 1 0 2 2 19 1 }} {SECT 0 {SECT 1 {PARA 256 "" 0 "" {TEXT -1 48 "Chapter 2\n\nThe First \+ Steps: Calculus on Numbers\n" }}{PARA 0 "" 0 "" {TEXT 301 31 "\251 Cop yright 2003 by Andr\351 Heck." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 0 "" 0 "" {TEXT 257 2 "1." }{TEXT -1 38 " Consider the f ollowing Maple session\n" }{TEXT 0 20 " > 3^2:\n > 4^2;\n" }{TEXT -1 13 " " }{TEXT 2 30 " " } {TEXT 20 2 "16" }{TEXT 2 1 "\n" }{TEXT 0 13 " > % + %%;\n" }{TEXT 256 78 "does the last instruction make sense? If so, what is the resul t? if not, why?\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "3^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "4^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "% + %%;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#D" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Yes. Maple adds the last power result (" }{TEXT 0 1 "%" }{TEXT -1 32 ") to the power before this one (" }{TEXT 0 2 "%%" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 88 "The first pow er is not displayed because the author has used a colon to hide the ou tput." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 258 2 "2." }{TEXT -1 67 " Explain the different \+ results of the following Maple commands.\n " }{TEXT 295 4 "(a) " } {TEXT 0 5 " x:y;" }{TEXT -1 4 "\n " }{TEXT 296 3 "(b)" }{TEXT -1 1 " " }{TEXT 0 5 " x/y;" }{TEXT -1 4 "\n " }{TEXT 297 3 "(c)" }{TEXT -1 1 " " }{TEXT 0 6 " x\\y;\n" }}{SECT 1 {PARA 0 "" 0 "" {TEXT 298 3 " (a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "x:y;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"yG" }}}{PARA 0 "" 0 "" {TEXT -1 33 "The command shows only the term \"" }{TEXT 260 1 "y" }{TEXT -1 22 "\" and hides t he term \"" }{TEXT 259 1 "x" }{TEXT -1 54 "\". Don't confuse the colon with the division operator " }{TEXT 0 1 "/" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 299 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "x/y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"xG\"\"\"%\"yG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 51 "The previous command shows a quotient of two terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 300 3 "(c)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "x\\y;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#xyG" }}} {PARA 0 "" 0 "" {TEXT -1 76 "The backslash is the continuation charact er; in fact, this input is read as " }{TEXT 263 1 "x" }{TEXT -1 2 "y. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 0 " " 0 "" {TEXT 261 2 "3." }{TEXT -1 82 " In this exercise you can practi ce your skills in using the help system of Maple.\n" }{TEXT 287 3 "(a) " }{TEXT -1 61 " Suppose that you can want to select from an equation, e.g., " }{XPPEDIT 18 0 "1=cos(x)^2+sin(x)^2" "6#/\"\"\",&*$-%$cosG6#% \"xG\"\"#F$*$-%$sinG6#F*F+F$" }{TEXT -1 69 ", only the left or right s ide. How can you easily do this in Maple?\n\n" }{TEXT 288 3 "(b)" } {TEXT -1 162 " Suppose that you want to compute the continued fraction approximation of the exponential function; can Maple do this for you? If yes, carry out the computation.\n" }{TEXT 289 3 "(c)" }{TEXT -1 49 " Suppose that you want to factor the polynomial " }{XPPEDIT 18 0 "x^8+x^6+10*x^3+8*x^2+2*x+8" "6#,.*$%\"xG\"\")\"\"\"*$F%\"\"'F'*&\"#5F '*$F%\"\"$F'F'*&F&F'*$F%\"\"#F'F'*&F0F'F%F'F'F&F'" }{TEXT -1 69 " modu lo 13. Can Maple do this? If yes, carry out this factorization.\n" } {TEXT 290 4 "\n(d)" }{TEXT -1 59 " Suppose that you want to determine \+ all subsets of the set " }{XPPEDIT 18 0 "\{1,2,3,4,5\}" "6#<'\"\"\"\" \"#\"\"$\"\"%\"\"&" }{TEXT -1 32 ". How can you do this in Maple?