The Reisner Papyri date to the reign of Senusret I , who was king of ancient Egypt in the 19th century BCE. The documents were discovered by G.A. Reisner during excavations in 1901–04 in Naga ed-Deir in southern Egypt. A total of four papyrus rolls were found in a wooden coffin in a tomb.
63-432: Several sections contain tables with mathematical content. Section G consistes of 19 lines of text. In the first line the column headings are given: length ( 3w ), width ( wsx ), thickness or depth ( mDwt ), units, product/volume ( sty ), and in the last column the calculations of the number of workers needed for the work of that day. The format of the table in section H is similar to that of section G. In this document only
126-766: A Scottish antiquarian, purchased two parts of the papyrus in 1858 in Luxor, Egypt ; it was stated to have been found in "one of the small buildings near the Ramesseum ", near Luxor. The British Museum, where the majority of the papyrus is now kept, acquired it in 1865 along with the Egyptian Mathematical Leather Roll , also owned by Henry Rhind. Fragments of the text were independently purchased in Luxor by American Egyptologist Edwin Smith in
189-465: A Scottish antiquarian, purchased two parts of the papyrus in 1858 in Luxor, Egypt ; it was stated to have been found in "one of the small buildings near the Ramesseum ", near Luxor. The British Museum, where the majority of the papyrus is now kept, acquired it in 1865 along with the Egyptian Mathematical Leather Roll , also owned by Henry Rhind. Fragments of the text were independently purchased in Luxor by American Egyptologist Edwin Smith in
252-465: A "quadruple ro". The quadruple heqat and the quadruple ro are units of volume derived from the simpler heqat and ro, such that these four units of volume satisfy the following relationships: 1 quadruple heqat = 4 heqat = 1280 ro = 320 quadruple ro. Thus, Problems 48–55 show how to compute an assortment of areas . Problem 48 is notable in that it succinctly computes the area of a circle by approximating π . Specifically, problem 48 explicitly reinforces
315-465: A "quadruple ro". The quadruple heqat and the quadruple ro are units of volume derived from the simpler heqat and ro, such that these four units of volume satisfy the following relationships: 1 quadruple heqat = 4 heqat = 1280 ro = 320 quadruple ro. Thus, Problems 48–55 show how to compute an assortment of areas . Problem 48 is notable in that it succinctly computes the area of a circle by approximating π . Specifically, problem 48 explicitly reinforces
378-507: A certain number of loaves of bread by 10 men and record the outcome in unit fractions. Problems 7–20 show how to multiply the expressions 1 + 1/2 + 1/4 = 7/4, and 1 + 2/3 + 1/3 = 2 by different fractions. Problems 21–23 are problems in completion, which in modern notation are simply subtraction problems. Problems 24–34 are ‘‘aha’’ problems; these are linear equations . Problem 32 for instance corresponds (in modern notation) to solving x + 1/3 x + 1/4 x = 2 for x. Problems 35–38 involve divisions of
441-507: A certain number of loaves of bread by 10 men and record the outcome in unit fractions. Problems 7–20 show how to multiply the expressions 1 + 1/2 + 1/4 = 7/4, and 1 + 2/3 + 1/3 = 2 by different fractions. Problems 21–23 are problems in completion, which in modern notation are simply subtraction problems. Problems 24–34 are ‘‘aha’’ problems; these are linear equations . Problem 32 for instance corresponds (in modern notation) to solving x + 1/3 x + 1/4 x = 2 for x. Problems 35–38 involve divisions of
504-594: A collection of 21 arithmetic and 20 algebraic problems. The problems start out with simple fractional expressions, followed by completion ( sekem ) problems and more involved linear equations ( aha problems ). The first part of the papyrus is taken up by the 2/ n table . The fractions 2/ n for odd n ranging from 3 to 101 are expressed as sums of unit fractions . For example, 2 15 = 1 10 + 1 30 {\displaystyle {\frac {2}{15}}={\frac {1}{10}}+{\frac {1}{30}}} . The decomposition of 2/ n into unit fractions
