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Egyptian mathematicians deve loped a fascinating system of numeration for fractions. To represent t he sum of " }{XPPEDIT 18 0 "1/3" "*&\"\"\"\"\"\"\"\"$!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "1/5" "*&\"\"\"\"\"\"\"\"&!\"\"" }{TEXT -1 40 " for example, they would simply write " }{XPPEDIT 18 0 "1/3 + 1/ 5" ",&*&\"\"\"\"\"\"\"\"$!\"\"F%*&\"\"\"F%\"\"&F'F%" }{TEXT -1 118 ". \+ \n\nActually, there's not a thing wrong with this, although\nit is te mpting to go ahead and add them together to get " }{XPPEDIT 18 0 "8/1 5" "*&\"\")\"\"\"\"#:!\"\"" }{TEXT -1 11 " . But " }{XPPEDIT 18 0 "1/3 + 1/5" ",&*&\"\"\"\"\"\"\"\"$!\"\"F%*&\"\"\"F%\"\"&F'F%" }{TEXT -1 31 " is a perfectly good name for " }{XPPEDIT 18 0 "8/15" "*&\"\") \"\"\"\"#:!\"\"" }{TEXT -1 51 " , a fact we recognize when we write th e equation\n " }{XPPEDIT 18 0 "1/3 + 1/5 = 8/15" "/,&*&\"\"\"\"\"\"\" \"$!\"\"F&*&\"\"\"F&\"\"&F(F&*&\"\")F&\"#:F(" }{TEXT -1 128 " . The \+ problem the Egyptians had was that although they had \n a notation for the unit fractions, i.e., fractions of the form " }{XPPEDIT 18 0 "1/ n" "*&\"\"\"\"\"\"%\"nG!\"\"" }{TEXT -1 256 " , they did not have a co mpact notation for the general fraction m/n . Some might say their n umeration system was faulty, but that would be overly critical, since \+ they were the first (as far as we know) to have any way of giving \+ names to fractions.\n\n" }{TEXT 257 8 "Question" }{TEXT -1 1221 " How do you suppose the Egyptians would have written\n 3/8 ? 3/5 ? \n\n Even though the Egyptian method of writing fractions stuck around for a long time, people were still very aware of its limitations. \n A good example of this is found in the Almagast , written by the Gr eek Mathematician, Geographer, Scientist, etc, Ptolemy in the first ce ntury AD. This book is sometimes referred to as the first trig book b ecause it contains tables of trigonometric ratios for the sine and cos ine function for use in astronomical calculations. Ptolemy explains that he is using the Babylonian method of writing fractions rather th an the Egyptian method because of the embarrassments that the Egyptian method often cause.\n\nMuch of what we know about Egyptian fractions \+ has been inferred\nfrom the so-called Rhind papyrus , written by the scribe Ahmes around 1650 BC. This book consists mainly of 84 word pro blems of a diverse nature, plus a few tables to aid the young scriblet s that Ahmes had charge of with their calculations.\nGenerally speakin g, Ahmes seemed to be happy to write the answer to a problem as a whol e number plus a sum of unit fractions. He did not use any unit fracti on more than once in the answer.\n\n\n" }{TEXT 256 7 "Problem" }{TEXT -1 78 " Establish the following algebraic identity: \n\nFor any n e xcept 0 and 1, " }{XPPEDIT 18 0 "1/n = 1/(n+1) + 1/(n*(n+1))" "/ *&\"\"\"\"\"\"%\"nG!