Fudametal Joural of Mathematics ad Mathematical Scieces Vol., Issue, 0, Pages -55 This paper is available olie at http://www.frdit.com/ Published olie November 9, 0 A SECOND SOLUTION FOR THE RHIND PAPYRUS UNIT FRACTION DECOMPOSITIONS Departmet of Mathematics Texas A&M Uiversity-Commerce Commerce, Texas 759 USA e-mail: charles.dorsett@tamuc.edu Abstract Withi this paper, a secod, greatly-improved systematic solutio, cosistet with the mathematics used i the Rhid Papyrus, is give for the log usolved mystery of a method for costructio of the uit fractio decompositio table i the Rhid Papyrus.. Itroductio Most of our kowledge of aciet Egyptia mathematics is derived from two sizable papyri: the Rhid Papyrus ad the Goleischev Papyrus. I 858, A. Hery Rhid purchased a partial papyrus i Luxor, Egypt. The papyrus was reportedly foud i Thebes, i the ruis of a buildig ear the Ramesseum. The Rhid Papyrus was writte i hieratic script about 650 B.C. by a scibe amed Ahmes. Sice much of what we kow about aciet Egyptia mathematics comes from the Rhid Papyrus, Ahmes work is both mathematically ad historically sigificat. Readers of the papyrus are assured that its cotet is a likeess of earlier work Keywords ad phrases: early Egyptia mathematics, uit fractio decompositios. 00 Mathematics Subject Classificatio: 0A6, A67, D68. Received October 8, 0; Accepted November 06, 0 0 Fudametal Research ad Developmet Iteratioal
datig back to the Twelfth Dyasty: 89-80 B.C. Early Egyptia mathematics with fractios, datig back to the Twelfth Dyasty, was made difficult by the computatioal practice of allowig oly uit fractios, i.e., fractios of the form, where is a atural umber. To overcome this difficulty, aciet Egyptia mathematicias costructed uit fractio decompositio tables for quick referece ad use. The table at the begiig of the Rhid Papyrus gives uit fractio decompositios of fractios of the form, where is a odd atural umber betwee ad 0, ad is the most extesive of the uit fractio decompositio tables to be foud ad preserved amog the aciet Egyptia papyri. As stated withi the Rhid Papyrus, the cotet of the papyrus was to give a thorough study of all thigs, isight ito all that exists, ad kowledge of all obscure secrets. However, o isight or kowledge of the costructio of the uit fractio decompositio table was give withi the papyrus, leavig its method of costructio a obscure secret. The lack of iformatio about the costructio of the table did, almost 700 years after its creatio, stimulate iterest i its method of costructio motivatig may to try to ed the mystery, all of which has made the table eve more historically ad mathematically importat. As idicated, through the years, may iterested people have determiedly tried to solve the mystery of the table s costructio. Because of the isights ad systematic processes icluded withi the Rhid Papyrus, most iterested people searched for a systematic process that would give the uit fractio decompositios, but, up to the fall 006, oly patters givig some of the etries withi the table were kow. For example, it was kow that fractios withi the table of the form follow the geeral patter. Thus some progress was made. However, k k 6k may differet observed patters ad uexpected etries withi the table raised questios about the existece of a systematic method makig the search for a systematic solutio very much like lookig for a eedle i a haystack, ot kowig if, i fact, there is a eedle i the haystack. I additio, prior to the fall 006, there were uit fractio decompositios that followed oe of the kow patters cotiuig to leavig the costructio of the table a ukow mystery. The 006 solutio, which was published i 008 [], provided a sigle systematic method that could be used to geerate all the etries i the table. The 008 k
A SECOND SOLUTION FOR THE RHIND PAPYRUS solutio [] gave a process that could be used, but fell short of the ideal solutio sice additioal work was required to determie each etry. However, the discoveries, isights, ad processes i the 008 solutio [] proved to be mathematically sigificat sice their use helped produce the more precise algorithm i this paper. Additioal iformatio cocerig the Rhid Papyrus uit fractio decompositios ad the difficulties i searchig for a systematic solutio to the mystery ca be foud i David Burto s book: The History of Mathematics: A Itroductio [], ad would serve as a excellet resource. As recogized i Burto s book [], there are two totally differet types of uit fractio decompositios i the table, the secod oe for, 0 which ulike the previous etries, abadoed the missio for the creatio of the table givig a oe mold fit all etry that was, with almost total certaity, a later additio to the table by someoe other tha the