567-594: A collection of 21 arithmetic and 20 algebraic problems. The problems start out with simple fractional expressions, followed by completion ( sekem ) problems and more involved linear equations ( aha problems ). The first part of the papyrus is taken up by the 2/ n table . The fractions 2/ n for odd n ranging from 3 to 101 are expressed as sums of unit fractions . For example, 2 15 = 1 10 + 1 30 {\displaystyle {\frac {2}{15}}={\frac {1}{10}}+{\frac {1}{30}}} . The decomposition of 2/ n into unit fractions
630-599: A full restoration of scribal division around 1906. In summary, the Reisner Papyri was built upon a method described in the Akhmim Wooden Tablet, and later followed by Ahmes writing the RMP. The Reisner calculations apparently follows our modern Occam's Razor rule, that the simplest method was the historical method; in this case remainder arithmetic, such that: n/10 = Q + R/10 where Q was a quotient and R
693-737: A major fragment of Akhmim Wooden Tablet and Reisner Papyrus remainder arithmetic. That is, Gillings' citation in the Reisner and RMP documented in the "Mathematics in the Time of the Pharaohs" only scratched the surface of scribal arithmetic. Had scholars dug a little deeper, academics may have found 80 years ago other reasons for the Reisner Papyrus 39/10 error. The Reisner Papyrus error may have been noted by Gillings as using quotients (Q) and remainders (R). Ahmes used quotients and remainders in
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#1732772600558756-520: A poor approximation to the correct value, reported Gillings. Gillings fairly reported that the scribe should have stated the problem and data as: Yet, all other the division by 10 problems and answers were correctly stated, points that Gillings did not stress. Table 22.2 data described the work done in the Eastern Chapel. Additional raw data was listed on lines G5, G6/H32, G14, G15, G16, G17/H33 and G18/H34, as follows: Chace and Shute had noted
819-434: A title page, the 2/n table, a tiny "1–9/10 table", and 91 problems, or "numbers". The latter are numbered from 1 through 87 and include four mathematical items which have been designated by moderns as problems 7B, 59B, 61B, and 82B. Numbers 85–87, meanwhile, are not mathematical items forming part of the body of the document, but instead are respectively: a small phrase ending the document, a piece of "scrap-paper" used to hold
882-434: A title page, the 2/n table, a tiny "1–9/10 table", and 91 problems, or "numbers". The latter are numbered from 1 through 87 and include four mathematical items which have been designated by moderns as problems 7B, 59B, 61B, and 82B. Numbers 85–87, meanwhile, are not mathematical items forming part of the body of the document, but instead are respectively: a small phrase ending the document, a piece of "scrap-paper" used to hold
945-572: Is never more than 4 terms long as in for example: This table is followed by a much smaller, tiny table of fractional expressions for the numbers 1 through 9 divided by 10. For instance the division of 7 by 10 is recorded as: After these two tables, the papyrus records 91 problems altogether, which have been designated by moderns as problems (or numbers) 1–87, including four other items which have been designated as problems 7B, 59B, 61B and 82B. Problems 1–7, 7B and 8–40 are concerned with arithmetic and elementary algebra. Problems 1–6 compute divisions of
1008-572: Is never more than 4 terms long as in for example: This table is followed by a much smaller, tiny table of fractional expressions for the numbers 1 through 9 divided by 10. For instance the division of 7 by 10 is recorded as: After these two tables, the papyrus records 91 problems altogether, which have been designated by moderns as problems (or numbers) 1–87, including four other items which have been designated as problems 7B, 59B, 61B and 82B. Problems 1–7, 7B and 8–40 are concerned with arithmetic and elementary algebra. Problems 1–6 compute divisions of
