{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 "Grey Emphasis" -1 257 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Green Emphasis" -1 262 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 264 "Times" 1 12 255 0 0 1 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 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 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 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 1 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 27 "Multiplication of fractions" }} {PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }} {PARA 0 "" 0 "" {TEXT -1 19 "Version: 27.3.2007" }}{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 29 "load procedures for fractions" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 257 11 "fractions.m" }{TEXT -1 98 ". \+ contains code for procedures which can be used to illustrate arithmeti c operations on fractions." }}{PARA 0 "" 0 "" {TEXT -1 128 "This file \+ can be read into a Maple session by a command similar to the one that \+ follows, where the file path gives its location." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 37 "read \"K:\\\\Maple/procdrs/fractions.m\";" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 51 "load extra colours and utility routines for colours" }} {PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 257 9 "colours. m" }{TEXT -1 49 ". contains code for utility routines for colours." }} {PARA 0 "" 0 "" {TEXT -1 128 "This file can be read into a Maple sessi on by a command similar to the one that follows, where the file path g ives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "read \+ \"K:\\\\Maple/procdrs/colours.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "A procedure for multiplication of fractions: " }{TEXT 0 13 "MultFractions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "MultFractio ns: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT -1 28 " MultFractions( f1, f2 ) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 68 " f1, f2 - two positiv e fractions given in the form a/b or a " }{TEXT 265 2 ".." }{TEXT -1 4 " b. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 14 "The procedure " }{TEXT 0 13 "MultFractions" }{TEXT -1 103 " adds two fractions and provides information concerning how the a ddition could be perfromed \"by hand\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 8 "Options:" }{TEXT -1 1 "\n" }} {PARA 0 "" 0 "" {TEXT -1 10 "step=1,2,3" }}{PARA 0 "" 0 "" {TEXT -1 101 "The calculation can be interrupted at different stages by includi ng the option \"step=1\" or \"step=2\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "info=true/false or info=0,1,2 " }} {PARA 0 "" 0 "" {TEXT -1 62 "The level of information given is control led with this option." }}{PARA 0 "" 0 "" {TEXT -1 73 "No information i s given with \"info=0\", which is the same as \"info=false\"." }} {PARA 0 "" 0 "" {TEXT -1 82 "Full details are given with the option \" info=2\", which is the same as \"info=true\"." }}{PARA 0 "" 0 "" {TEXT -1 130 "With the option \"info=1\" details concerning the determ ination of the lowest common multiple (LCM) of the denominators are om itted." }}{PARA 0 "" 0 "" {TEXT -1 28 "The default is \"info=true\". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "outpu t=mixed/MIXED/improper/IMPROPER" }}{PARA 0 "" 0 "" {TEXT -1 85 "This o ption controls the form of the value returned for the sum of the two f ractions." }}{PARA 0 "" 0 "" {TEXT -1 105 "In the case \"output=mixed \", which is the default, if the sum is greater than 1, it is returned as a pair " }{XPPEDIT 18 0 "a,b" "6$%\"aG%\"bG" }{TEXT -1 7 " where \+ " }{TEXT 266 1 "a" }{TEXT -1 55 " is an integer giving the integer par t of the sum, and " }{TEXT 267 1 "b" }{TEXT -1 66 " is a fraction less than 1 giving the fractional part of the sum. " }}{PARA 0 "" 0 "" {TEXT -1 97 "In the case \"output=improper\", the sum is given as an i mproper fraction if it is greater than 1. " }}{PARA 0 "" 0 "" {TEXT -1 46 "The upper-case modifications are also allowed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "draw=true/false" }} {PARA 0 "" 0 "" {TEXT -1 73 "This option