creator of the above etries.. The Motivatio ad Goal for the Costructio of the Table The motivatio for costructig a mathematics table is eed. As idicated above, there was a eed amog aciet Egyptia mathematicias for a uit fractio decompositio table. The goal would be to costruct a easily used table that best served the eed, without extraeous etries. The creator of the uit fractio decompositios i the table for less tha or equal to 99 would be highly motivated sice he or she would be the first to use the table ad would ot wat to further complicate a already difficult otatio restrictio with aythig less tha the best possible uit fractio decompositios.. Developmet of a Strategy The strategy for the costructio of a table is determied by the eed, kow iformatio, kow strategies, the difficulties of the costructio, ad the cleveress of the creator, together, i this case, with a determied resolve to fid that best choice. As witessed by the absece of fractios of the form, where is eve, the creator of the table was aware of the multiplicatio ad cacellatio properties for fractios. Thus iclusio of fractios, where is eve ad or bigger, would be
extraeous iformatio ot to be icluded i the table. The objective was to determie the best uit fractio decompositios for further calculatios that would work best with the required otatio. To accomplish these objectives, the umber of terms i the decompositio would be a importat cosideratio as would be the size of the deomiators, particularly the deomiators of the first ad last fractio i the decompositio. Thus the ideal choice for a etry would be a two term decompositio with smallest possible first ad secod fractio deomiators that meets the eed. However, the creator of the etries realized that a three term or four term uit fractio decompositio could possibly better serve the eeds ad added those possibilities i the search for the best uit fractio decompositio. I additio, eve though the focus was o the idividual etries, ad decisios were made at the etry level, uiformity withi the etries would make use of the table easier ad more productive. etry betwee As was the case for the Rhid Papyrus uit fractio decompositios table, for a i the table, obtaiig uit fractio decompositio for multiples of ad would be a importat related calculatio. With the best uit fractio decompositio for a etry, for a eve iteger k j betwee ad, k j k j betwee ad, k j. this objective could be best accomplished: ad for a odd iteger Thus a key i the search for a table etry is the uit fractio decompositio that best gives iteger multiplies of. For example cosider the table etry for :. Usig that etry, 5 0 0 5 6,, 5 5 5 5 5 5 0,, ad 0 0 5 6 5 8 5 5 makig the give uit fractio decompositio for decompositio for 5 5 5 5 5 5 5 6 5 5 5 the ideal uit fractio based o eed. Of all the may possibilities, how did the
A SECOND SOLUTION FOR THE RHIND PAPYRUS 5 creator of the table come up with the 0 ad 0? The creator would have to develop a strategy that idetified the 0 ad 0 for prime cosideratio ad the establish that choice as the best choice. For cosistecy ad better compariso ad use, each etry should be writte usig the same format. The etries i the table are writte i icreasig orders of the deomiators. To maximize the use of the table, the deomiator of the first fractio i the decompositio of should be less tha, allowig better ad greater use of table. This could be, ad, was doe, by cosiderig oly values of p startig at ( ) ad goig to. For each p, let o p. The values for o will be icreasig odd itegers startig with up to ad icludig givig p ( o) o o. The objective is to fid the o that has a p p p p p p uit fractio decompositio with smallest last term deomiator that best simplifies related calculatios. If <, the p havig a factor of would immediately reduce to a 6 sum of uit fractios. If <, the p havig aother factor of would 6 8 immediately reduce to a sum of uit fractios. If <, p havig a third factor of would immediately reduce 8 to a sum of uit fractios. The process could be cotiued makig values of p with factors of highly attractive for use i costructig the uit fractio decompositios. However, selectig values of p with may factors of makes the deomiators big, defeatig a goal i the creatio of the table. The best etry would cotai oe factor of ad, quickly ad easily, regeerate the desired factor of withi the calculatios. Havig a factor of with 6 the would be helpful i calculatig. If o divides p, cacelig o i the umerator ad deomiator of the secod fractio gives a easily-calculated, highly desirable two uit fractio decompositio cadidate for. Whether or ot o divides p, if o ca be writte as a decreasig sum of two or three factors of p, the ca be reduced to a sum of uit fractios.