\"\",&*&\"\"\"F%,&F&F%\"\"\"F%F'F%*&\"\"\"F%*&F&F% ,&F&F%\"\"\"F%F%F'F%" }{TEXT -1 7 " . " }}{PARA 0 "" 0 "" {TEXT -1 155 "With this identity, you can see that a fraction can always be \+ written in lots\nof different ways.\n\nThe largest table in the Rhind \+ Papyrus is the so-called " }{XPPEDIT 18 0 "2/n" "*&\"\"#\"\"\"%\"nG! \"\"" }{TEXT -1 223 " table, where Ahmes gives decompositions of thes e fractions into sums of unit fractions. Most of the entries in the ta ble come from the decomposition you get by multiplying the above ident ity on both sides by two. Thus\n " }{XPPEDIT 18 0 " 2/7 = 2/8 + \+ 2/(7*8) ,`=`, 1/4 + 1/28" "6%/*&\"\"#\"\"\"\"\"(!\"\",&*&\"\"#F&\" \")F(F&*&\"\"#F&*&\"\"(F&\"\")F&F(F&%\"=G,&*&\"\"\"F&\"\"%F(F&*&\"\"\" F&\"#GF(F&" }{TEXT -1 6 " . \n" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 30 " Egyptian Fractions with Maple" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 " ~\nMaple acts very naturally with Egyptian fractions. It si mply adds them up and reduces to lowest terms. Thus if you enter " } {XPPEDIT 18 0 "1/3 + 1/5" ",&*&\"\"\"\"\"\"\"\"$!\"\"F%*&\"\"\"F%\"\"& F'F%" }{TEXT -1 25 " , the\noutput will be " }{XPPEDIT 18 0 "8/15" "*&\"\")\"\"\"\"#:!\"\"" }{TEXT -1 159 ". We have to do something to \+ keep a record of the decomposition, and the simplest way to do that is to make it into a ||list|. Thus an egyptian fraction for " } {XPPEDIT 18 0 "8/15" "*&\"\")\"\"\"\"#:!\"\"" }{TEXT -1 12 " would be \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " ef := [1/3,1/5]; " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#efG7$#\"\"\"\"\"$#F'\"\"&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "\nIn order to convert to the usual form we could typ e \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " convert(ef,`+`);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##\"\")\"#:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Note: the \+ quotes around the + are backquotes ` not forward '. " }}{PARA 0 "" 0 "" {TEXT -1 459 "\nOne thing that would be nice to do is take an egy ptian fraction and print out an equation whose left hand side is the u sual form of the fraction and whose right hand side is the fraction wr itten as a sum of unit fractions. Because Maple is so intent on writin g fractions in the usual form, we must play a trick on it. First we wi ll convert each of the unit fractions in the egyptian fraction to a st ring. This can be done using the Maple word map , like so\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " ef2 := map(convert,ef,string);" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "N ow we can write the desired equation." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 " convert(ef,`+`)=convert(e f2,`+`); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 258 11 " Problems: " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "Try these problems from the Rhi nd Papyrus. Write the answer in unit fraction form." }}{PARA 0 "" 0 " " {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 24 " Find the product of " }{XPPEDIT 18 0 "1/16 + 1/2" ",&*&\"\"\"\"\"\"\"#;!