1071-519: Is one of the best known examples of ancient Egyptian mathematics . It is one of two well-known mathematical papyri, along with the Moscow Mathematical Papyrus . The Rhind Papyrus is the larger, but younger, of the two. In the papyrus' opening paragraphs Ahmes presents the papyrus as giving "Accurate reckoning for inquiring into things, and the knowledge of all things, mysteries ... all secrets". He continues: This book
1134-439: Is one of the best known examples of ancient Egyptian mathematics . It is one of two well-known mathematical papyri, along with the Moscow Mathematical Papyrus . The Rhind Papyrus is the larger, but younger, of the two. In the papyrus' opening paragraphs Ahmes presents the papyrus as giving "Accurate reckoning for inquiring into things, and the knowledge of all things, mysteries ... all secrets". He continues: This book
1197-780: Is reported as follows: The solution to the problem is given as the ratio of half the side of the base of the pyramid to its height, or the run-to-rise ratio of its face. In other words, the quantity found for the seked is the cotangent of the angle to the base of the pyramid and its face. The third part of the Rhind papyrus consists of the remainder of the 91 problems, being 61, 61B, 62–82, 82B, 83–84, and "numbers" 85–87, which are items that are not mathematical in nature. This final section contains more complicated tables of data (which frequently involve Horus eye fractions), several pefsu problems which are elementary algebraic problems concerning food preparation, and even an amusing problem (79) which
1260-728: Is reported as follows: The solution to the problem is given as the ratio of half the side of the base of the pyramid to its height, or the run-to-rise ratio of its face. In other words, the quantity found for the seked is the cotangent of the angle to the base of the pyramid and its face. The third part of the Rhind papyrus consists of the remainder of the 91 problems, being 61, 61B, 62–82, 82B, 83–84, and "numbers" 85–87, which are items that are not mathematical in nature. This final section contains more complicated tables of data (which frequently involve Horus eye fractions), several pefsu problems which are elementary algebraic problems concerning food preparation, and even an amusing problem (79) which
1323-452: Is suggestive of geometric progressions, geometric series, and certain later problems and riddles in history. Problem 79 explicitly cites, "seven houses, 49 cats, 343 mice, 2401 ears of spelt, 16807 hekats." In particular problem 79 concerns a situation in which 7 houses each contain seven cats, which all eat seven mice, each of which would have eaten seven ears of grain, each of which would have produced seven measures of grain. The third part of
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#17327726005581386-452: Is suggestive of geometric progressions, geometric series, and certain later problems and riddles in history. Problem 79 explicitly cites, "seven houses, 49 cats, 343 mice, 2401 ears of spelt, 16807 hekats." In particular problem 79 concerns a situation in which 7 houses each contain seven cats, which all eat seven mice, each of which would have eaten seven ears of grain, each of which would have produced seven measures of grain. The third part of
1449-485: Is thought to describe events during the " Hyksos domination", a period of external interruption in ancient Egyptian society which is closely related with its second intermediary period. With these non-mathematical yet historically and philologically intriguing errata, the papyrus's writing comes to an end. Much of the Rhind Papyrus's material is concerned with Ancient Egyptian units of measurement and especially
1512-422: Is thought to describe events during the " Hyksos domination", a period of external interruption in ancient Egyptian society which is closely related with its second intermediary period. With these non-mathematical yet historically and philologically intriguing errata, the papyrus's writing comes to an end. Much of the Rhind Papyrus's material is concerned with Ancient Egyptian units of measurement and especially