allows a pictorial illustrati on of the addition to be drawn. " }}{PARA 0 "" 0 "" {TEXT -1 27 "The d efault is \"draw=true\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 20 "divisions=true/false" }}{PARA 0 "" 0 "" {TEXT -1 144 "This option controls the drawing of vertical subdividing lines in the diagramatic representation of the two fractions to be added and t heir sum." }}{PARA 0 "" 0 "" {TEXT -1 209 "With the option \"divisions =false\" each of the two fractions to be added is represented by a rec tangular region as the corresponding fraction of the area of a larger \+ rectangle, with no extra subdividing lines. " }}{PARA 0 "" 0 "" {TEXT -1 316 "With the option \"divisions=true\" each rectangle representing the two fractions to be added is subdivided into a number of smaller \+ rectangles corresponding the denominator of the fraction, and the sum \+ is subdivided into a number of smaller rectangles corresponding to the LCM of the denominators of the two fractions." }}{PARA 0 "" 0 "" {TEXT -1 112 "The default is \"divisions=true\", when the LCM of the d enominators is less than 82; otherwise \"divisions=false\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "colour/color=[c 1,c2] " }}{PARA 0 "" 0 "" {TEXT -1 98 "The colours of the rectangles u sed to illustrate the two fractions is controlled with this option." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 227 "light=t \nWhen a particular colour is used for the area which represents a fra ction, a lightened form of this colour is used to represent the residu al area. This option controls the lightness, the default value being \+ \"light=2/3\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 16 "How to activate:" } {TEXT 256 1 "\n" }{TEXT -1 154 "To make the procedure active open the \+ subsection, place the cursor anywhere after the prompt [ > and press \+ [Enter].\nYou can then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "MultFractions: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14926 "MultFract ions := proc(f1,f2)\n local n1,d1,n2,n3,n4,d2,d3,d4,stp,Options,t1,t 2,\n v,clr,clr1,clr2,clr3,divs,e1,e2,e3,e4,\n p1,p2,p3,p4,p5 ,p6,p7,p8,p9,c1,c2,c3,drw,nd1,nd2,nd3,nd4,\n sep1,sep2,sep3,sep4, txt1,txt2,spc,i,v1,v2,v3,d,u1,u2,u3,\n prntflg,outpt,dots,j,k,nm1 ,nm2,dm1,dm2,mthd,intpart,rmdr,\n np1,dp1,np2,dp2,np3,dp3,lght;\n \n if not assigned(`type/color`) or not assigned(lighten) then\n \+ error \"the procedure, %1, requires the type 'color' to be defined a nd also uses the procedure 'lighten' among the utility routines which \+ enhance Maple's colours\",'procname';\n end if;\n\n if nargs<2 or \+ not type(f1,rational) or not type(f2,rational) then\n error \"usu al syntax: '%1(a/b,c/d)', where a,b,c and d are integers (whole number s)\",procname;\n end if;\n if type(f1,integer) then\n error \+ \"the 1st argument must not be an integer (whole number)\"\n end if; \n if type(f2,integer) then\n error \"the 2nd argument must not be an integer (whole number)\"\n end if; \n if f1<=0 then \n \+ error \"the 1st fraction,%1, must be positive\",f1;\n end if;\n i f f2<=0 then \n error \"the 2nd fraction,%1, must be positive\",f 2;\n end if;\n\n n1 := op(1,f1);\n d1 := op(2,f1);\n n2 := op( 1,f2);\n d2 := op(2,f2);\n n3 := n1*n2;\n d3 := d1*d2;\n v := \+ f1*f2;\n n4 := op(1,v);\n d4 := op(2,v);\n k := iquo(d3,d4);\n \+ \n stp := 4;\n if true then\n if d3<=121 then\n divs \+ := 2\n elif d4<=121 then\n divs := 1\n else\n \+ divs := 0\n end if;\n else\n if d4<=81 then\n di vs := 2;\n elif max(d1,d2)<=81 then\n divs := 1;\n e lse\n divs := 0;\n end if;\n end if;\n if f1<1 and f2 <1 then mthd := 3 else mthd := 2 end if;\n \n clr := [red,blue,COL OR(RGB,0,.93,0),COLOR(RGB,.75,.65,1)];\n if assigned(_FractionProcCo lour) then\n for j to min(4,nops(_FractionProcColour)) do\n \+ clr[j] := _FractionProcColour[j];\n end do;\n end if;\n if assigned(_FractionProcOutput) then outpt := _FractionProcOutput \n \+ else outpt := mixed end if;\n if assigned(_FractionProcDraw) then dr w := _FractionProcDraw \n else drw := true end if;\n if assigned(_ FractionProcInfo) then prntflg := _FractionProcInfo \n else prntflg \+ := 2 end if;\n lght := evalf(2/3);\n if nargs>=2 