\"\"F%*&\"\"\"F% \"\"#F'F%" }{TEXT -1 8 " with " }{XPPEDIT 18 0 "1 + 1/2 + 1/4" ",(\" \"\"\"\"\"*&\"\"\"F$\"\"#!\"\"F$*&\"\"\"F$\"\"%F(F$" }{TEXT -1 3 " . \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 " A quantity and its two thirds and its half" }} {PARA 0 "" 0 "" {TEXT -1 60 "and its it one seventh together make 33. \+ Find the quantity." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 47 " Divide 100 loaves of bread among \+ 5 men so that" }}{PARA 0 "" 0 "" {TEXT -1 60 "the shares are in arithm etic progression and 1/7 of the sum" }}{PARA 0 "" 0 "" {TEXT -1 56 "o f the three largest shares is the sum of two smallest. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 " Find the quotient when 100 is divided by " }{XPPEDIT 18 0 "8 + 1/2 + 1/4 + 1/8" ",*\"\")\"\"\"*&\" \"\"F$\"\"#!\"\"F$*&\"\"\"F$\"\"%F(F$*&\"\"\"F$\"\")F(F$" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 21 "Fibonnaci's theorem " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 625 "The Egyptian system of writing fractions\nas s ums of unit fractions stuck around even after much more efficient syst ems were developed. Fibonnaci was aware of the system in 1200 AD and i ncluded in his book ||Liber Abaci| a method for writing any fraction as a sum of unit fractions. His method was very natural: \n\nTake th e fraction you wish to express in the Egyptian manner and subtract fr om it the largest unit fraction which is not larger than it. If the r emainder is not itself a unit fraction, then repeat the process on the remainder. Continue this until the remainder is a unit fraction.\n~ \nFor example, suppose \n " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 " a := 4/5 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG#\"\"%\"\"&" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 49 "The largest unit fraction not larger tha n it is " }{XPPEDIT 18 0 "1/2" "*&\"\"\"\"\"\"\"\"#!\"\"" }{TEXT -1 15 ". Subtract it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " a := a - 1/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"aG#\"\"$\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 44 " The largest unit fraction not larger than " }{XPPEDIT 18 0 "3/10" "*&\"\"$\"\"\"\"#5!\"\"" }{TEXT -1 6 " \+ is " }{XPPEDIT 18 0 "1/4" "*&\"\"\"\"\"\"\"\"%!\"\"" }{TEXT -1 20 " \+ . Subtract\nthat.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " a := a - 1/4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG#\"\"\"\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Fibonnaci's method leads to the decomposition " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 285 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 " 4 / 5 = 1 / 2 + 1 \+ / 4 + 1 / 20" "/*&\"\"%\"\"\"\"\"&!