6 For a fixed such p, give a choice of decreasig sums, the sum with largest last term would make the deomiator of the last two fractios as small as possible givig a cadidate for the uit fractio decompositio. For each, either there exists a odd iteger o greater tha that divides p or o such o exists. Cosider the first case, where is composite ad o divides p. If o divides, the oe atural place to look for the best etry would be the largest factor of ad, i may cases, gives the desired two term uit fractio decompositio. If o divides p ad ot, the, as kow by the creator of the table based o use of the property i the costructio of the table ad prove i the 008 paper [], each factor of o is a factor of. Thus, to obtai the desired uit fractio decompositio, the focus falls o the factors of greater tha or equal to ad o the questio of whether or ot the focus could be exteded to additioal umbers. If replacig a prime factor f of of a factor o ( ) of by the square of the smallest prime factor m of gives a product less tha ad m > f, let k be the m ( ) larger such value off, ad let (, ) ( o ) o k. Let q (, k) be the iteger such k o(, k) o(, k( ) ) that ( q (, k) ) o(, k). The q(, k) ad ( q(, k) ) reduces to a sum of two uit fractios givig a cadidate for the used uit fractio decompositio. If o (, k) ca be writte as a sum of factors of q (, k), the the sum with the largest last term would give a additioal cadidate for the uit fractio decompositio. The smallest prime factor was selected i order to get a smaller value of p. I this case, to allow the possibility of a two term uit fractio decompositio, oly those values of o greater tha for which o divides p, where the p is the correspodig value for o, as discussed above, will be cosidered. Because of the desire for a small deomiator, for each o, the focus is o the deomiator of the last fractio i the correspodig decompositio or decompositios. If oe of the deomiators for the last uit fractio is much smaller tha ad cotais factors located earlier i the table, the the uit fractio decompositio for that choice would be the etry i the table. Give two possible choices, if oe gives a eve value for p, because of the preferece for eve umbers, the uit fractio decompositio with eve p would be used i the table. Give two choices, for ease of calculatios withi the etry itself, a factor of ad would be a importat cosideratio ad if the
A SECOND SOLUTION FOR THE RHIND PAPYRUS 7 prime factors of are also icluded, the to make calculatios for multiples of easier, that choice would be a serious cosideratio. If the p value is a multiple of a prime factor of, calculatios of multiples of would be simplified ad thus a cosideratio whe choosig the uit fractio decompositio. The focus is o ad o, but the correspodig p ca be calculated usig ( o) p. Below iformatio is give to help make the best choice. I some cases there is a clear choice. I the cases where there is o clear choice amogst the two best choices, the creator of the table, determied to obtai the best choice, would have used each to do calculatios ad the select the oe that worked best. Below the calculatios eeded i the search for the best etry is give for each composite value of usig the format (the value of o, the correspodig value for p, deomiator of last term) or (the value of o cosidered, the correspodig value for p, the sum used, deomiator of last term). For 9, there is oly oe choice: (, 6, ( )) givig the uit fractio decompositio. 9 6 8 For 5 ( 5), the choices are (, 9, ( )( 5)), ( 5, 0, ( )( 5) ), ( 9,, ( 5)), ad ( 9,, 5, ( 5)) givig a choice betwee ad. Use the criteria above, would be the oe selected givig the uit fractio decompositio. 5 0 0 For ( 7), the choices are (,, ( )( 7)), ( 7,, ( )( 7) ), ( 9, 5, 5( 7) ), ad ( 9, 5, 5, ( 5)( 7)). Choice would require use of a etry i the table followig the etry for ad would be excluded. Usig the criteria above, would be selected givig the uit fractio decompositio. For 5 5, there is oly oe choice ( 5, 5, ( 5 )) givig 5. 5 75