then\n Optio ns:=[args[3..nargs]];\n if not type(Options,list(equation)) then \n error \"each optional argument must be an equation\"\n \+ end if;\n if hasoption(Options,step,'stp','Options') then\n \+ if not type(stp,posint) then \n error \"\\\"steps\\\" m ust be a positive integer\"\n end if;\n end if;\n if hasoption(Options,method,'mthd','Options') then\n if not memb er(mthd,\{1,2,3,cancel_before,CANCEL_BEFORE,CancelBefore,\n \+ cancel_after,CANCEL_AFTER,CancelAfter,both,BOTH,Both\}) then \n \+ error \"\\\"method\\\" must be 1,2, 'cancel_before','CANCEL_B EFORE','CancelBefore',\n 'cancel_after','CANCEL_AFTER' o r 'CancelAfter'\"\n end if;\n if member(mthd,\{cancel_ before,CANCEL_BEFORE,CancelBefore\}) then\n mthd := 2;\n \+ end if;\n if member(mthd,\{cancel_after,CANCEL_AFTER,Can celAfter\}) then \n mthd := 1;\n end if;\n \+ if member(mthd,\{both,BOTH,Both\}) then mthd := 3 end if;\n end i f;\n if hasoption(Options,output,'outpt','Options') then\n \+ if not member(outpt,\{mixed,MIXED,improper,IMPROPER\}) then \n \+ error \"\\\"output\\\" must be 'mixed','MIXED','improper' or 'IM PROPER'\"\n end if;\n if outpt=MIXED then outpt := mix ed end if;\n if outpt=IMPROPER then outpt := improper end if; \n end if;\n if hasoption(Options,info,'prntflg','Options') \+ then\n if not member(prntflg,\{true,false,0,1,2\}) then\n \+ error \"\\\"info\\\" must be false=0, 1 or true=2\"\n e nd if;\n if prntflg=false then prntflg := 0\n elif prn tflg=true then prntflg := 2 end if; \n end if;\n if hasoptio n(Options,draw,'drw','Options') then\n if drw<>true then drw : = false end if;\n end if;\n if hasoption(Options,divisions,' divs','Options') then\n if not member(divs,\{true,false,0,1,2 \}) then\n error \"\\\"info\\\" must be false=0, 1 or true= 2\"\n end if;\n if divs=false then divs := 0\n \+ elif divs=true then divs := 2 end if; \n end if;\n if hasop tion(Options,color,'tp','Options') or \n hasoption(Options,col our,'tp','Options') then\n if type(tp,list(color)) then\n \+ for j to min(nops(clr),nops(tp)) do clr[j] := tp[j] end do;\n \+ elif type(tp,color) then\n clr[1] := tp;\n e lse\n error \"color option must be colour or a list of colo urs\"\n end if;\n end if;\n if hasoption(Options,lig ht,'lght','Options') then\n lght := evalf(lght);\n if \+ lght>1 or lght<0 then\n error \"\\\"light\\\" must evaluate to a floating point number between 0 and 1\"\n end if;\n \+ end if;\n if nops(Options)>0 then\n error \"%1 is not a \+ valid option for %2\",op(1,Options),procname;\n end if;\n end i f;\n\n if member(mthd,\{1,3\}) and prntflg>0 then\n print(``); \n if k>1 then\n print(`METHOD I .. cancelling after mult iplication`);\n print(``);\n print(``(n1/d1)*`.`*``(n2 /d2)=``(n3)/``(d3));\n if prntflg>1 then \n \+ np3 := ifactor(n3);\n dp3 := ifactor(d3);\n if prntflg>1 then\n if op(1,dp3)<>d3 or op(1,np3)<>n3 then print(``) end if;\n if op(1,dp3)<>d3 then\n \+ print(`The numerator `||n3||` has the prime factorisation .. `,n p3)\n end if;\n if op(1,np3)<>n3 then\n \+ print(`The denominator `||d3||` has the prime factorisa tion .. `,dp3);\n end if;\n end if;\n \+ end if;\n print(``);\n print(`The numerator `||n3||` \+ and denominator `||d3||` can be divided by the common factor `||k||` s o that ..`);\n print(``);\n print(``(n1/d1)*`.`*``(n2/ d2)=v);\n else\n if f1<>f2 then\n print(``(n1/ d1)*`.`*``(n2/d2)=v);\n else\n print(``(n1/d1)^2=v) ;\n end if;\n end if;\n print(``);\n if stp=1 t hen return NULL end if;\n end if;\n \n \n if drw and member(mthd ,\{1,3\}) then\n if f1>=1 or f2>=1 then\n WARNING(`cannot illustrate this method when either fraction is greater than 1`);\n \+ goto(1111);\n end if;\n for i to 3 do\n if not type(clr[i],color) then\n error \"incorrect colour data fo r %-1 colour\",i;\n end if;\n end do;\n clr1 := ligh ten(clr[1],lght);\n clr2 := lighten(clr[2],lght);\n clr3 := \+ lighten(clr[3],lght);\n\n e1 := evalf(1/d1);\n c1 := n1*e1; \n e2 := evalf(f1/d2);\n c2 := n2*e2;\n e3 := evalf(1/d 3); \n c3 := n3*e3;\n p1 := plots[polygonplot]([[0,3],[c1,3] ,[c1,4],[0,4]],\n style=PATCHNOGRID,color=clr[1]);\n p2 \+ := plots[polygonplot]([[c1,3],[1,3],[1,4],[c1,4]],\n style=PA TCHNOGRID,color=clr1);\n p3 := plots[polygonplot]([[0,1.5],[c2,1. 