1575-562: The English Egyptologist T. Eric Peet and contains a discussion of the text that followed Francis Llewellyn Griffith 's Book I, II and III outline. Chace published a compendium in 1927–29 which included photographs of the text. A more recent overview of the Rhind Papyrus was published in 1987 by Robins and Shute. The Rhind Mathematical Papyrus dates to the Second Intermediate Period of Egypt . It
1638-411: The English Egyptologist T. Eric Peet and contains a discussion of the text that followed Francis Llewellyn Griffith 's Book I, II and III outline. Chace published a compendium in 1927–29 which included photographs of the text. A more recent overview of the Rhind Papyrus was published in 1987 by Robins and Shute. The Rhind Mathematical Papyrus dates to the Second Intermediate Period of Egypt . It
1701-461: The RMP's first six problems. Gillings may have forgotten to summarize his findings in a rigorous manner, showing that several Middle Kingdom texts had used quotients and remainders. Seen in a broader sense the Reisner Papyrus data should be noted as: such that: with 9/10 being converted to a unit fraction series following rules set down in the AWT, and followed in RMP and other texts. Confirmation of
1764-528: The Reisner Papyrus division by 10 method, also applied in the RMP. Chace, nor Shute, clearly cite the quotients and remainders that were used by Ahmes. Other additive scholars have also muddled the reading the first 6 problems of the Rhind Mathematical Papyrus , missing its use of quotient and remainders. Gillings, Chace and Shute apparently had not analyzed the RMP data in a broader context, and reported its older structure, thereby missing
1827-422: The Reisner and RMP to convert vulgar fractions to unit fraction series look similar to the conversion methods used in the Egyptian Mathematical Leather Roll . Gillings repeated a common and incomplete view of the Reisner Papyrus. He analyzed lines G10, from table 22.3B, and line 17 from Table 22.2 on page 221, in the "Mathematics in the Time of the Pharaohs", citing these Reisner Papyrus facts: divide 39 by 10 = 4,
1890-538: The Rhind papyrus is therefore a kind of miscellany, building on what has already been presented. Problem 61 is concerned with multiplications of fractions. Problem 61B, meanwhile, gives a general expression for computing 2/3 of 1/n, where n is odd. In modern notation the formula given is The technique given in 61B is closely related to the derivation of the 2/n table. Problems 62–68 are general problems of an algebraic nature. Problems 69–78 are all pefsu problems in some form or another. They involve computations regarding
1953-538: The Rhind papyrus is therefore a kind of miscellany, building on what has already been presented. Problem 61 is concerned with multiplications of fractions. Problem 61B, meanwhile, gives a general expression for computing 2/3 of 1/n, where n is odd. In modern notation the formula given is The technique given in 61B is closely related to the derivation of the 2/n table. Problems 62–68 are general problems of an algebraic nature. Problems 69–78 are all pefsu problems in some form or another. They involve computations regarding
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2016-413: The amount of feed necessary for various animals, such as fowl and oxen. However, these problems, especially 84, are plagued by pervasive ambiguity, confusion, and simple inaccuracy. The final three items on the Rhind papyrus are designated as "numbers" 85–87, as opposed to "problems", and they are scattered widely across the papyrus's back side, or verso. They are, respectively, a small phrase which ends
2079-413: The amount of feed necessary for various animals, such as fowl and oxen. However, these problems, especially 84, are plagued by pervasive ambiguity, confusion, and simple inaccuracy. The final three items on the Rhind papyrus are designated as "numbers" 85–87, as opposed to "problems", and they are scattered widely across the papyrus's back side, or verso. They are, respectively, a small phrase which ends