\"\",(*&\"\"\"F%\"\"#F'F%*&\"\"\"F% \"\"%F'F%*&\"\"\"F%\"#?F'F%" }{TEXT -1 6 " . \n" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 75 "It is not clear whether Fibonnaci had a proof that his method always works." }}{PARA 0 "" 0 "" {TEXT -1 32 "But it does, and here's a proof." }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 259 7 " Lemma " }{TEXT -1 8 " Let " }{XPPEDIT 18 0 "p/q" "*&%\"pG\"\"\"%\" qG!\"\"" }{TEXT -1 52 " be any fraction which is not a unit fraction. \+ Let " }{XPPEDIT 18 0 "1/n" "*&\"\"\"\"\"\"%\"nG!\"\"" }{TEXT -1 42 " \+ be the largest unit fraction less than " }{XPPEDIT 18 0 "q/p" "*&%\" qG\"\"\"%\"pG!\"\"" }{TEXT -1 9 " . Then " }{XPPEDIT 18 0 "p/q-1/n" " ,&*&%\"pG\"\"\"%\"qG!\"\"F%*&\"\"\"F%%\"nGF'F'" }{TEXT -1 15 " is a fr action " }{XPPEDIT 18 0 "r/s" "*&%\"rG\"\"\"%\"sG!\"\"" }{TEXT -1 7 " \+ with " }{XPPEDIT 18 0 " r < p" "2%\"rG%\"pG" }{TEXT -1 28 " .\n \nS o, for example, if " }{XPPEDIT 18 0 "p/q=3/5" "/*&%\"pG\"\"\"%\"qG!\" \"*&\"\"$F%\"\"&F'" }{TEXT -1 12 " , then " }{XPPEDIT 18 0 "n=2" " /%\"nG\"\"#" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "3/5 - 1/2=1/10" "/, &*&\"\"$\"\"\"\"\"&!\"\"F&*&\"\"\"F&\"\"#F(F(*&\"\"\"F&\"#5F(" }{TEXT -1 12 " \nand so " }{XPPEDIT 18 0 " 3 / 5 = 1 / 2 + 1 / 10 " "/*&\"\"$\"\"\"\"\"&!\"\",&*&\"\"\"F%\"\"#F'F%*&\"\"\"F%\"#5F'F%" } {TEXT -1 2 " \n" }}{PARA 0 "" 0 "" {TEXT -1 24 "Another example: Take " }{XPPEDIT 18 0 "p/q" "*&%\"pG\"\"\"%\"qG!\"\"" }{TEXT -1 7 " to be " }{XPPEDIT 18 0 "3/7" "*&\"\"$\"\"\"\"\"(!\"\"" }{TEXT -1 21 ". The n n is 3 and " }{XPPEDIT 18 0 "3/7 - 1/3" ",&*&\"\"$\"\"\"\"\"(!\"\" F%*&\"\"\"F%\"\"$F'F'" }{TEXT -1 6 " is " }{XPPEDIT 18 0 "2/21" "*& \"\"#\"\"\"\"#@!\"\"" }{TEXT -1 4 " . " }}{PARA 0 "" 0 "" {TEXT -1 41 "Repeating the process with the remainder " }{XPPEDIT 18 0 "2/21" " *&\"\"#\"\"\"\"#@!\"\"" }{TEXT -1 22 " , we see n is 11 and " } {XPPEDIT 18 0 "2/21 -1/11" ",&*&\"\"#\"\"\"\"#@!\"\"F%*&\"\"\"F%\"#6F' F'" }{TEXT -1 4 "\nis " }{XPPEDIT 18 0 "1/(21*11) " "*&\"\"\"\"\"\"*& \"#@F$\"#6F$!\"\"" }{TEXT -1 59 " . So Fibonnaci's method leads to \+ the decomposition\n " }}}{EXCHG {PARA 286 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 " 3 / 7 = 1 / 3 + 1 / 11 + 1 / 231" "/*&\"\"$\"\" \"\"\"(!\"\",(*&\"\"\"F%\"\"$F'F%*&\"\"\"F%\"#6F'F%*&\"\"\"F%\"$J#F'F% " }{TEXT -1 6 " . \n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "\n" } {TEXT 260 6 "Proof " }{TEXT -1 58 " The argument for the lemma is s imple. First write\n\n " }{XPPEDIT 18 0 "p/q - 1 / n = (n*p - q)/( n*q) " "/,&*&%\"pG\"\"\"%\"qG!\"\"F&*&\"\"\"F&%\"nGF(F(*&,&*&F+F&F%F&F &F'F(F&*&F+F&F'F&F(" }{TEXT -1 13 " . Now if " }{XPPEDIT 18 0 "n*p \+ - q >= p" "1%\"pG,&*&%\"nG\"\"\"F#F'F'%\"qG!\"\"" }{TEXT -1 15 " , t hen adding " }{XPPEDIT 18 0 "q-p" ",&%\"qG\"\"\"%\"pG!\"\"" }{TEXT -1 45 " to both sides of the inequality gives that " }{XPPEDIT 18 0 "n*p - p >= q" "1%\"qG,&*&%\"nG\"\"\"%\"pGF'F'F(!