5],[c2,2.5],[0,2.5]],\n style=PATCHNOGRID,color=clr[2]);\n \+ p4 := plots[polygonplot]([[c2,1.5],[c1,1.5],[c1,2.5],[c2,2.5]],\n \+ style=PATCHNOGRID,color=clr2);\n p5 := plots[polygonplot] ([[0,0],[c3,0],[c3,1],[0,1]],\n style=PATCHNOGRID,color=clr[3 ]);\n p6 := plots[polygonplot]([[c3,0],[1,0],[1,1],[c3,1]],\n \+ style=PATCHNOGRID,color=clr3);\n\n v1 := c1/2;\n nd1 : = length(d1);\n sep1 := cat(`_`$nd1);\n v2 := c2/2;\n n d2 := length(d2);\n sep2 := cat(`_`$nd2);\n\n txt1 := plots[ textplot]([[v1,4.35,`n1`],[v1,4.15,`d1`],[v1,4.31,sep1],\n \+ [v2,2.85,`n2`],[v2,2.65,`d2`],[v2,2.81,sep2]],\n font=[HEL VETICA,9],color=black,linestyle=1);\n\n v3 := c3/2;\n nd3 := length(d3);\n sep3 := cat(`_`$nd3);\n\n if k>1 then \n \+ nd4 := length(d4);\n sep4 := cat(`_`$nd4);\n spc : = (nd3+nd4)/100;\n d := max(spc-v3,0);\n u1 := v3-spc+ d;\n u2 := v3+d;\n u3 := v3+spc+d;\n txt2 := p lots[textplot]([[u1,1.35,`n3`],\n [u1,1.15,`d3`],[u1,1.31, sep3],\n [u3,1.35,`n4`],[u3,1.15,`d4`],\n [u3, 1.31,sep4],[u2,1.25,`=`]],\n font=[HELVETICA,9],color=black ,linestyle=1);\n else\n txt2 := plots[textplot]([[v3,1.35 ,`n3`],\n [v3,1.15,`d3`],[v3,1.31,sep3]],\n fon t=[HELVETICA,9],color=black,linestyle=1);\n end if;\n\n if d ivs>0 then\n e4 := evalf(1/d4);\n p7 := plot([seq([[i* e1,3],[i*e1,4]],i=0..d1),\n [[0,3],[1,3],[1,4],[0,4],[0, 3]],\n seq([[i*e2,1.5],[i*e2,2.5]],i=0..d2),\n \+ [[0,1.5],[c1,1.5],[c1,2.5],[0,2.5],[0,1.5]],\n seq ([[i*e4,0],[i*e4,1]],i=0..d4),\n [[0,0],[1,0]],[[0,1],[1, 1]]],color=black,linestyle=1);\n if k>1 and divs>1 then\n \+ dots := [];\n for j from 1 to d3-1 do\n \+ if irem(j,k)<>0 then\n dots := [op(dots),[[j*e3,0], [j*e3,1]]];\n end if;\n end do;\n \+ p8 := plot(dots,color=black,linestyle=2); \n else\n \+ p8 := NULL;\n end if;\n else \n p7 := plot([[[c1 ,3],[c1,4]],[[0,3],[1,3],[1,4],[0,4],[0,3]],\n [[c2,1.5], [c2,2.5]],[[0,1.5],[c1,1.5],[c1,2.5],[0,2.5],[0,1.5]],\n \+ [[c3,0],[c3,1]],[[0,0],[1,0],[1,1],[0,1],[0,0]]],\n color=b lack,linestyle=1);\n p8 := NULL;\n end if;\n p9 := p lot([[[c2,3],[c2,2.5]],[[c2-0.01,2.93],[c2,3],[c2+0.01,2.93]]],\n \+ linestyle=[2,1],color=COLOR(RGB,.3,.3,.3));\n print(`` );\n print(plots[display]([txt1,txt2,p1,p2,p3,p4,p5,p6,p7,p8,p9], \n axes=none));\n print(``);\n if stp=2 then \+ return NULL end if;\n end if;\n 1111:\n\n if mthd>1 then\n \+ t1 := igcd(n1,d2);\n t2 := igcd(n2,d1);\n if t1<>1 or t2<>1 \+ then\n if prntflg>0 then\n print(``);\n \+ print(`METHOD II .. cancelling before multiplication`);\n p rint(``);\n print(`Performing the multiplication .. `,``(n1 /d1)*`.`*``(n2/d2));\n end if;\n else\n nm1 := n1 ;\n dm1 := d1;\n nm2 := n2;\n dm2 := d2;\n \+ if prntflg>0 then\n print(``);\n print(`No \+ cancellation is possible .. `,``(n1/d1)*`.`*``(n2/d2)=v);\n \+ print(``); \n end if;\n end if;\n\n if t1<>1 then \n np1 := ifactor(n1);\n dp2 := ifactor(d2);\n \+ if prntflg>1 then\n if op(1,np1)<>n1 or op(1,dp2)<>d2 then print(``) end if;\n if op(1,np1)<>n1 then\n \+ print(`The 1st numerator `||n1||` has the prime factorisation .. `,np1 )\n end if;\n if op(1,dp2)<>d2 then\n \+ print(`The 2nd denominator `||d2||` has the prime factorisation .. `,dp2);\n end if;\n end if;\n nm1 := iquo( n1,t1);\n dm2 := iquo(d2,t1);\n if prntflg>0 then\n \+ print(``);\n print(`The 1st numerator `||n1||` and the 2nd denominator `||d2||` have the common factor `||t1||`.`);\n \+ print(`Dividing the numerator of the 1st fraction and the den ominator of the 2nd fraction by `||t1||` gives ..`);\n prin t(``(n1/d1)*`.`*``(n2/d2)=``(nm1/d1)*`.`*``(n2/dm2));\n end if ;\n n1 := nm1;\n d2 := dm2;\n else\n nm1 \+ := n1;\n dm2 := d2;\n end if;\n\n if t2<>1 then\n \+ dp1 := ifactor(d1);\n np2 := ifactor(n2);\n if p rntflg>1 then\n if op(1,dp1)<>d1 or op(1,np2)<>n2 then prin t(``) end if;\n if op(1,dp1)<>d1 then\n print (`The 1st denominator `||d1||` has the prime factorisation .. `,dp1)\n end if;\n if op(1,np2)<>n2 then\n \+ print(`The 2nd numerator `||n2||` has the prime factorisation .. `,np 2);\n end if;\n end if;\n dm1 := iquo(d1,t2 );\n nm2 := iquo(n2,t2);\n if prntflg>0 then\n \+ print(``);\n print(`The 1st denominator `||d1||` and th e 2nd numerator `||n2||` have the common factor `||t2||`.