2142-571: The column heading product/volume is used however, and there is no column recording the number of workers required. Section I closely resembles section H. Columns recording the length, width, height and product/volume are presented. In this case there are no column headings written down by the scribe. The text is damaged in places but can be reconstructed. The units are cubits except where the scribe mentions palms. The square brackets indicate added or reconstructed text. Gillings and other scholars accepted 100-year-old views of this document, with several of
2205-423: The convention (used throughout the geometry section) that "a circle's area stands to that of its circumscribing square in the ratio 64/81." Equivalently, the papyrus approximates π as 256/81, as was already noted above in the explanation of problem 41. Other problems show how to find the area of rectangles, triangles and trapezoids. The final six problems are related to the slopes of pyramids . A seked problem
2268-423: The convention (used throughout the geometry section) that "a circle's area stands to that of its circumscribing square in the ratio 64/81." Equivalently, the papyrus approximates π as 256/81, as was already noted above in the explanation of problem 41. Other problems show how to find the area of rectangles, triangles and trapezoids. The final six problems are related to the slopes of pyramids . A seked problem
2331-413: The dimensional analysis used to convert between them. A concordance of units of measurement used in the papyrus is given in the image. This table summarizes the content of the Rhind Papyrus by means of a concise modern paraphrase. It is based upon the two-volume exposition of the papyrus which was published by Arnold Buffum Chace in 1927, and in 1929. In general, the papyrus consists of four sections:
2394-413: The dimensional analysis used to convert between them. A concordance of units of measurement used in the papyrus is given in the image. This table summarizes the content of the Rhind Papyrus by means of a concise modern paraphrase. It is based upon the two-volume exposition of the papyrus which was published by Arnold Buffum Chace in 1927, and in 1929. In general, the papyrus consists of four sections:
2457-406: The document (and has a few possibilities for translation, given below), a piece of scrap paper unrelated to the body of the document, used to hold it together (yet containing words and Egyptian fractions which are by now familiar to a reader of the document), and a small historical note which is thought to have been written some time after the completion of the body of the papyrus's writing. This note
2520-406: The document (and has a few possibilities for translation, given below), a piece of scrap paper unrelated to the body of the document, used to hold it together (yet containing words and Egyptian fractions which are by now familiar to a reader of the document), and a small historical note which is thought to have been written some time after the completion of the body of the papyrus's writing. This note
2583-421: The document together (having already contained unrelated writing), and a historical note which is thought to describe a time period shortly after the completion of the body of the papyrus. These three latter items are written on disparate areas of the papyrus's verso (back side), far away from the mathematical content. Chace therefore differentiates them by styling them as numbers as opposed to problems , like
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2646-421: The document together (having already contained unrelated writing), and a historical note which is thought to describe a time period shortly after the completion of the body of the papyrus. These three latter items are written on disparate areas of the papyrus's verso (back side), far away from the mathematical content. Chace therefore differentiates them by styling them as numbers as opposed to problems , like