\"\"" }{TEXT -1 14 " . \+ But then " }{XPPEDIT 18 0 "1/( n*p - p) <= 1/q " "1*&\"\"\"\"\"\",&*&% \"nGF%%\"pGF%F%F)!\"\"F**&\"\"\"F%%\"qGF*" }{TEXT -1 41 ". Now multip ly both sides by p and get " }{XPPEDIT 18 0 "1/(n-1) <= p/q " "1*&\" \"\"\"\"\",&%\"nGF%\"\"\"!\"\"F)*&%\"pGF%%\"qGF)" }{TEXT -1 8 ".\nSinc e " }{XPPEDIT 18 0 "p/q" "*&%\"pG\"\"\"%\"qG!\"\"" }{TEXT -1 58 " is n ot a unit fraction, but is less than 1, we have that " }{XPPEDIT 18 0 " n > 2" "2\"\"#%\"nG" }{TEXT -1 7 " , and " }{XPPEDIT 18 0 "1/(n-1)" "*&\"\"\"\"\"\",&%\"nGF$\"\"\"!\"\"F(" }{TEXT -1 32 " is a unit fracti on larger than " }{XPPEDIT 18 0 "1/n" "*&\"\"\"\"\"\"%\"nG!\"\"" } {TEXT -1 23 " which is smaller than " }{XPPEDIT 18 0 "p/q" "*&%\"pG\" \"\"%\"qG!\"\"" }{TEXT -1 58 ". This contradicts the choice of n, and \+ we conclude that " }{XPPEDIT 18 0 "n*p - q < p" "2,&*&%\"nG\"\"\"%\"p GF&F&%\"qG!\"\"F'" }{TEXT -1 11 " . qed.\n \n" }{TEXT 261 7 "Theorem " }{TEXT -1 47 " Fibbonaci's method works for all fractions " } {XPPEDIT 18 0 "p/q" "*&%\"pG\"\"\"%\"qG!\"\"" }{TEXT -1 3 ".\n\n" } {TEXT 262 6 "Proof " }{TEXT -1 16 " Starting with " }{XPPEDIT 18 0 "p /q" "*&%\"pG\"\"\"%\"qG!\"\"" }{TEXT -1 17 " and subtracting " } {XPPEDIT 18 0 "1/n" "*&\"\"\"\"\"\"%\"nG!\"\"" }{TEXT -1 27 " we get a smaller fraction " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "( n*p-q)/(n*q)= p2/q2" "/*&,&*&%\"nG\"\"\"%\"pGF'F'%\"qG!\"\"F'*&F&F'F)F 'F**&%#p2GF'%#q2GF*" }{TEXT -1 31 " . Further, the numerator of " } {XPPEDIT 18 0 "p2/q2" "*&%#p2G\"\"\"%#q2G!\"\"" }{TEXT -1 112 " is sma ller than p by the Lemma. If it is 1, we are done. If not, then cle arly we will be after\nno more than " }{XPPEDIT 18 0 "p2 - 1" ",&%#p2G \"\"\"\"\"\"!\"\"" }{TEXT -1 179 " more applications of the Lemma. qed .\n\nWith just a little more work, we can modify Fibbonaci's method to apply to any rational number. In fact, we have the following theor em.\n\n" }{TEXT 263 8 "Theorem " }{TEXT -1 53 " Given any rational num ber p/q and any unit fraction " }{XPPEDIT 18 0 "1/n" "*&\"\"\"\"\"\"% \"nG!\"\"" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "p/q" "*&%\"pG\"\"\"%\"qG !\"\"" }{TEXT -1 65 " can be written as a sum of distinct unit fractio ns smaller than " }{XPPEDIT 18 0 "1/n" "*&\"\"\"\"\"\"%\"nG!\"\"" } {TEXT -1 4 ".\n \n" }{TEXT 264 6 "Proof " }{TEXT -1 20 " Since the se ries " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Sum(1/i,i=n+1..infinity); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*$%\"iG!\"\"/F';,&%\"nG\" \"\"F-F-%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "diverges, \+ there is a positive integer k so that " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 " Sum(1/i,i=n+1..n+k) <= p/q , `and`, p/q < Sum(1/i,i=n+1..n+k +1) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%1-%$SumG6$*$%\"iG!\"\"/F(;,&% \"nG\"\"\"F.F.,&F-F.%\"kGF.*&%\"pGF.