`);\n \+ print(`Dividing the denominator of the 1st fraction and the numera tor of the 2nd fraction by `||t2||` gives ..`);\n print(``( n1/d1)*`.`*``(n2/d2)=``(n1/dm1)*`.`*``(nm2/d2));\n end if;\n \+ else\n dm1 := d1;\n nm2 := n2;\n end if;\n\n \+ if t1<>1 or t2<>1 then\n if prntflg>0 then\n p rint(` `=v);\n print(``);\n end if;\n end if;\n n1 := op(1,f1);\n d2 := op(2,f2);\n if mthd>1 an d stp=1 or stp=3 then return NULL end if;\n end if;\n\n if drw and mthd>1 and d4<=149 and dm1<>1 and dm2<>1 then\n if prntflg>0 the n\n print(`--------------`);\n print(`The product`,``( nm1/dm1)*`.`*``(nm2/dm2)=v,` can be found by first obtaining the produ ct`,``(1/dm1)*`.`*``(1/dm2)=1/d4,``);\n print(`and then multip lying by`,``(nm1)*`.`*``(nm2)=n4); \nprint(`The multiplication of the fractions`,1/dm1,` and`,1/dm2,` can be illustrated by ..`);\n \+ end if;\n if mthd=2 then j := 1 else j := 4 end if;\n i f not type(clr[j],color) then\n error \"incorrect colour data \+ for %-j colour\",j\n end if;\n p1 := plots[polygonplot]([[0, 0],[1,0],[1,1],[0,1]],\n style=PATCHNOGRID,color=clr[j]);\n\n nd1 := length(dm1);\n sep1 := cat(`_`$nd1);\n nd2 := l ength(dm2);\n sep2 := cat(`_`$nd2);\n nd4 := length(d4);\n \+ sep4 := cat(`_`$nd4);\n v2 := .5;\n spc := (nd1+nd2)/100 ;\n u1 := v2-spc;\n v1 := u1-spc;\n spc := (nd2+nd4)/10 0;\n u2 := v2+spc;\n v3 := u2+spc;\n txt1 := plots[text plot]([[v1,1.35,`1`],[v1,1.15,`dm1`],\n [v1,1.31,sep1],[u1,1.2 5,`.`],[v2,1.35,`1`],\n [v2,1.15,`dm2`],[v2,1.31,sep2],[u2,1.2 5,`=`],\n [v3,1.35,`1`],[v3,1.15,`d4`],[v3,1.31,sep4]],\n \+ font=[HELVETICA,9],color=black,linestyle=1);\n e1 := evalf (1/dm1);\n p2 := plot([seq([[i*e1,0],[i*e1,1]],i=0..dm1),\n \+ [[0,0],[1,0]],[[0,1],[1,1]]],color=black,linestyle=1);\n dots \+ := [];\n e4 := evalf(1/d4);\n for j from 1 to d4-1 do\n \+ if irem(j,dm2)<>0 then\n dots := [op(dots),[[j*e4,0],[j *e4,1]]];\n end if;\n end do;\n p3 := plot(dots,colo r=black,linestyle=2); \n\n print(``);\n print(plots[display] ([txt1,p1,p2,p3],axes=none,view=[0..1,-0.7..1.7]));\n print(``); \n end if;\n\n if outpt=mixed then\n intpart := iquo(n3,d3,'r mdr');\n if intpart>0 and rmdr>0 then\n if prntflg>0 then \n print(`Converting`,v,` from an improper fraction to a mi xed number gives ..`);\n end if;\n intpart,rmdr/d3;\n \+ else\n v;\n end if;\n else\n v\n end if; \n end proc: # of MultFractions" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 64 "convert(..,RGB), convert(..,HSV),convert(..,HLS): implementation" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6790 "`convert/RGB` := proc(cc)\n local h,s,v,synOK,clr,i,f,p,q,t,l, r,g,b;\n\n if type(cc,\{list,set\}) then \n return map('procnam e',cc)\n end if;\n\n clr := cc;\n\n if type(clr,function) and me mber(op(0,clr),\{'COLOR','COLOUR'\}) then\n if member(op(1,clr), \{'RGB','HSV','HLS'\}) and nops(clr)=4 \n and type([op(2..4,clr )],list(numeric)) then synOK := true\n elif op(1,clr)='HUE' and n ops(clr)=2 \n and type(op(2,clr),numeric) then synOK := true\n \+ else synOK := false end if;\n else\n synOK := false\n en d if; \n\n if synOK then\n if op(1,clr)='RGB' then\n\011 \+ r := max(min(evalf(op(2,clr)),1.),0.);\n\011 g := max(min(eval f(op(3,clr)),1.),0.);\n \011 b := max(min(evalf(op(4,clr)),1.),0 .);\n return COLOR(RGB,r,g,b);\n elif op(1,clr)='HUE' the n\n h := evalf(op(2,clr));\n h := h-floor(h);\n \+ clr := COLOR(HSV,h,0.9,1.0)\n end if;\n end if;\n\n if not \+ synOK then\n try\n clr := `plot/color`(clr);\n re turn clr; \n catch: error \"unable to convert %1 to an RG B colour\",clr;\n end try;\n end if;\n\n Digits := 8;\n if \+ op(1,clr)='HSV' then\n\011 h := evalf(op(2,clr));\n h := h-fl oor(h);\n\011 s := max(min(evalf(op(3,clr)),1.),0.);\n \011 v : = max(min(evalf(op(4,clr)),1.),0.);\n h := h*6;\n i := floor (h);\n\011 \011\011f := h-i;\n\011 \011 p := v*(1-s);\n\011 \011 q := v*(1-s*f) ;\n\011 \011 t := v*(1-s*(1-f));\n \n \011\011 if i=0 then\n COLOR('RGB',v,t,p)\n\011\011\011 elif i=1 the n\n\011\011\011 COLOR('RGB',q,v,p)\n elif i=2 then\n \+ COLOR('RGB',p,v,t)\n elif i=3 then\n COLOR('RGB',p,q,v) \n elif i=4 then\n COLOR('RGB',t,p,v)\n\011\011\011 els e # i=5 then\n COLOR('RGB',v,p,q)\n end if;\n else # op (1,clr)='HLS'\n\011 