2709-4601: The following multiplications, write the product as an Egyptian fraction. 9 : ( 1 2 + 1 14 ) S = 1 ; 10 : ( 1 4 + 1 28 ) S = 1 2 ; 11 : 1 7 S = 1 4 {\displaystyle 9:{\bigg (}{\frac {1}{2}}+{\frac {1}{14}}{\bigg )}S=1\;\;\;;\;\;\;10:{\bigg (}{\frac {1}{4}}+{\frac {1}{28}}{\bigg )}S={\frac {1}{2}}\;\;\;;\;\;\;11:{\frac {1}{7}}S={\frac {1}{4}}} 12 : 1 14 S = 1 8 ; 13 : ( 1 16 + 1 112 ) S = 1 8 ; 14 : 1 28 S = 1 16 {\displaystyle 12:{\frac {1}{14}}S={\frac {1}{8}}\;\;\;;\;\;\;13:{\bigg (}{\frac {1}{16}}+{\frac {1}{112}}{\bigg )}S={\frac {1}{8}}\;\;\;;\;\;\;14:{\frac {1}{28}}S={\frac {1}{16}}} 15 : ( 1 32 + 1 224 ) S = 1 16 ; 16 : 1 2 T = 1 ; 17 : 1 3 T = 2 3 {\displaystyle 15:{\bigg (}{\frac {1}{32}}+{\frac {1}{224}}{\bigg )}S={\frac {1}{16}}\;\;\;;\;\;\;16:{\frac {1}{2}}T=1\;\;\;;\;\;\;17:{\frac {1}{3}}T={\frac {2}{3}}} 18 : 1 6 T = 1 3 ; 19 : 1 12 T = 1 6 ; 20 : 1 24 T = 1 12 {\displaystyle 18:{\frac {1}{6}}T={\frac {1}{3}}\;\;\;;\;\;\;19:{\frac {1}{12}}T={\frac {1}{6}}\;\;\;;\;\;\;20:{\frac {1}{24}}T={\frac {1}{12}}} 22 : ( 2 3 + 1 30 ) + x = 1 → x = 1 5 + 1 10 {\displaystyle 22:{\bigg (}{\frac {2}{3}}+{\frac {1}{30}}{\bigg )}+x=1\;\;\;\rightarrow \;\;\;x={\frac {1}{5}}+{\frac {1}{10}}} 23 : ( 1 4 + 1 8 + 1 10 + 1 30 + 1 45 ) + x = 2 3 → x = 1 9 + 1 40 {\displaystyle 23:{\bigg (}{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{10}}+{\frac {1}{30}}+{\frac {1}{45}}{\bigg )}+x={\frac {2}{3}}\;\;\;\rightarrow \;\;\;x={\frac {1}{9}}+{\frac {1}{40}}} 24 : x + 1 7 x = 19 → x = 16 + 1 2 + 1 8 {\displaystyle 24:x+{\frac {1}{7}}x=19\;\;\;\rightarrow \;\;\;x=16+{\frac {1}{2}}+{\frac {1}{8}}} 25 : x + 1 2 x = 16 → x = 10 + 2 3 {\displaystyle 25:x+{\frac {1}{2}}x=16\;\;\;\rightarrow \;\;\;x=10+{\frac {2}{3}}} 26 : x + 1 4 x = 15 → x = 12 {\displaystyle 26:x+{\frac {1}{4}}x=15\;\;\;\rightarrow \;\;\;x=12} 27 : x + 1 5 x = 21 → x = 17 + 1 2 {\displaystyle 27:x+{\frac {1}{5}}x=21\;\;\;\rightarrow \;\;\;x=17+{\frac {1}{2}}} 28 : ( x + 2 3 x ) − 1 3 ( x + 2 3 x ) = 10 → x = 9 {\displaystyle 28:{\bigg (}x+{\frac {2}{3}}x{\bigg )}-{\frac {1}{3}}{\bigg (}x+{\frac {2}{3}}x{\bigg )}=10\;\;\;\rightarrow \;\;\;x=9} 29 : 1 3 ( ( x + 2 3 x ) + 1 3 ( x + 2 3 x ) ) = 10 → x = 13 + 1 2 {\displaystyle 29:{\frac {1}{3}}{\Bigg (}{\bigg (}x+{\frac {2}{3}}x{\bigg )}+{\frac {1}{3}}{\bigg (}x+{\frac {2}{3}}x{\bigg )}{\Bigg )}=10\;\;\;\rightarrow \;\;\;x=13+{\frac {1}{2}}} Rhind Mathematical Papyrus The Rhind Mathematical Papyrus ( RMP ; also designated as papyrus British Museum 10057, pBM 10058, and Brooklyn Museum 37.1784Ea-b)
2772-564: The heqat, which is an ancient Egyptian unit of volume. Beginning at this point, assorted units of measurement become much more important throughout the remainder of the papyrus, and indeed a major consideration throughout the rest of the papyrus is dimensional analysis . Problems 39 and 40 compute the division of loaves and use arithmetic progressions . The second part of the Rhind papyrus, being problems 41–59, 59B and 60, consists of geometry problems. Peet referred to these problems as "mensuration problems". Problems 41–46 show how to find
2835-564: The heqat, which is an ancient Egyptian unit of volume. Beginning at this point, assorted units of measurement become much more important throughout the remainder of the papyrus, and indeed a major consideration throughout the rest of the papyrus is dimensional analysis . Problems 39 and 40 compute the division of loaves and use arithmetic progressions . The second part of the Rhind papyrus, being problems 41–59, 59B and 60, consists of geometry problems. Peet referred to these problems as "mensuration problems". Problems 41–46 show how to find
2898-580: The mid 1860s, were donated by his daughter in 1906 to the New York Historical Society, and are now held by the Brooklyn Museum . An 18 cm (7.1 in) central section is missing. The papyrus began to be transliterated and mathematically translated in the late 19th century. The mathematical-translation aspect remains incomplete in several respects. The first part of the Rhind papyrus consists of reference tables and
2961-435: The mid 1860s, were donated by his daughter in 1906 to the New York Historical Society, and are now held by the Brooklyn Museum . An 18 cm (7.1 in) central section is missing. The papyrus began to be transliterated and mathematically translated in the late 19th century. The mathematical-translation aspect remains incomplete in several respects. The first part of the Rhind papyrus consists of reference tables and