%\"qGF)%$andG2F1-F%6$F'/F(;F,,(F-F .F0F.F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "If the leftmost ineq uality is strict, then the remainder is a fraction " }{XPPEDIT 18 0 " r/s" "*&%\"rG\"\"\"%\"sG!\"\"" }{TEXT -1 20 " which is less than " } {XPPEDIT 18 0 "1/(n+k+1)" "*&\"\"\"\"\"\",(%\"nGF$%\"kGF$\"\"\"F$!\"\" " }{TEXT -1 181 ". Consequently, Fibonacci's method can be used from here on to express r/s as a sum of distinct unit fractions which are \+ all distinct\nfrom the consecutive ones already used. qed." }}{PARA 0 "" 0 "" {TEXT -1 3 "\n\n " }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 60 " A Maple word to carry out (the modified) Fibonnaci method. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 355 "Here is the definition of a Maple word fibo which implements Fibonnaci' s method for decomposing a fraction into a sum of unit fractions. It \+ uses two Maple words in its definition, numer , which gives the numer ator of its argument, and trunc , which returns the\ninteger part of \+ its argument. It includes the modification established above which \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 325 "fibo := proc(x,m)\n lo cal z,l,i,n,k;\n k := m+1;\n z := x;\n \+ l := NULL;\n for i while 1 < numer(z) do\n \+ n := max(k,trunc(1/z)+1);\n k := k+1;\n \+ z := z-1/n;\n l := l,1/n;\n od;\n \+ l := [l,z];\n end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Ex ecute it. Now you have added a word to the Maple vocabulary. To get a decomposition of " }{XPPEDIT 18 0 "5/23 " "*&\"\"&\"\"\"\"#B!\"\" " }{TEXT -1 27 " into unit fractions, enter" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " ef := fibo(5/23,1);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#efG7%#\"\"\"\"\"&#F'\"#e#F'\"%qm" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 64 "To print out an equation showing the decomposition nice ly, enter" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " convert(ef,`+`) = convert(map(convert,ef,string), `+`); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"\"&\"#B,(%$1/5G\"\"\"%%1 /58GF)%'1/6670GF)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Here is a us eful word to display the decomposition." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 123 " exhibit := proc(r,k)\n local ef;\n ef := fibo(r,k);\n convert(ef,`+`) = convert(map(convert,ef,string),`+`); \n end:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " exhibit(13/11,1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"#8\"#6,,%$1/2G\"\"\"%$1/3GF)%$1/4G F)%%1/11GF)%&1/132GF)" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 15 " A vi sual Fibo" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "How might we define \+ a more visual version of fibo? One which shows the decomposition as br eaking up a rectangle with area " }{XPPEDIT 18 0 "p/q" "*&%\"pG\"\"\"% \"qG!\"\"" }{TEXT -1 21 " into rectangles with" }}{PARA 0 "" 0 "" {TEXT -1 6 " area " }{XPPEDIT 18 0 "1/n\n" "*&\"\"\"\"\"\"%\"nG!