\n" } }{SECT 1 {PARA 0 "" 0 "" {TEXT 262 3 "(a)" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 24 "1 = cos(x)^2 + sin(x)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"\",&*$)-%$cosG6#%\"xG\"\"#F$F$*$)-%$sinGF*F,F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "lhs(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " rhs(%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)-%$cosG6#%\"xG\"\"#\" \"\"F+*$)-%$sinGF(F*F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 9 "The name " } {TEXT 0 3 "lhs" }{TEXT -1 12 " stands for " }{TEXT 0 1 "l" }{TEXT -1 4 "eft-" }{TEXT 0 1 "h" }{TEXT -1 4 "and " }{TEXT 0 1 "s" }{TEXT -1 5 "ide; " }{TEXT 0 3 "rhs" }{TEXT -1 12 " stands for " }{TEXT 0 1 "r" } {TEXT -1 5 "ight-" }{TEXT 0 1 "h" }{TEXT -1 4 "and " }{TEXT 0 1 "s" } {TEXT -1 4 "ide." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 0 "" 0 "" {TEXT 264 3 "(b)" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 28 "convert(exp(x), confrac, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*&%\"xGF$,&F$F$*&F&F$,&\"\"#!\"\"*&F&F$,&\"\" $F+*&F&F$,&F*F$*&\"\"&F+F&F$F$F+F$F+F$F+F$F+F$" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 265 3 "(c) " }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Factor(x^8+x^6+10*x^3+8 *x^2+2*x+8) mod 13;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&%\"xG\"\"\" \"\"(F&F&,(*$)F%\"\"#F&F&*&\"#6F&F%F&F&\"#7F&F&,(*$)F%\"\"$F&F&*&\"\"' F&F%F&F&\"\"%F&F&,(F)F&*&\"\")F&F%F&F&\"\"*F&F&" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 266 3 "(d) " }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(combinat):" }} {PARA 7 "" 1 "" {TEXT -1 67 "Warning, the protected name Chi has been \+ redefined and unprotected\n" }}}{PARA 0 "" 0 "" {TEXT -1 43 "Two ways \+ to compute the set of all subsets:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "powerset(5);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "powerset(\{1,2,3,4,5\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 74 "S := subsets(\{1,2,3,4,5\}):\nwhile not S[finished] do S[nextvalue ]() end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#\"\"&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<$\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$\"\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$\"\" \"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$\"\"\"\"\"&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<$\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# <$\"\"#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$\"\"#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$\"\"$\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$\"\"%\"\"&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<%\"\"\"\"\"#\"\"$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<%\"\"\"\"\"#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #<%\"\"\"\"\"#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%\"\"\"\"\"$\" \"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%\"\"\"\"\"$\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%\"\"\"\"\"%\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%\"\"#\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<% \"\"#\"\"$\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%\"\"#\"\"%\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%\"\"$\"\"%\"\"&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<&\"\"\"\"\"#\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&\"\"\"\"\"#\"\"$\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&\" \"\"\"\"#\"\"%\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&\"\"\"\"\"$\" \"%\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&\"\"#\"\"$\"\"%\"\"&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<'\"\"\"\"\"#\"\"$\"\"%\"\"&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 0 "" 0 " " {TEXT 267 2 "4." }{TEXT -1 10 " Load the " }{TEXT 0 9 "numtheory" } {TEXT -1 21 " package by entering " }{TEXT 0 16 "with(numtheory);" } {TEXT -1 140 " You may recognize some functions from number theory; so me of the routines in this package are useful in answering the followi ng questions.