3024-1910: The other 88 numbered items. 4 10 = 1 3 + 1 15 ; 5 10 = 1 2 ; 6 10 = 1 2 + 1 10 {\displaystyle {\frac {4}{10}}={\frac {1}{3}}+{\frac {1}{15}}\;\;\;;\;\;\;{\frac {5}{10}}={\frac {1}{2}}\;\;\;;\;\;\;{\frac {6}{10}}={\frac {1}{2}}+{\frac {1}{10}}} 7 10 = 2 3 + 1 30 ; 8 10 = 2 3 + 1 10 + 1 30 ; 9 10 = 2 3 + 1 5 + 1 30 {\displaystyle {\frac {7}{10}}={\frac {2}{3}}+{\frac {1}{30}}\;\;\;;\;\;\;{\frac {8}{10}}={\frac {2}{3}}+{\frac {1}{10}}+{\frac {1}{30}}\;\;\;;\;\;\;{\frac {9}{10}}={\frac {2}{3}}+{\frac {1}{5}}+{\frac {1}{30}}} 6 10 = 1 2 + 1 10 ; 7 10 = 2 3 + 1 30 {\displaystyle {\frac {6}{10}}={\frac {1}{2}}+{\frac {1}{10}}\;\;\;;\;\;\;{\frac {7}{10}}={\frac {2}{3}}+{\frac {1}{30}}} 8 10 = 2 3 + 1 10 + 1 30 ; 9 10 = 2 3 + 1 5 + 1 30 {\displaystyle {\frac {8}{10}}={\frac {2}{3}}+{\frac {1}{10}}+{\frac {1}{30}}\;\;\;;\;\;\;{\frac {9}{10}}={\frac {2}{3}}+{\frac {1}{5}}+{\frac {1}{30}}} S = 1 + 1 / 2 + 1 / 4 = 7 4 {\displaystyle S=1+1/2+1/4={\frac {7}{4}}} and T = 1 + 2 / 3 + 1 / 3 = 2 {\displaystyle T=1+2/3+1/3=2} . Then for
3087-1910: The other 88 numbered items. 4 10 = 1 3 + 1 15 ; 5 10 = 1 2 ; 6 10 = 1 2 + 1 10 {\displaystyle {\frac {4}{10}}={\frac {1}{3}}+{\frac {1}{15}}\;\;\;;\;\;\;{\frac {5}{10}}={\frac {1}{2}}\;\;\;;\;\;\;{\frac {6}{10}}={\frac {1}{2}}+{\frac {1}{10}}} 7 10 = 2 3 + 1 30 ; 8 10 = 2 3 + 1 10 + 1 30 ; 9 10 = 2 3 + 1 5 + 1 30 {\displaystyle {\frac {7}{10}}={\frac {2}{3}}+{\frac {1}{30}}\;\;\;;\;\;\;{\frac {8}{10}}={\frac {2}{3}}+{\frac {1}{10}}+{\frac {1}{30}}\;\;\;;\;\;\;{\frac {9}{10}}={\frac {2}{3}}+{\frac {1}{5}}+{\frac {1}{30}}} 6 10 = 1 2 + 1 10 ; 7 10 = 2 3 + 1 30 {\displaystyle {\frac {6}{10}}={\frac {1}{2}}+{\frac {1}{10}}\;\;\;;\;\;\;{\frac {7}{10}}={\frac {2}{3}}+{\frac {1}{30}}} 8 10 = 2 3 + 1 10 + 1 30 ; 9 10 = 2 3 + 1 5 + 1 30 {\displaystyle {\frac {8}{10}}={\frac {2}{3}}+{\frac {1}{10}}+{\frac {1}{30}}\;\;\;;\;\;\;{\frac {9}{10}}={\frac {2}{3}}+{\frac {1}{5}}+{\frac {1}{30}}} S = 1 + 1 / 2 + 1 / 4 = 7 4 {\displaystyle S=1+1/2+1/4={\frac {7}{4}}} and T = 1 + 2 / 3 + 1 / 3 = 2 {\displaystyle T=1+2/3+1/3=2} . Then for
3150-583: The scribal remainder arithmetic is found in other hieratic texts. The most important text is the Akhmim Wooden Tablet . The AWT defines scribal remainder arithmetic in term of another context, a hekat (volume unit) . Oddly, Gillings did not cite AWT data in "Mathematics in the Time of the Pharaohs". Gillings and the earlier 1920s scholars had missed a major opportunity to point out a multiple use of scribal remainder arithmetic built upon quotient and remainders. The modern looking remainder arithmetic
3213-407: The strength of bread and beer, with respect to certain raw materials used in their production. Problem 79 sums five terms in a geometric progression . Its language is strongly suggestive of the more modern riddle and nursery rhyme " As I was going to St Ives ". Problems 80 and 81 compute Horus eye fractions of hinu (or heqats). The last four mathematical items, problems 82, 82B and 83–84, compute
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#17327726005583276-407: The strength of bread and beer, with respect to certain raw materials used in their production. Problem 79 sums five terms in a geometric progression . Its language is strongly suggestive of the more modern riddle and nursery rhyme " As I was going to St Ives ". Problems 80 and 81 compute Horus eye fractions of hinu (or heqats). The last four mathematical items, problems 82, 82B and 83–84, compute
3339-413: The value of π as being 3.1605..., an error of less than one percent. Problem 47 is a table with fractional equalities which represent the ten situations where the physical volume quantity of "100 quadruple heqats" is divided by each of the multiples of ten, from ten through one hundred. The quotients are expressed in terms of Horus eye fractions, sometimes also using a much smaller unit of volume known as