8 For 7, the choices are (, 5, ( 5)), ( 9, 8, ( )), ad ( 9, 8, 9 6, ( )). Of the choices, is selected givig the table etry 7 8. 5 For ( ), the choices are (, 8, ( )( )) ad (,, ( )( )). The clear choice is givig the decompositio. 66 For 5 5( 7), the choices are ( 5, 0, ( 5)( 7)), ( 7,, ( 5)( 7) ), ( 5, 0, ( )( 7) ), ad ( 5, 0, 5 0, ( 5)( 7) ). Usig the criteria above, would be selected givig. 5 0 For 9 ( ), the choices are (,, ( 7)( ) ), (, 6, ( )( ) ). Of the two, is the choice givig. 9 6 78 For 5 ( 5), the choices are (,, ( )( 5)), ( 5, 5, ( 5 )), ( 9, 7, ( 5 )), ad ( 5, 0, ( )( 5)). Of those choices, is selected givig. 5 0 90 For 9 7, there is oly oe choice ( ( 7 7, 8, )) givig 9. 8 96 For 5 ( 7), the choices are (, 7, ( 7)) ad ( 7,, ( )( 7)), Choice would be selected givig. 5 0 For 55 5( ), the choices are ( 5, 0, ( )( 5)( ) ), (,, ( 5)( )), ad ( 5, 5, 7( ) ). Sice gives a eve p, is the selectio givig 55 0. 0 For 57 ( 9), the choices are (, 0, ( )( 5)( 9)) ad ( 9, 8, ( )( 9)) with the best choice for table etry. 57 8
A SECOND SOLUTION FOR THE RHIND PAPYRUS 9 For 6 ( 7), the choices are (,, ( 7)( )), ( 7, 5, ( 5)( 7)), ( 9, 6, ( )( 7)), (,, ( )( 7)), 5 ( 7, 5, ( 5)( 7) ), ad 6 ( 7, 5, 5 9, ( 5)( 7)) ad the oe selected is :. 6 6 For 65 5( ), the choices are ( 5, 5, 5( 7)( ) ), (, 9, ( 5)( 7)), ( 5, 5, ( )), ad ( 5, 5, 9, ( 5 )( )). There are o eve values for p ad has the smallest last term makig the selectio givig 65 9. 05 For 69 ( ), the choices are (, 6, ( )( )) ad (, 6, ( )( )) givig selectio ad decompositio. 69 6 8 For 75 ( 5 ), the choices are (, 9, ( 5 )( )), ( 5, 0, ( )( 5)), ( 5, 5, ( 5 ), ad ( 5, 50, ( )( 5 )) givig selectio ad decompositio. 75 50 50 For 77 7( ), the choices are ( 7,, ( )( 7)( )), (,, ( 7)( )), ( 9, 6, ( )), ad ( 9, 6, 9 7, ( 7)( )). This is a tough call betwee ad, but for, p, is a multiple of makig calculatios easier ad givig the decompositio. 77 08 For 8, the choices are (,, ( )( 7)), ( 9, 5, ), ad ( 7, 5, ( )). Followig the criteria, the selectio is givig the decompositio. 8 5 6 For 85 5( 7), the choices are ( 5, 5, ( 5)( 7)), ( 7, 5, ( 5)( 7)), ad ( 5, 55, ( 7)). Sice 5 is a multiple of 7, would be selected givig decompositio. 85 5 55
50 For 87 ( 9), the choices are (, 5, ( 5)( 9)) ad ( 9, 58, ( )( 9)) givig selectio ad table etry. 87 58 7 For 9 7( ), the choices are ( 7, 9, 7 ( )), (, 5, ( 7)( )), ( 9, 70, ( 5)( )), ad ( 9, 70, 5, 5( 7)( )). Selected would be givig decompositio. 9 70 0 For 9 ( ), the choices are (, 8, ( )( )) ad (, 6, ( )( )). Clearly is the choice givig table etry. 9 6 86 For 95 5( 9), the choices are ( 5, 50, ( 5 )( 9)), ( 9, 57, ( 5)( 7)), ( 5, 60, ( )( 9)), ad ( 5, 60, 5 0, ( )( 5)( 9)). Usig the criteria, would be selected givig decompositio. 95 95 80 570 For 99 ( ), the choices are (, 5, ( )( 7)), ( 9, 5, ( )( )), (, 55, ( 5)( )), (, 66, ( )( )), ad 5 (, 66,, ( )( )). Selected would be givig decompositio. 99 66 98 Thus cosider, where is prime ad ot 0. As above, i the case of a composite umber, the focus is o each with objective obtaiig the best uit fractio decompositio for performig eeded calculatios. For each there are values for p ad for each of those p values, there could be may possible uit fractio decompositios for cosideratio, all of which must be kow ad cosidered to determie the best choice. Oce the uit fractio decompositios are kow for each p, from those the oe that is best suited for the eeded calculatios is the uit fractio decompositio for i the table. To do all this for oe, without a pla, is difficult, time cosumig, ad filled with ucertaity about the required completeess. To have to repeat the etire process for the ext value of perhaps requirig eve more determiatios ca be overwhelmig. Thus a strategy to make
A SECOND SOLUTION FOR THE RHIND PAPYRUS 5 the selectio process more easily maaged ad to esure completeess would be i order. I this case, because is prime, oly values of o, with correspodig value p, are cosidered for which o divides p or o ca be writte as a decreasig sum of two or three factors of p. Below a partial strategy is give itetioally leavig its completio for the iterested reader. The strategy is applied to the values of p for a fixed, but arbitrary, prime umber. Thus, all the cases below are withi the same odd prime. Cosider the case that p is the product of two or more distict odd primes. The p is the product of exactly odd primes