\n" }{TEXT 291 4 "\n(a)" }{TEXT -1 64 " Build a list of a ll integers that divide 9876543210123456789.\n\n" }{TEXT 292 3 "(b)" } {TEXT -1 64 " Find the prime number that is closest to 987654321012345 6789.\n\n" }{TEXT 293 3 "(c)" }{TEXT -1 36 " What is the prime factori zation of " }{XPPEDIT 18 0 "5^(5^(5^5)):" "6#)\"\"&)F$*$F$F$" }{TEXT -1 3 "?\n\n" }{TEXT 294 3 "(d)" }{TEXT -1 95 " Expand the base exp(1) \+ of the natural logarithm as a continued fraction up to 10 levels deep. \n" }}{SECT 1 {PARA 0 "" 0 "" {TEXT 268 3 "(a)" }{TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 16 "with(numtheory);" }}{PARA 7 "" 1 "" {TEXT -1 69 "Warning, the protected name order has been redefined and unprot ected\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7Q%&GIgcdG%)bigomegaG%&cfra cG%)cfracpolG%+cyclotomicG%)divisorsG%)factorEQG%*factorsetG%'fermatG% )imagunitG%&indexG%/integral_basisG%)invcfracG%'invphiG%*issqrfreeG%'j acobiG%*kroneckerG%'lambdaG%)legendreG%)mcombineG%)mersenneG%(migcdexG %*minkowskiG%(mipolysG%%mlogG%'mobiusG%&mrootG%&msqrtG%)nearestpG%*nth converG%)nthdenomG%)nthnumerG%'nthpowG%&orderG%)pdexpandG%$phiG%#piG%* pprimrootG%)primrootG%(quadresG%+rootsunityG%*safeprimeG%&sigmaG%*sq2f actorG%(sum2sqrG%$tauG%%thueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "divisors(9876543210123456789);" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#\"&2p#\"&xr&\"&@2) \"&*e#)\"'(f;\"\"'nxC\"'\"z\\$\"',Lu\"(t$\\5\"4*ycM75Kaw)*\"2\",fxxxx&\"-xKLLL<\"/zjabbBG\"/P\"Rmm1Z)\"04V.uSNA\"\"0FH5AA1n$\"1\"y 3jm'=,6\"/*3.uS(G8\"/n#4AAi)R\"3^KZ&p(pWKD\"3`(>k3$4M(f(\"4@ur8+p$R(4 \"\"4jA:T+2\"=#H$\"0,yimme>\"\"0d,C'HOF<\"0r/s)))3#=&\"189;mmia:\")d+) p&\"*r,%4<\"*80#G^\"*T2uS(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "sort(convert(%, list));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7R\"\" \"\"\"$\"\"*\"#8\"#R\"$<\"\"%`j\"%p*)\"&f!>\"&2p#\"&xr&\"&@2)\"&*e#)\" '(f;\"\"'nxC\"'\"z\\$\"',Lu\"(t$\\5\")d+)p&\"*r,%4<\"*80#G^\"*T2uS(\"+ \"[\"[\"[\"\"+BAAAA\"+VWWWW\"+pmmmm\",HLLLL\"\",`#f#f#>\",fxxxx&\"-xKL LL<\".$z[=&=T*\"/*3.uS(G8\"/zjabbBG\"/n#4AAi)R\"/P\"Rmm1Z)\"0,yimme>\" \"04V.uSNA\"\"0d,C'HOF<\"0FH5AA1n$\"0r/s)))3#=&\"1\"y3jm'=,6\"189;mmia :\"2k3$4M(f(\"4@ur8+p$R(4\"\"4jA:T+2\"=# H$\"4*ycM75Kaw)*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 0 "" 0 "" {TEXT 269 3 "(b)" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "with(numtheory);" }}{PARA 7 "" 1 "" {TEXT -1 69 "Wa rning, the protected name order has been redefined and unprotected\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7Q%&GIgcdG%)bigomegaG%&cfracG%)cfrac polG%+cyclotomicG%)divisorsG%)factorEQG%*factorsetG%'fermatG%)imagunit G%&indexG%/integral_basisG%)invcfracG%'invphiG%*issqrfreeG%'jacobiG%*k roneckerG%'lambdaG%)legendreG%)mcombineG%)mersenneG%(migcdexG%*minkows kiG%(mipolysG%%mlogG%'mobiusG%&mrootG%&msqrtG%)nearestpG%*nthconverG%) nthdenomG%)nthnumerG%'nthpowG%&orderG%)pdexpandG%$phiG%#piG%*pprimroot G%)primrootG%(quadresG%+rootsunityG%*safeprimeG%&sigmaG%*sq2factorG%(s um2sqrG%$tauG%%thueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "p : = 9876543210123456789;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG\"4*yc M75Kaw)*" }}}{PARA 0 "" 0 "" {TEXT -1 50 "We compute the smallest prim e number greater than " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT -1 30 " an d calculate the difference." