The Rhind Mathematical Papyrus dates to the Second Intermediate Period of Egypt and is the best example of Egyptian mathematics. It 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 king Amenemhat III (12th dynasty). Written in the hieratic script, this Egyptian manuscript is 33 cm tall and over 5 meters long, and began to be transliterated and mathematically translated in the late 19th century. In 2008, the mathematical translation aspect is incomplete in several respects. The document is dated to Year 33 of the Hyksos king Apophis and also contains a separate later Year 11 on its verso likely from his successor, Khamudi.
In the opening paragraphs of the papyrus, Ahmes presents the papyrus as giving “Accurate reckoning for inquiring into things, and the knowledge of all things, mysteries...all secrets”.
* = = = + +
Ahmes often listed the additive (19 + 3 + 2) numerator of his shorthand info, omitting other data, thereby confusing math historians for 100 years (with several suggesting that 2/n table conversions had only been additive based).
The RMP's 84 problems begin with six division-by-10 problems, the central subject of the Reisner Papyrus. There are 15 problems dealing with addition, and 18 algebra problems. There are 15 algebra problems of the same type. They ask the reader to find x and a fraction of x such that the sum of x and its fraction equals a given integer. Problem #24 is the easiest, and asks the reader to solve this equation, x + 1/7x = 19. Ahmes, the author of the RMP, worked the problem this way:
(8/7)x = 19, or x = 133/8 = 16 + 5/8,
with 133/8 being the initial vulgar fraction find 16 as the quotient and 5/8 as the remainder term. Ahmes converted 5/8 to an Egyptian fraction series by (4 + 1)/8 = 1/2 + 1/8, making his final quotient plus remainder based answer x = 16 + 1/2 + 1/8.
Each of the RMP's other 14 algebra problems produced increasingly difficult vulgar fractions. Yet, all were easily converted to an optimal (short and small last term) Egyptian fraction series.
Two arithmetical progressions (A.P.) were solved, one being RMP 64. The method of solution followed the method defined in the Kahun Papyrus. The problem solved sharing 10 hekats of barley, between 10 men, by a difference of 1/8th of a hekat finding 1 7/16 as the largest term.
The second A.P. was RMP 40, the problem divided 100 loaves of bread between five men such that the smallest two shares (12 1/2) were 1/7 of the largest three shares' sum (87 1/2). The problem asked Ahmes to find the shares for each man, which he did without finding the difference (9 1/6) or the largest term (38 1/3). All five shares 38 1/3, 29 1/6, 20, 10 2/3 1/6, and 1 1/3) were calculated by first finding the five terms from a proportional A.P. that summed to 60. The median and the smallest term, x1, were used to find the differential and each term. Ahmes then multiplied each term by 1 2/3 to obtain the sum to 100 A.P. terms. In reproducing the problem in modern algebra, Ahmes also found the sum of the first two terms by solving x + 7x = 60.
The RMP continues with 5 hekat division problems from the Akhmim Wooden Tablet, 15 problems similar to ones from the Moscow Mathematical Papyrus, 23 problems from practical weights and measures, especially the hekat, and three problems from recreational diversion subjects, the last the famous multiple of 7 riddle, written in the Medieval era as, "Going to St. Ives".
The Rhind Mathematical Papyrus also contains the following problem related to trigonometry:
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 he found for the seked is the cotangent of the angle to the base of the pyramid and its face.
The papyrus calculates π as (a margin of error of less than 1%). Two viable theories offering some insight into a possible motivation for such an accurate derivation have been proposed:
The papyrus also demonstrates knowledge of weights and measures, business, and recreational diversions.
The Egyptian use of arithmetic proportions in the Rhind Papyrus, problems 40 and 64, and the Kahun Papyrus, are briefly discussed by Gillings. In particular the use of the Remen, which has two values, is reflected in the foot which has two values, (the second being the nibw or ell which is two feet), and the cubit which has two values. Doubling is also seen in the subdivisions such as fingers and palms. Since doubling seems to have been the basis of most of the unit fraction calculations, which it was not (multiples were) up to and including the calculations of circles with dimensions given in khet (see Ancient Egyptian units of measurement), looking at how the remen and seked were used provided many insights to Greek and Roman geometers and architects. The actual and proposed readings/decodings of the RMP and Kahun 2/n tables is required to be fairly interjected.
In the Rhind Papyrus we first encounter the remen which is defined as the proportion of the diagonal of a rectangle to its sides when its other sides are whole units. Yet, a singular arithmetic proportion formula reported in the RMP and Kahun Papyrus offer an additional example beyond the remen's diagonal of a square, with its sides a cubit. We also find problems using the seked or unit rise to run proportion. Typical of the Classical orders of the Greeks and Romans, it was built upon the canon of proportions derived from the inscription grids of the Egyptians.
This document is one of the main sources of our knowledge of Egyptian mathematics.