h := evalf(op(2,clr));\n h := h-floor(h); \n\011 l := max(min(evalf(op(3,clr)),1.),0.);\n \011 s := max(m in(evalf(op(4,clr)),1.),0.);\n h := h*6;\n p := `if`(l<=.5,l +l*s,l+s-l*s);\n q := 2*l-p;\n t := p-q;\n if s=0 then \n COLOR('RGB',l,l,l);\n else\n if h<1 then\n \+ COLOR('RGB',p,q+t*h,q)\n elif h<2 then\n CO LOR('RGB',q+t*(2-h),p,q)\n elif h<3 then\n COLOR('R GB',q,p,q+t*(h-2))\n elif h<4 then \n COLOR('RGB',q ,q+t*(4-h),p)\n elif h<5 then\n COLOR('RGB',q+t*(h- 4),q,p)\n else\n COLOR('RGB',p,q,q+t*(6-h))\n \+ end if;\n end if;\n end if;\nend proc: # `convert/RGB`\n\n`c onvert/HSV` := proc(cc)\n local clr,synOK,r,g,b,h,s,v,d,l,u,t;\n\n \+ if type(cc,\{list,set\}) then \n return map('procname',cc)\n e nd if;\n\n clr := cc;\n if type(clr,function) and member(op(0,clr) ,\{'COLOR','COLOUR'\}) then\n if member(op(1,clr),\{'RGB','HSV',' HLS'\}) and nops(clr)=4 \n and type([op(2..4,clr)],list(numeric )) then synOK := true\n elif op(1,clr)='HUE' and nops(clr)=2 \n \+ and type(op(2,clr),numeric) then synOK := true\n else synOK := false end if;\n else\n synOK := false\n end if;\n\n if \+ synOK then\n if op(1,clr)='HSV' then\n\011 h := evalf(op(2 ,clr));\n h := h-floor(h);\n\011 s := max(min(evalf(op( 3,clr)),1.),0.);\n \011 v := max(min(evalf(op(4,clr)),1.),0.);\n return COLOR('HSV',h,s,v);\n elif op(1,clr)='HUE' then\n \011 h := evalf(op(2,clr));\n h := h-floor(h);\n \+ return COLOR('HSV',h,0.9,1.0);\n end if;\n end if;\n\n if n ot synOK then\n try clr := `plot/color`(clr);\n catch: error \"unable to convert %1 to an HSV colour\",clr;\n end try;\n en d if;\n\011\n Digits := 8;\n if op(1,clr)='RGB' then\n\011 r : = max(min(evalf(op(2,clr)),1.),0.);\n\011 g := max(min(evalf(op(3, clr)),1.),0.);\n \011 b := max(min(evalf(op(4,clr)),1.),0.);\n\n \+ v := max(r,g,b); \n d := v-min(r,g,b);\n if d=0 then # r = g = b\n return COLOR('HSV',evalf(2/3),0.,r);\n end i f;\n\n s := d/v; \n\011 if v=r then\n\011\011 h := (g -b)/d;\011# between yellow & magenta\n\011 elif v=g then\n\011\011 h := 2+(b-r)/d;\011# between cyan & yellow\n\011 else\n\011 \011 h := 4+(r-g)/d;\011# between magenta & cyan\n end if; \n h := h/6;\n\011 if h<0 then h := h+1. end if;\n COLOR ('HSV',h,s,v);\n else # op(1,clr)=HLS\n\011 h := evalf(op(2,clr) );\n h := h-floor(h);\n\011 l := max(min(evalf(op(3,clr)),1.) ,0.);\n \011 s := max(min(evalf(op(4,clr)),1.),0.);\n u := 2*l ;\n d := `if`(u<1,u*s,(2-u)*s);\n t := u+d;\n v := t/2; \n s := `if`(v<>0,d/v,0.);\n COLOR('HSV',h,s,v);\n end if; \nend proc: # `convert/HSV`\n\n`convert/HLS` := proc(cc)\n local clr ,synOK,r,g,b,h,s,l,d,u,v,m;\n\n if type(cc,\{list,set\}) then \n \+ return map('procname',cc)\n end if;\n\n clr := cc;\n if type(c lr,function) and member(op(0,clr),\{'COLOR','COLOUR'\}) then\n if member(op(1,clr),\{'RGB','HSV','HLS'\}) and nops(clr)=4 \n and type([op(2..4,clr)],list(numeric)) then synOK := true\n elif op( 1,clr)='HUE' and nops(clr)=2 \n and type(op(2,clr),numeric) the n synOK := true\n else synOK := false end if;\n else\n syn OK := false\n end if;\n\n if synOK then\n if op(1,clr)='HLS' \+ then\n\011 h := evalf(op(2,clr));\n h := h-floor(h);\n \011 l := max(min(evalf(op(3,clr)),1.),0.);\n \011 s := m ax(min(evalf(op(4,clr)),1.),0.);\n return COLOR(HSV,h,l,s);\n \+ elif op(1,clr)='HUE' then\n\011 h := evalf(op(2,clr));\n \+ h := h-floor(h);\n return COLOR('HLS',h,0.55,1.0);\n \+ end if;\n end if;\n\n if not synOK then\n try clr := `plot /color`(clr);\n catch: error \"unable to convert %1 to an HLS col our\",clr;\n end try;\n end if;\n\011\n Digits := 8;\n if o p(1,clr)='RGB' then\n\011 r := max(min(evalf(op(2,clr)),1.),0.);\n \011 g := max(min(evalf(op(3,clr)),1.),0.);\n \011 b := max(min (evalf(op(4,clr)),1.),0.);\n\n # compute the lightness\n v : = max(r,g,b);\n m := min(r,g,b);\n u := v+m;\n d := v-m ;\n l := u/2;\n\n if d=0 then # r = g = b\n return COLOR('HLS',evalf(2/3),l,0.);\n end if;\n\n # compute the H LS saturation\n s := `if`(l<=.5,d/u,d/(2-u));\n\n # compute \+ the hue\n\011 if v=r then\n\011\011 h := (g-b)/d;\011# betwe en yellow & magenta\n\011 elif v=g then\n\011\011 h := 2+(b- r)/d;\011# between cyan & yellow\n\011 else\n\011\011 h := 4 +(r-g)/d;\011# between magenta & cyan\n end if;\n h := h/6; \n if h<0 then h := h+1. end if;\n COLOR('HLS',h,l,s);\n e lse # op(1,clr)=HSV\n\011 h := evalf(op(2,clr));\n h := h-flo or(h);\n\011 s := max(min(evalf(op(3,clr)),1.),0.);\n \011 v := max(min(evalf(op(4,clr)),1.),0.);\n d := s*v;\n m := v-d;\n u := v+m;\n l := u/2;\n\n # compute the HLS saturation \n if d=0 then\n s := 0.