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "nextprime(p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"4.oXB,@Vl()*" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "% - p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#9" }}}{PARA 0 "" 0 "" {TEXT -1 46 "We compute the la rgest prime number less than " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT -1 30 " and calculate the difference." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "prevprime(p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"4 \"ycM75Kaw)*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "p - %;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 68 "We conclude that 9876543210123456781 is the prime number closest to \+ " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 17 "In one statement:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "if nextprime(p)-p > p-prevprime(p) then \n closest := prevprim e(p) \nelse \n closest := nextprime(p) \nend if;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(closestG\"4\"ycM75Kaw)*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 270 3 "(c)" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(numtheory);" }}{PARA 7 "" 1 "" {TEXT -1 69 "Warning, the protected name order has been rede fined and unprotected\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7Q%&GIgcdG% )bigomegaG%&cfracG%)cfracpolG%+cyclotomicG%)divisorsG%)factorEQG%*fact orsetG%'fermatG%)imagunitG%&indexG%/integral_basisG%)invcfracG%'invphi G%*issqrfreeG%'jacobiG%*kroneckerG%'lambdaG%)legendreG%)mcombineG%)mer senneG%(migcdexG%*minkowskiG%(mipolysG%%mlogG%'mobiusG%&mrootG%&msqrtG %)nearestpG%*nthconverG%)nthdenomG%)nthnumerG%'nthpowG%&orderG%)pdexpa ndG%$phiG%#piG%*pprimrootG%)primrootG%(quadresG%+rootsunityG%*safeprim eG%&sigmaG%*sq2factorG%(sum2sqrG%$tauG%%thueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "5^(5^(5^5)):" }}{PARA 8 "" 1 "" {TEXT -1 35 "Err or, numeric exception: overflow\n" }}}{PARA 0 "" 0 "" {TEXT -1 123 "A \+ typical case where thinking before acting helps. The attempted input i s a power of the prime number 5 to a high exponent " }{XPPEDIT 18 0 "e " "6#%\"eG" }{TEXT -1 40 ". Therefore, the prime factorization is " } {XPPEDIT 18 0 "5^e" "6#)\"\"&%\"eG" }{TEXT -1 85 ". For your interest, the exponent is an integer with 2185 digits in decimal notation." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "e := 5^(5^5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "length(e);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%&=#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{SECT 1 {PARA 0 "" 0 "" {TEXT 271 3 "(d)" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "with(numtheory);" }}{PARA 7 "" 1 "" {TEXT -1 69 "Wa rning, the protected name order has been redefined and unprotected\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7Q%&GIgcdG%)bigomegaG%&cfracG%)cfrac polG%+cyclotomicG%)divisorsG%)factorEQG%*factorsetG%'fermatG%)imagunit G%&indexG%/integral_basisG%)invcfracG%'invphiG%*issqrfreeG%'jacobiG%*k roneckerG%'lambdaG%)legendreG%)mcombineG%)mersenneG%(migcdexG%*minkows kiG%(mipolysG%%mlogG%'mobiusG%&mrootG%&msqrtG%)nearestpG%*nthconverG%) nthdenomG%)nthnumerG%'nthpowG%&orderG%)pdexpandG%$phiG%#piG%*pprimroot G%)primrootG%(quadresG%+rootsunityG%*safeprimeG%&sigmaG%*sq2factorG%(s um2sqrG%$tauG%%thueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "cfr ac(exp(1), 10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"#\"\"\"*&F%F% ,&F%F%*&F%F%,&F$F%*&F%F%,&F%F%*&F%F%,&F%F%*&F%F%,&\"\"%F%*&F%F%,&F%F%* &F%F%,&F%F%*&F%F%,&\"\"'F%*&F%F%,&F%F%*&F%F%,&F%F%%$...GF%!\"\"F%F=F%F =F%F=F%F=F%F=F%F=F%F=F%F=F%F=F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 272 2 "5." }{TEXT -1 42 " \+ In Maple, what is the difference between " }{XPPEDIT 18 0 "1/3+1/3+1/3 " "6#,(*&\"\"\"F%\"\"$!