sice, otherwise, p ( 5)( 7) 05, which is too large for cosideratio i the table. Let a ad b be two distict odd primes such that b a ad p ab. The the oly possible way to form a uit fractio decompositio from p would be use of or b or a or the sum a b, where obviously the larger possibility is the a b. Kowig the largest possibility m is useful sice, if the correspodig o value for p is greater tha m, p ca be immediately removed from cosideratio. If o m, the the resultig uit fractio decompositio would be saved for further cosideratio ad all other possibilities ca be discarded. For this p, ad all others, there is mathematically at most oe two term uit fractios saved for further cosideratio, which is easily determied by seeig if oe of the possible factors is the value of o. All other possibilities must be checked savig oly those whose value is o. If p is prime ad o is ot, for p to be further cosidered, the correspodig o would have to be p, p 0 p, ad p, which is impossible. Thus all prime values for p, with o ot, ca be immediately discarded. Cosider the case where p is a power of a odd prime with expoet or more, k give by p a. As above, the oly possible two term uit fractio decompositio from p is the possible case where o or, because of the restrictios o o, o a, where m < k. If a 5, the, because of the limitatios i the table, k ad there are o possible three or four term uit fractio decompositios. For a, k. For, there would be o two term or three term decreasig sums for cosideratio. For, if o or o or o or if 5 ad m
5 o, there would be a two or a four term uit fractio decompositio from p for further cosideratio. For, if o is or or 9 or 7 or or or, there would two or four term uit decompositios to cosider. As above, if o >, the cosideratio. p ca be discarded from For p k, k, the maximum value of k is 6 ad there would be the possibility of two or three or four term uit fractio decompositios for further cosideratio all of which the last term is. p For example, if k ad o is or or or or or or, there would be a two or four term uit fractio decompositio for further cosideratio. If p a, where a is a odd prime, for further cosideratio of p, the correspodig value for o, would have to be a or a. Obviously, havig a strategy makes the process more maageable ad also gives ot before observed, importat isights ito the problem. As idicated above, the remaiig cases iclude values of p with may factors that possibly ca be used to create may sums for cosideratio i the search for uit fractio decompositios ad are left as a exercise for the iterested reader. As i the case above where is composite, the process is labor itesive, but producig the best i almost all istaces requires a greater effort. For example, for, 6 there are possible uit fractio decompositios saved from use of the above completed algorithm. The cadidates are give usig the format (p, factor or sum, deomiator of last term): (,, 6( )), (,, 6( )), ( 6, 9, 6( 8)), ( 6, 6, 6( 8)), 5 ( 6, 6, 6( 6)), 6 ( 9,, 6( 9)), 7 ( 0, 0 5, 6( 0)), 8 ( 0, 0 8, 6( 0)), 9 (,, 6( )), 0 (, 6, 6( )), ( 5, 5 9 5, 6( 9)), ( 8, 8, 6( 6)), ad ( 5, 7 8, 6( 7)). Choosig the two with the smallest last deomiators give the uit fractio decompositios a ad 0 6( 0) 6( 8) 6( 0) b 5 6( ) 6( 5) 6( 9). Both
A SECOND SOLUTION FOR THE RHIND PAPYRUS 5 cadidates have positives to cosider, but because of the power of as discussed above, decompositio a is used for the table etry. After the developed algorithm has bee applied to a odd prime ad uit fractio decompositios have bee saved, the, as i the composite case, ad the example above, size matters. Also, eve umbers cotiue to be good choices makig calculatios easier. However, each etry presets its ow special circumstaces that impact the selectio process makig a oe fit all selectio process impossible whe searchig for the best choice. As a geeral rule, from the choices pick the two with smallest last term deomiators. This was the case for. 6 If oe of the selectios is a two term uit fractio decompositio ad the deomiator of the last term of the two term uit fractio is the smaller, the, as a geeral rule, that two term uit fractio decompositio is the table etry. Otherwise the other uit fractio decompositio is the table etry. For example, for, 5 5 7 the two selected are ad the table etry is the first oe. For, 0 0 7 5 ad 5 the choices are ad, the last of which is the etry i the table. 