\n else\n s := `if`( l<=.5,d/u,d/(2-u));\n end if;\n COLOR('HLS',h,l,s); \n end if;\nend proc: # `convert/HLS`" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "ShowColours: implementation" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2853 "ShowColours := proc(ff)\n local i,clrs,clrnames,cc,ci,t,h,pp,r ect,txt,start,finish,\n jF,jS,jM,n,c1,k,j,under,over,ord;\n\n c lrnames := map(op,[indices(`plot/colortable`)]);\n cc := [];\n for i to nops(clrnames) do\n ci := clrnames[i];\n if not String Tools[IsUpper](ci) then\n t := COLOR(RGB,op(`plot/colortable` [ci]));\n h := convert(t,HSV);\n cc := [op(cc),[op(2 ..4,h),ci,t]]\n end if;\n end do;\n ord := proc(_u,_v)\n \+ if op(1,_u)1 do\n jM := trunc((jF+jS)/2); \n if not ord(h,cc[jM]) then jF := jM \n else jS := jM end if;\n end do;\n end if;\n\n if nargs>=2 a nd type(args[2],posint) then \n k := iquo(min(args[2],n),2);\n else\n k := 5;\n end if;\n under := min(n+jF-k ,n);\n start := max(jF-k,1);\n finish := min(jF+k,n);\n \+ over := max(jF+k-n,1);\n end if;\n\n pp := NULL;\n j := 1;\n \+ if under1 then\n for i from 1 to ove r do\n rect := plots[polygonplot]([[0,1-j],[1,1-j],[1,-j],[0,- j]],\n style=patchnogrid,color=op(4,cc[i]));\n tx t := plots[textplot]([1.2,0.5-j,op(4,cc[i])],color=black,align=RIGHT); \n pp := pp,plots[display]([rect,txt]);\n j := j+1;\n \+ end do;\n end if;\n plots[display]([pp],axes=NONE);\nend proc : # ShowColours" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "lighten, dar ken: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2775 "lighten := proc(cc::\{color,list(color),s et(color)\})\n local c,r,c2,drw,optn,p1,p2;\n\n if type(cc,\{list, set\}) then \n return map('procname',cc,args[2..nargs])\n end i f;\n\n if op(1,cc)='HLS' then\n c := cc;\n else\n c := c onvert(cc,'HLS'); \n end if; \n\n Digits := 8;\n if nargs>1 \+ and type(args[2],numeric) then\n r := evalf(args[2]);\n if r >1 or r<0 then\n error \"expecting the 2nd argument to evaluat e to a floating point number between 0 and 1, but received %1\",args[2 ];\n end if;\n else\n r := evalf(1/3);\n end if;\n\n i f nargs=2 and not type(args[2],numeric) then\n optn := args[2]\n \+ elif nargs=3 then\n optn := args[3]\n end if;\n\n drw := fa lse;\n if assigned(optn) then\n if type(optn,`=`) then \n \+ if op(1,optn)=draw then drw := op(2,optn) end if;\n if drw< >true then drw := false end if;\n elif optn=draw then drw := true ;\n end if;\n end if; \n \n # increase the 'L' component t owards 1\n c2 := COLOR(HLS,op(2,c),r+(1-r)*op(3,c),op(4,c));\n if \+ drw then\n p1 := plots[polygonplot]([[0,0],[1,0],[1,1],[0,1]],col or=c);\n p2 := plots[polygonplot]([[0,0],[1,0],[1,-1],[0,-1]],col or=c2);\n print(plots[display]([p1,p2],axes=NONE));\n end if;\n if op(1,cc)='RGB' then\n convert(c2,'RGB')\n elif op(1,cc)=' HSV' then\n convert(c2,HSV)\n else c2 end if;\nend proc: # of l ighten\n\ndarken := proc(cc::\{color,list(color),set(color)\})\n loc al c,r,c2,drw,optn,p1,p2;\n\n if type(cc,\{list,set\}) then \n \+ return map('procname',cc,args[2..nargs])\n end if;\n\n if op(1,cc) ='HLS' then\n c := cc;\n else\n c := convert(cc,'HLS'); \+ \n end if; \n\n Digits := 8;\n if nargs>1 and type(args[2],num eric) then\n r := evalf(args[2]);\n if r>1 or r<0 then\n \+ error \"expecting the 2nd argument to evaluate to a floating poin t number between 0 and 1, but received %1\",args[2];\n end if;\n \+ else\n r := evalf(1/3);\n end if;\n\n if nargs=2 and not ty pe(args[2],numeric) then\n optn := args[2]\n elif nargs=3 then \n optn := args[3]\n end if;\n\n drw := false;\n if assigne d(optn) then\n if type(optn,`=`) then \n if op(1,optn)=dr aw then drw := op(2,optn) end if;\n if drw<>true then drw := f alse end if;\n elif optn=draw then drw := true;\n end if;\n \+ end if;\n\n # decrease the 'L' component towards 0\n c2 := COLOR (HLS,op(2,c),(1-r)*op(3,c),op(4,c));\n if drw