\"\"F%*&F%F%F&F'F%*&F%F%F&F'F%" }{TEXT -1 5 " a nd " }{XPPEDIT 18 0 "`1.0`/`3.0`+`1.0`/`3.0`+`1.0`/`3.0`;" "6#,(*&%$1. 0G\"\"\"%$3.0G!\"\"F&*&F%F&F'F(F&*&F%F&F'F(F&" }{XPPEDIT 18 0 "%?;" "6 #%#%?G" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "re start;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "1/3 + 1/3 + 1/3; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 28 "1.0/3.0 + 1.0/3.0 + 1.0/3.0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+**********!#5" }}}{PARA 0 "" 0 "" {TEXT -1 275 "In \+ the first calculation, Maple computes with rational numbers doing exac t arithmetic. In the second calculation it computes with floating-poin t number doing approximate computations (with round-off errors). A fur ther illustration of approximate computation is the following:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "% + 1.0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++?!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 273 2 "6." }{TEXT -1 42 " Fin d the floating-point approximation of " }{XPPEDIT 18 0 "exp(1)^(Pi*sqr t(163))-262537412640768744;" "6#,&)-%$expG6#\"\"\"*&%#PiGF(-%%sqrtG6# \"$j\"F(F(\"3W(o2k7u`i#!\"\"" }{TEXT -1 59 " using a precision of 15, \+ 25, and 35 digits, respectively.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "exp(Pi*sqrt(163))-262537412640768744;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#*&%#PiG\"\"\"\"$j\"#F)\"\"#F)\"3W(o2k7u`i#!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf[15](%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$!\"&\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf[25](%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!# O!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf[35](%%%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$!&?](!#<" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 274 2 "7." } {TEXT -1 11 " Calculate " }{XPPEDIT 18 0 "Pi^(Pi^Pi))" "6#)%#PiG)F$F$ " }{TEXT -1 25 " to nine decimal places.\n" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "x := Pi^(Pi^Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG)%#PiG)F &F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf[9](x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*mj,M\"\"#5" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "evalf[10](x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+SU;S8\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "seq(ev alf[k](x), k=11..15);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6'$\",W=k,M\"\" \")$\"-+$=k,M\"\"\"($\".GI=k,M\"\"\"'$\"/m+$=k,M\"\"\"&$\"0)f+$=k,M\" \"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 126 "Apparently, you have to raise \+ the precision up to 12 or 13 to see convergeance in the first nine dei mals of the approximation." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 279 2 "8." }{TEXT -1 107 " Comput e this exercise in a floating-point precision of eight decimal places. What is the result of \n310.0 " }{TEXT 275 1 "\264" }{TEXT -1 7 " 320 .0 " }{TEXT 276 1 "\264" }{TEXT -1 7 " 330 - " }{XPPEDIT 18 0 "sqrt(`3 10.0`*` 320.0`);" "6#-%%sqrtG6#*&%&310.0G\"\"\"%(~~320.0GF(" }{TEXT 277 2 " \264" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(`320.0 330.0`);" " 6#-%%sqrtG6#%-320.0~~330.0G" }{TEXT 278 2 " \264" }{TEXT -1 1 " " } {XPPEDIT 18 0 "sqrt(`330.0 310.0`);" "6#-%%sqrtG6#%-330.0~~310.0G" } {TEXT -1 2 "?\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Digits := 8:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "310.0 * 320.0 * 330.0 -\nsqrt(310.0 *320.0) * sqrt(320.0*330.0) * sqrt(330.0*310.0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 280 2 "9." }{TEXT -1 38 " Do yo u remember which of the numbers " }{XPPEDIT 18 0 "19/6" "6#*&\"#>\"\" \"\"\"'!