\"\"" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "1/n" "*&\"\"\"\"\"\"%\"nG!\"\"" }{TEXT -1 42 " occurs in the Fibbonaci decomposition of " }{XPPEDIT 18 0 "p/q" "*&%\"pG\"\"\"%\"qG!\"\"" }{TEXT -1 42 ".\n\nHere is an imp lementation of this idea." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "block := (llc,x,clr) -> plots[polygonplot]([llc,[llc[1]+1,llc[2]],[llc[1]+1 ,llc[2]+x],[llc[1],llc[2]+x]],scaling=constrained,color=clr) ;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%&blockG:6%%$llcG%\"xG%$clrG6\"6$%)op eratorG%&arrowGF*-&%&plotsG6#%,polygonplotG6%7&9$7$,&&F56#\"\"\"F:F:F: &F56#\"\"#7$F7,&F;F:9%F:7$F8F?/%(scalingG%,constrainedG/%&colorG9&F*F* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 422 "cutcube := proc(llc,n,clr) local tmp,j,t op,pl1;\n tmp:= NULL: \n for j from 1 to n-1 do tmp := tmp,\n \+ block([llc[1] ,llc[2]+(j-1)/n],1/n,scaling=constrained,\n color=clr [j mod nops(clr)+1]) od;\n top := [[llc[1] ,llc[2]+(n-1)/n],[llc[1 ]+1,llc[2]+(n-1)/n],[llc[1]+1,llc[2]+1],\n [llc[1],llc[2]+1]] ;\n \+ pl1 := plots[textplot]([llc[1]+.5,1.5 ,cat(n,` equal pieces`)]);\n \+ [top,plots[display]([pl1,tmp])]; end;\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(cutcubeG:6%%$llcG%\"nG%$clrG6&%$tmpG%\"jG%$topG%$pl1 G6\"F/C'>8$%%NULLG?(8%\"\"\"F6,&9%F6!\"\"F6%%trueG>F26$F2-%&blockG6&7$ &9$6#F6,&&FB6#\"\"#F6*&,&F5F6F9F6F6F8F9F6*$F8F9/%(scalingG%,constraine dG/%&colorG&9&6#,&-%$modG6$F5-%%nopsG6#FQF6F6F6>8&7&7$FA,&FEF6*&F7F6F8 F9F67$,&FAF6F6F6Fhn7$F[o,&FEF6F6F67$FAF]o>8'-&%&plotsG6#%)textplotG6#7 %,&FAF6$\"\"&F9F6$\"#:F9-%$catG6$F8%.~equal~piecesG7$Fen-&Fco6#%(displ ayG6#7$F`oF2F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2318 " vfibo := proc(x,m,clr)\nl ocal z,l,i,n,k,film,frame,top,box,t,lbox,fboxes,tmp,rbox,tbox,pl1;\n \+ lbox := block([0,0],x,clr[1]);\n film := lbox;\n k := m+1;\n \+ z := x;\n l := NULL;\n fboxes := NULL;\n for i while 1 < nu mer(z) do\n n := max(k,trunc(1/z)+1);\n if n < 200 then \n tmp := cutcube([3,0],n,clr);\n top := tmp[1]; \n rbox := tmp[2];\n film := film,plots[display] ([fboxes,lbox,rbox])\n else top := [[3,(n-1)/n],[4,(n-1)/n],[4, 1],[3,1]]\n fi;\n for t from 0 by 1/3 to 1 do\n \+ tbox := plots[polygonplot]([\n [(1-t)*top[1][1],(1-t )*top[1][2]+t*(x-z)],\n [(1-t)*top[2][1]+t,(1-t)*top[2] [2]+t*(x-z)],\n [(1-t)*top[3][1]+t,(1-t)*top[3][2]+t*(x -z+1/n)],\n [(1-t)*top[4][1],(1-t)*top[4][2]+t*(x-z+1/n )]],\n color = clr[(i mod nops(clr))+1]);\n \+ if t < 1 then\n film :=\n film,plots [display]([tbox,fboxes,lbox,rbox])\n else\n \+ fboxes := fboxes,tbox;\n film := film,plots[display]([f boxes,lbox,rbox])\n fi\n od;\n k := k+1;\n \+ z := z-1/n;\n l := l,1/n\n od;\n n := 1/z;\n if n < 200 then\n tmp := cutcube([3,0],n,clr);\n top := tmp[ 1];\n rbox := tmp[2];\n film := film,plots[display]([fbo xes,lbox,rbox])\n else top := [[3,(n-1)/n],[4,(n-1)/n],[4,1],[3,1]] \n fi;\n for t from 0 by 1/3 