9 5 7 5 68 A exceptio occurs whe the compared deomiator is smaller, but odd, ad the deomiators of the two term uit fractio decompositio are eve, i which case the two term uit fractio decompositio is the table etry. For example, for, the two 7 choices are 7 ad 7 with the first oe the table etry. 6 Whe the two selected are three term uit fractio decompositios, to esure easiest use for calculatios, the focus is o the deomiators of the last two terms for each decompositio. If the deomiator of the last term with smaller deomiator is eve ad, for the other, is odd, the the oe with the smaller last deomiator is selected. For example, for, 9 the choices are ad 9( ) 9( 6) ad the first decompositio is the etry i the table. I a 9( ) 9( 7) cosistet maer, the same thig happes whe the two selected decompositios are
5 a three term ad a four term decompositio as see for. decompositios are ad ( 6) ( 8) 6 The two selected ad the ( ) ( ) ( 9) first is the etry i the table. Whe comparig a three ad four term decompositio both havig eve last deomiator, the smallest secod from last deomiator becomes a factor for cosideratio. For example, for, 7 0 ad 7( ) 7( 0) 0 the two choices are with the first the table 7( ) 7( 5) 7( 8) etry. A similar situatio occurs whe comparig two four term uit fractio decompositios. The remaiig etries are left as a exercise to give the iterested reader the opportuity to retrace the process used by the creator of the table. The remaiig etries i the table are give below for the iterested readers to check their aswers:,,,, 6 6 66 8 5 0 76 58 7,, 0 55 7 96,,, 86 9 0 5 0 8 795 59 6 6 5,,, 5 56 7 0 568 70 7 60 9 9 65,, 7 6 790 8 60 5 98, ad. 890 97 56 679 776 89 60 56 9 67 79 Because the method of costructio of the table was so time cosumig ad labor itesive, the creator recorded his or her creatio i writte form to avoid havig to recreate it. As witessed above, the creator s focus was to obtai the best uit fractio decompositio for each fractio i the table based o the eed at that time. Uder such circumstaces, to try to reverse the process by usig the table to explai the idividual etries i the table, as may have tried, is self defeatig resultig oly with patters for some of the etries withi the table. Ulike may moder day creators that require all to fit oe mold quickly ad 0 60 5
A SECOND SOLUTION FOR THE RHIND PAPYRUS 55 easily, whether or ot it is a good fit, the creator of the table withi the Rhid Papyrus made sure each part was the best fit for the eed, makig the whole a masterpiece ot oly for his or her time, but for all time. For the creator of the table to make a writte record of the method of costructio of the table would have bee extremely difficult: the otatio of the period was totally iadequate for such a major udertakig ad much space would be required for the explaatio of how the cadidates for each etry were determied ad the, eve more space for a explaatio of how the cadidates were arrowed to the best choice, makig eve the availability of writig material a obstacle. As a result, most likely, a very small umber of people kew through verbal discussios the method of costructio of the table ad, oce those people were goe, the method of costructio of the table was a mystery to eve the faithful users of the table. As is true for all ma-made creatios, with of without explaatio, the true test of value for the creatio is efficiet, depedable, lastig utility of the creatio for as log as there is a eed for the creatio. Almost 00 years after its creatio, the creator s uit fractio decompositio table cotiued to be the best of the uit fractio decompositio tables. To recogize the table s importace ad to esure its cotiued availability, Ahmes icluded it i his papyrus ad, as a result, preserved the work of a mathematical master for all time. Refereces [] D. Burto, The History of Mathematics: A Itroductio, McGraw Hill, 6th ed., 007. [] C. Dorsett, A solutio for the Rhid Papyrus uit fractio decompositios, Texas College Mathematics Joural 5() (008), -.