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "22/7" "6#*&\"#A\"\"\"\" \"(!\"\"" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "25/8" "6#*&\"#D\"\"\"\" \")!\"\"" }{TEXT -1 44 " is a fairly good rational approximation of " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 90 "? Use Maple to find the be st of these three numbers. Find the best rational approximation " } {XPPEDIT 18 0 "a/b" "6#*&%\"aG\"\"\"%\"bG!\"\"" }{TEXT -1 4 " of " } {XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "a " "6#%\"aG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 87 " are natural numbers less than 1000 (Hint: look at the continue d fraction expansion of " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 4 ") .\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "convert(evalf(Pi), confrac, cvgts); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"$\"\"(\"#:\"\"\"\"$$H\"#6" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "cvgts;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"$#\"#A\"\"(#\"$L$\"$1\"#\"$b$\"$8\"#\"'[V5\"&:K$# \"($=[6\"'yaO" }}}{PARA 0 "" 0 "" {TEXT -1 4 "So, " }{XPPEDIT 18 0 "22 /7" "6#*&\"#A\"\"\"\"\"(!\"\"" }{TEXT -1 49 " is the best approximatio n of the given ones and " }{XPPEDIT 18 0 "355/113" "6#*&\"$b$\"\"\"\"$ 8\"!\"\"" }{TEXT -1 42 " is the best with integers less than 1000." }} {PARA 0 "" 0 "" {TEXT -1 64 "The same result you get with the appropri ate procedues from the " }{TEXT 0 9 "numtheory" }{TEXT -1 9 " package. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(numtheory):" }} {PARA 7 "" 1 "" {TEXT -1 69 "Warning, the protected name order has bee n redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "pi := cfrac(Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#piG,&\"\" $\"\"\"*&F'F',&\"\"(F'*&F'F',&\"#:F'*&F'F',&F'F'*&F'F',&\"$#HF'*&F'F', &F'F'*&F'F',&F'F'*&F'F',&F'F'*&F'F',&\"\"#F'*&F'F',&F'F'*&F'F',&F&F'%$ ...GF'!\"\"F'FAF'FAF'FAF'FAF'FAF'FAF'FAF'FAF'FAF'" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 30 "seq(nthconver(pi,k), k=1..10);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6,#\"#A\"\"(#\"$L$\"$1\"#\"$b$\"$8\"#\"'$*R5\"&-J$ #\"'[V5\"&:K$#\"'T$3#\"&P$)\"'\"Ql##\"(3k9\"\"' 8\\O#\"(VHF%\"(?,O\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }}{SECT 1 {PARA 0 "" 0 "" {TEXT 281 3 "10." }{TEXT -1 12 " Check that \+ " }{XPPEDIT 18 0 "sqrt(2*sqrt(19549)+286)" "6#-%%sqrtG6#,&*&\"\"#\"\" \"-F$6#\"&\\&>F)F)\"$'GF)" }{TEXT -1 13 " is equal to " }{XPPEDIT 18 0 "sqrt(113)+sqrt(173)" "6#,&-%%sqrtG6#\"$8\"\"\"\"-F%6#\"$t\"F(" } {TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "'sqrt(2*sqrt(19549)+286)' ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%sqrtG6#,&*&\"\"#\"\"\"-F$6#\"& \\&>F)F)\"$'GF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "%;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$\"$t\"#\"\"\"\"\"#F'*$\"$8\"F&F'" }}}{PARA 0 "" 0 "" {TEXT -1 8 "Maple's " }{TEXT 0 4 "sqrt" }{TEXT -1 185 " function automatically simplifies nested square roots of numbers . But if you had used fractional powers instead of the sqrt function i t would have been neccessary to simplify with the " }{TEXT 0 8 "simpli fy" }{TEXT -1 11 " procedure." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "(2*(19549)^(1/2)+286)^(1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# *$,&*&\"\"#\"\"\"\"&\\&>#F'F&F'\"$'GF'F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$ \"$t\"#\"\"\"\"\"#F'*$\"$8\"F&F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 282 3 "11." } {TEXT -1 21 " In Maple, transform " }{XPPEDIT 18 0 "1/(sqrt(3)+1)" "6# *&\"\"\"F$,&-%%sqrtG6#\"\"$F$F$F$!