3402-413: The value of π as being 3.1605..., an error of less than one percent. Problem 47 is a table with fractional equalities which represent the ten situations where the physical volume quantity of "100 quadruple heqats" is divided by each of the multiples of ten, from ten through one hundred. The quotients are expressed in terms of Horus eye fractions, sometimes also using a much smaller unit of volume known as
3465-463: The views being incomplete and misleading. Two of the documents, reported in Tables 22.2 and 22.2, a detail a division by 10 method, a method that also appears in the Rhind Mathematical Papyrus . Labor efficiencies were monitored by applying this method. For example, how deep did 10 workmen dig in one day as calculated in the Reisner Papyrus, and by Ahmes 150 years later? In addition, the methods used in
3528-500: The volume of both cylindrical and rectangular granaries. In problem 41 Ahmes computes the volume of a cylindrical granary. Given the diameter d and the height h, the volume V is given by: In modern mathematical notation (and using d = 2r) this gives V = ( 8 / 9 ) 2 d 2 h = ( 256 / 81 ) r 2 h {\displaystyle V=(8/9)^{2}d^{2}h=(256/81)r^{2}h} . The fractional term 256/81 approximates
3591-500: The volume of both cylindrical and rectangular granaries. In problem 41 Ahmes computes the volume of a cylindrical granary. Given the diameter d and the height h, the volume V is given by: In modern mathematical notation (and using d = 2r) this gives V = ( 8 / 9 ) 2 d 2 h = ( 256 / 81 ) r 2 h {\displaystyle V=(8/9)^{2}d^{2}h=(256/81)r^{2}h} . The fractional term 256/81 approximates
3654-440: Was a remainder. The Reisner, following this Occam's Razor rule, says that 10 workmen units were used to divide raw data using a method that was defined in the text, a method that also begins the Rhind Mathematical Papyrus , as noted in its first six problems. Rhind Mathematical Papyrus The Rhind Mathematical Papyrus ( RMP ; also designated as papyrus British Museum 10057, pBM 10058, and Brooklyn Museum 37.1784Ea-b)
3717-503: Was copied by the scribe Ahmes (i.e., Ahmose; Ahmes is an older transcription favoured by historians of mathematics) from a now-lost text from the reign of the 12th dynasty king Amenemhat III . It dates to around 1550 BC. The document is dated to Year 33 of the Hyksos king Apophis and also contains a separate later historical note on its verso likely dating from "Year 11" of his successor, Khamudi . Alexander Henry Rhind ,
3780-451: Was copied by the scribe Ahmes (i.e., Ahmose; Ahmes is an older transcription favoured by historians of mathematics) from a now-lost text from the reign of the 12th dynasty king Amenemhat III . It dates to around 1550 BC. The document is dated to Year 33 of the Hyksos king Apophis and also contains a separate later historical note on its verso likely dating from "Year 11" of his successor, Khamudi . Alexander Henry Rhind ,
3843-546: Was copied in regnal year 33, month 4 of Akhet , under the majesty of the King of Upper and Lower Egypt, Awserre, given life, from an ancient copy made in the time of the King of Upper and Lower Egypt Nimaatre. The scribe Ahmose writes this copy. Several books and articles about the Rhind Mathematical Papyrus have been published, and a handful of these stand out. The Rhind Papyrus was published in 1923 by
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#17327726005583906-415: Was copied in regnal year 33, month 4 of Akhet , under the majesty of the King of Upper and Lower Egypt, Awserre, given life, from an ancient copy made in the time of the King of Upper and Lower Egypt Nimaatre. The scribe Ahmose writes this copy. Several books and articles about the Rhind Mathematical Papyrus have been published, and a handful of these stand out. The Rhind Papyrus was published in 1923 by
3969-443: Was later found by others by taking a broader view of the 39/10 error, as corrected as the actual Eastern Chapel data reports. Gillings and the academic community therefore had inadvertently omitted a critically important discussion of fragments of remainder arithmetic. Remainder arithmetic, as used in many ancient cultures to solve astronomy and time problems, is one of several plausible historical division methods that may have allowed
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