then\n p1 := plot s[polygonplot]([[0,0],[1,0],[1,1],[0,1]],color=c);\n p2 := plots[ polygonplot]([[0,0],[1,0],[1,-1],[0,-1]],color=c2);\n print(plots [display]([p1,p2],axes=NONE));\n end if;\n if op(1,cc)='RGB' then \n convert(c2,'RGB')\n elif op(1,cc)='HSV' then\n convert( c2,HSV)\n else c2 end if;\nend proc: # of darken" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "FractionOption s: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1924 "FractionOptions := proc()\n local Options ,clr,i,tp;\n global _FractionProcOutput,_FractionProcInfo,\n _F ractionProcDraw,_FractionProcColour;\n\n if args[1]='defaults' then \n unassign('_FractionProcOutput','_FractionProcInfo',\n \+ '_FractionProcColour','_FractionProcDraw');\n return;\n end \+ if;\n \n Options:=[args[1..nargs]];\n if not type(Options,list(equ ation)) then\n error \"each option must be an equation\"\n end \+ if;\n if hasoption(Options,output,'_FractionProcOutput','Options') t hen\n if not member(_FractionProcOutput,\{mixed,MIXED,improper,IM PROPER\}) then \n error \"\\\"output\\\" must be 'mixed','MIXE D','improper' or 'IMPROPER'\"\n end if;\n if _FractionProcOu tput=MIXED then _FractionProcOutput := mixed end if;\n if _Fracti onProcOutput=IMPROPER then _FractionProcOutput := improper end if;\n \+ end if;\n if hasoption(Options,info,'_FractionProcInfo','Options') \+ then\n if not member(_FractionProcInfo,\{true,false,0,1,2\}) then \n error \"\\\"info\\\" must be false=0, 1 or true=2\"\n \+ end if;\n if _FractionProcInfo=false then _FractionProcInfo := 0 \n elif _FractionProcInfo=true then _FractionProcInfo := 2 end if ; \n end if;\n if hasoption(Options,draw,'_FractionProcDraw','Opti ons') then\n if _FractionProcDraw<>true then _FractionProcDraw := false end if;\n end if;\n if hasoption(Options,color,'tp','Option s') or \n hasoption(Options,colour,'tp','Options') then\n if not type(tp,list) then\n error \"colour option must be a list of colours\"\n end if;\n _FractionProcColour := [];\n \+ for i to nops(tp) do\n if not type(tp[i],color) then\n \+ error \"incorrect colour data for %-1 colour\",i;\n end if ;\n _FractionProcColour := [op(_FractionProcColour),tp[i]];\n \+ end do;\n end if;\n if nops(Options)>0 then\n error \"%1 is not a valid option\",op(1,Options);\n end if;\nend proc: " }}} {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 ";" }}}{PARA 0 "" 0 "" {TEXT -1 39 "Examples are given \+ in the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 13 "MultFractions" }{TEXT -1 11 ": examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "3/4" "6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 1 " " } {TEXT 269 1 "." }{TEXT -1 1 " " }{XPPEDIT 18 0 "2/3 = 1/2" "6#/*&\"\"# \"\"\"\"\"$!\"\"*&F&F&F%F(" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "MultFractions(3/4, 2/3,info=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%LMETHOD~I~..~cancelling~after~multiplicationG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/*(-%!G6##\"\"$\"\"%\"\"\"%\".GF+-F&6##\"\"#F)F+*&-F&6#\"\"'F+-F&6#\" #7!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%^pThe~numerator~6~and~denominator~12~can~be~divided~by ~the~common~factor~6~so~that~..G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(-%!G6##\"\"$\"\"%\"\"\"%\".GF+ -F&6##\"\"#F)F+#F+F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 13 "" 1 "" {GLPLOT2D 568 379 379 {PLOTDATA 2 "6T-%%TEXTG6'7$$\"++++]P!#5$\"$N%!\"#Q\"36\"-%'COLOURG 6&%$RGBG\"\"!F3F3-%*LINESTYLEG6#\"\"\"-%%FONTG6$%*HELVETICAG\"\"*-F$6' 7$F'$\"$:%F,Q\"4F.F/F4F8-F$6'7$F'$\"$J%F,Q\"_F.F/F4F8-F$6'7$$\"+++++DF )$\"$&GF,Q\"2F.F/F4F8-F$6'7$FL$\"$l#F,F-F/F4F8-F$6'7$FL$\"$\"GF,FHF/F4 F8-F$6'7$$\"+++++AF)$\"$N\"F,Q\"6F.F/F4F8-F$6'7$Fhn$\"$:\"F,Q#12F.F/F4 F8-F$6'7$Fhn$\"$J\"F,Q#__F.F/F4F8-F$6'7$$\"+++++GF)FjnQ\"1F.F/F4F8-F$6 '7$F\\pF`oFPF/F4F8-F$6'7$F\\pFfoFHF/F4F8-F$6'7$FL$\"$D\"F,Q\"=F.F/F4F8 -%)POLYGONSG6%7&7$$F3F3$\"\"$F37$$\"+++++vF)Faq7$Fdq$\"\"%F37$F`qFgq-F 06&F2$\"*++++\"!\")F`qF`q-%&STYLEG6#%,PATCHNOGRIDG-F\\q6%7&Fcq7$$F7F3F aq7$FgrFgqFfq-F06&%$HSVGF`q$\"++MLLLF)$\"+++++5!\"*F_r-F\\q6%7&7$F`q$ \"#:!\"\"7$$\"+++++]F)Fes7$Fis$\"#DFgs7$F`qF\\t-F06&F2F`qF`qF\\rF_r-F 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