\"\"" }{TEXT -1 32 " into an express ion of the form " }{XPPEDIT 18 0 "a+b*sqrt(3)" "6#,&%\"aG\"\"\"*&%\"bG F%-%%sqrtG6#\"\"$F%F%" }{TEXT -1 24 ", with rational numbers " } {XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b" "6# %\"bG" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "re start;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "1/(sqrt(3)+1);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&*$\"\"$#F$\"\"#F$F$F$!\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "rationalize(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"\"\"\"\"#!\"\"*&F&F'\"\"$#F%F&F% " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 " " 0 "" {TEXT 283 3 "12." }{TEXT -1 5 " Let " }{XPPEDIT 18 0 "theta" "6 #%&thetaG" }{TEXT -1 29 " be a root of the polynomial " }{XPPEDIT 18 0 "theta^3-theta-1" "6#,(*$%&thetaG\"\"$\"\"\"F%!\"\"F'F(" }{TEXT -1 66 " and consider the extension of the field of rational numbers with \+ " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 42 ". So, we consider \+ expressions of the form " }{XPPEDIT 18 0 "a+b*theta+c*theta^2" "6#,(% \"aG\"\"\"*&%\"bGF%%&thetaGF%F%*&%\"cGF%*$F(\"\"#F%F%" }{TEXT -1 8 ", \+ where " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 " b" "6#%\"bG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 88 " are rational numbers, and in calculations with these expressio ns we apply the identity " }{XPPEDIT 18 0 "theta^3=theta+1" "6#/*$%&th etaG\"\"$,&F%\"\"\"F(F(" }{TEXT -1 23 ". Transform with Maple " } {XPPEDIT 18 0 "1/(theta^2+1)" "6#*&\"\"\"F$,&*$%&thetaG\"\"#F$F$F$!\" \"" }{TEXT -1 32 " into an expression of the form " }{XPPEDIT 18 0 "a+ b*theta+c*theta^2" "6#,(%\"aG\"\"\"*&%\"bGF%%&thetaGF%F%*&%\"cGF%*$F( \"\"#F%F%" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 23 " are rational numbers.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 43 "alias(theta=RootOf(theta^3-theta-1,theta));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%&thetaG" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "1/(theta^2+ 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&*$)%&thetaG\"\"#F$F $F$F$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(#\"\"%\"\"&\"\"\"*&#\"\"#F&F'*$) %&thetaGF*F'F'!\"\"*&#F'F&F'F-F'F'" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "sort(%, theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"#\"\"&\"\"\"*$)%&thetaGF&F( F(!\"\"*&#F(F'F(F+F(F(#\"\"%F'F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 284 3 "13." } {TEXT -1 5 " Let " }{XPPEDIT 18 0 "alpha=sqrt(2)" "6#/%&alphaG-%%sqrtG 6#\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "beta=sqrt(3)" "6#/%%betaG-%% sqrtG6#\"\"$" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "gamma=sqrt(5)" "6#/ %&gammaG-%%sqrtG6#\"\"&" }{TEXT -1 20 ". Use the procedure " }{TEXT 0 9 "Primfield" }{TEXT -1 32 " to compute a primitive element " } {XPPEDIT 18 0 "zeta" "6#%%zetaG" }{TEXT -1 25 " for the field extensio n " }{XPPEDIT 18 0 "Q(alpha, beta, gamma)" "6#-%\"QG6%%&alphaG%%betaG% &gammaG" }{TEXT -1 55 " and compare the result with the last example o f \2472.5.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "alpha := convert(sqrt(2),Roo tOf);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG-%'RootOfG6$,&*$)%#_ ZG\"\"#\"\"\"F-F,!\"\"/%&indexGF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "beta := convert(sqrt(3),RootOf);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG-%'RootOfG6$,&*$)%#_ZG\"\"#\"\"\"F-\"\"$!\"\"/% &indexGF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "unprotect(gamm a):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "gamma := convert(sqr t(5),RootOf);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&gammaG-%'RootOfG6$ ,&*$)%#_ZG\"\"#\"\"\"F-\"\"&!