to 1 do\n tbox := plots[poly gonplot]([\n [(1-t)*top[1][1],(1-t)*top[1][2]+t*(x-z)],\n \+ [(1-t)*top[2][1]+t,(1-t)*top[2][2]+t*(x-z)],\n [( 1-t)*top[3][1]+t,(1-t)*top[3][2]+t*(x-z+1/n)],\n [(1-t)*top [4][1],(1-t)*top[4][2]+t*(x-z+1/n)]],\n color = clr[(i mod \+ nops(clr))+1]);\n if t < 1 then\n film := film,plots [display]([tbox,fboxes,lbox,rbox])\n else\n fboxes : = fboxes,tbox;\n film := film,plots[display]([fboxes,lbox,r box])\n fi\n od;\n l := [l,z];\n fnt := [HELVETICA,BOL D,24 ]:\n pl1 := plots[textplot]([2,1.2*x,convert(\n convert (l,`+`) = convert(map(convert,l,string),`+`),string\n )],font=f nt);\n film := film,plots[display]([pl1,fboxes,lbox,rbox]);\n [l ,plots[display]([film],insequence = true)]\nend:" }}{PARA 7 "" 1 "" {TEXT -1 43 "Warning, `fnt` is implicitly declared local" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 54 " vfibo(1/2+1/3 ,1,[ye llow,red,blue,yellow,white])[2] ;" }}{PARA 13 "" 1 "" {INLPLOT "6#-%(A NIMATEG6.7$-%)POLYGONSG6$7&7$\"\"!F,7$$\"\"\"F,F,7$F.$\"+LLLL$)!#57$F, F1-%'COLOURG6&%$RGBG$\"*++++\"!\")F9F,-%(SCALINGG6#%,CONSTRAINEDG7&F'- %%TEXTG6$7$$\"#N!\"\"$\"#:FG%/2~equal~piecesG-F(6#7&7$$\"\"$F,F,7$$\" \"%F,F,7$FR$\"+++++]F37$FOFUF<7'-F(6$7&FWFT7$FRF.7$FOF.-F66&F8F9F,F,F' FAFKF<7'-F(6$7&7$$\"\"#F,$\"+LLLLLF37$FOFao7$FOF17$F_oF1FhnF'FAFKF<7'- F(6$7&7$F.$\"+nmmm;F37$F_oF[p7$F_o$\"+nmmmmF37$F.F_pFhnF'FAFKF<7'-F(6$ 7&F+F-7$F.FU7$F,FUFhnF'FAFKF<7(FcpF'-FB6$FD%/3~equal~piecesG-F(6#7&FNF Q7$FRFaoFco-F(6#7&FcoF_q7$FRF_p7$FOF_pF<7)-F(6$7&FdqFcqFfnFgn-F66&F8F, F,F9FcpF'FipF\\qF`qF<7)-F(6$7&7$F_o$\"+6666hF37$FOF`r7$FO$\"+WWWW%*F37 $F_oFdrFiqFcpF'FipF\\qF`qF<7)-F(6$7&7$F.$\"+cbbbbF37$F_oF\\s7$F_o$\"+* )))))))))F37$F.F`sFiqFcpF'FipF\\qF`qF<7)Fcp-F(6$7&FgpFfpF0F4FiqF'FipF \\qF`qF<7*-FB6%7$F_o$\"+++++5!\"*%25/6~=~`1/2`+`1/3`G-%%FONTG6%%*HELVE TICAG%%BOLDG\"#CFcpFdsF'FipF\\qF`qF<" 2 377 377 377 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 100 1 }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 48 "Prob lems. Investigate these problems. Use fibo ." }}{EXCHG {PARA 14 "" 0 " " {TEXT -1 107 "Use fibo to print out decompositions of the frac tions with an odd numerator and a denominator of 100." }}{PARA 14 "" 0 "" {TEXT -1 364 " Find a fraction which fibo decomposes into t he sum of six unit fractions.\n Find a fraction which fibo deco mposes into the sum of eight unit fractions.\nThink of an interesting question about decompositions of fractions.\n\n Use Fibonacci's theor em to argue that there are\ninfinitely many ways to write a given frac tion as a sum of unit fractions. \n " }}{PARA 14 "" 0 "" {TEXT -1 85 " Burton, David M., History of Mathematics: an introduction, 3rd Ed ition, WCB, 1995.\n " }}}}{EXCHG {PARA 15 "" 0 "" {HYPERLNK 17 "Class \+ 20" 1 "class20.mws" "" }}}{EXCHG {PARA 15 "" 0 "" {HYPERLNK 17 "Conten ts" 1 "ma503.mws" "" }}}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0 " 0 }{VIEWOPTS 1 1 0 1 1 1803 }