\"\"/%&indexGF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "evala(Primfield(\{alpha, beta, gamma\}));" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7$7#/-%'RootOfG6#,,*$)%#_ZG\"\")\"\"\" F.*&\"#SF.)F,\"\"'F.!\"\"*&\"$_$F.)F,\"\"%F.F.*&\"$g*F.)F,\"\"#F.F3\"$ w&F.,(-F'6$,&*$F:F.F.\"\"&F3/%&indexGF.F.-F'6$,&FAF.\"\"$F3FCF.-F'6$,& FAF.F;F3FCF.7%/F>,**&#\"#&*\"#OF.*$)F&FHF.F.F.*&#\"#(*\"$)GF.*$)F&FBF. F.F3*&#FBF " 0 "" {MPLTEXT 1 0 17 "op([1,1,1,1], %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*$)%#_ZG\"\")\"\"\"F(*&\"#SF()F&\"\"'F(!\" \"*&\"$_$F()F&\"\"%F(F(*&\"$g*F()F&\"\"#F(F-\"$w&F(" }}}{PARA 0 "" 0 " " {TEXT -1 62 "The primitive element is the root of the following pol ynomial" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(_Z=zeta, %); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*$)%%zetaG\"\")\"\"\"F(*&\"#SF() F&\"\"'F(!\"\"*&\"$_$F()F&\"\"%F(F(*&\"$g*F()F&\"\"#F(F-\"$w&F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 " " {TEXT 285 3 "14." }{TEXT -1 176 " Show that Maple knows that the exp onential power of a complex number can be written in terms of cosine a nd sine of the real and imaginary parts of that number. Also calculate " }{XPPEDIT 18 0 "exp(Pi*I/12)" "6#-%$expG6#*(%#PiG\"\"\"%\"IGF(\"#7! \"\"" }{TEXT -1 15 " in that form.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "exp(x+I*y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#,&%\"xG\"\"\" *&%\"yGF(^#F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalc(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$expG6#%\"xG\"\"\"-%$cosG6 #%\"yGF)F)*(F%F)-%$sinGF,F)^#F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "exp(Pi*I/12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$e xpG6#*&^##\"\"\"\"#7F)%#PiGF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalc(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$cosG6#,$*&\"#7! \"\"%#PiG\"\"\"F,F,*&-%$sinGF&F,^#F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "convert(%, radical);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(\"\"%!\"\"\"\"##\"\"\"F',&F)F)*$\"\"$F(F)F)F)*(^##F)F%F)F'F(,&F +F)F)F&F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 286 3 "15." }{TEXT -1 22 " Show with Maple tha t " }{XPPEDIT 18 0 "tanh(z/2) = (sinh(x)+I*sin(y))/(cosh(x)+cos(y));" "6#/-%%tanhG6#*&%\"zG\"\"\"\"\"#!\"\"*&,&-%%sinhG6#%\"xGF)*&%\"IGF)-%$ sinG6#%\"yGF)F)F),&-%%coshG6#F1F)-%$cosG6#F7F)F+" }{TEXT -1 25 ", for \+ any complex number " }{XPPEDIT 18 0 "z=x+y*I" "6#/%\"zG,&%\"xG\"\"\"*& %\"yGF'%\"IGF'F'" }{TEXT -1 11 " with real " }{XPPEDIT 18 0 "x" "6#%\" xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 2 ".\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "tanh(z/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%tanhG6#,$*&\"\"#!\"\"%\"zG\"\"\"F+" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " expand(%, sincos);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&-%%coshG6#% \"zG\"\"\"F)!\"\"F)-%%sinhGF'F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(z=x+I*y, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&-%%co shG6#,&%\"xG\"\"\"*&%\"yGF*^#F*F*F*F*F*!\"\"F*-%%sinhGF'F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(,&*&-%%sinhG6#%\"xG\"\"\"-%$cosG6#%\"yGF+F+*(-%%cos hGF)F+-%$sinGF.F+^#F+F+F+!\"\"F1F+F,F+F+**F%F6F'F+F3F+F5F+F+*&F+F+F%F6 F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalc(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,***-%%sinhG6#%\"xG\"\"\"-%$cosG6#%\"yG\"\"#,&* &)F%F.F))F*F.F)F)*&)-%%coshGF'F.F))-%$sinGF,F.F)F)!\"\"F5F)F)**F5F)F8F .F/F:F%F)F)*(F%F)F*F)F/F:F:*&,(**F5F.F8F)F/F:F*F)F:**F%F.F*F)F/F:F8F)F )*(F5F)F8F)F/F:F)F)^#F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&-%%sinhG6#%\"x G\"\"\"*&-%$sinG6#%\"yGF)^#F)F)F)F),&-%$cosGF-F)-%%coshGF'F)!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }