Mathematics in Ancient Egypt

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MATHEMATICS IN ANCIENT EGYPT

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The Egyptians used a base-10 system of numbers. Sometimes they used different symbols for 1s, 10s, 100s, 1,000s, 10,000s, 100,000s and 1,000,000s. Other times they used a group of hieroglyphic symbols for individual numbers. Egyptian arithmetic was based on a system binary system like that used to power computers today. Multiplication was done by duplicating numbers and adding the sums together. Division was the same operation performed backwards.

The Egyptians had a decimal system using seven different symbols. 1 is shown by a single stroke. 10 is shown by a drawing of a hobble for cattle. 100 is represented by a coil of rope. 1,000 a drawing of a lotus plant. 10,000 is represented by a finger. 100,000 a tadpole or frog. 1,000,000 figure of a god with arms raised above his head. The conventions for reading and writing numbers is quite simple; the higher number is always written in front of the lower number and where there is more than one row of numbers the reader should start at the top. [Source: Mark Millmore, discoveringegypt.com]

Peter Schjeldahl wrote in The New Yorker: “The Egyptians were perniciosus keepers of lists, and recorders of methods and procedures. They meant for things they made to last forever.”To rule effectively, the ancient Egyptians set up an efficient and extensive administration that levied taxes, conducted censuses, and maintained and supplied a large army — all of which required some mathematics. At first they used counting glyphs. By 2000 B.C., hieratic glyphs were in use.

Use of Mathematics in Ancient Egypt

Mathematics served a practical purpose for the ancient Egyptians. They solved the problems of everyday life and generally didn't formulate and work out problems for their own sake. How certain foodstuffs were to be divided as payments of wages; how, in the exchange of bread for beer the respective value was to be determined when converted into a quantity of grain; how to reckon the size of a field; how to determine whether a given quantity of grain would go into a granary of a certain size — these and similar problems were taught in the arithmetic book. [Source: Adolph Erman, “Life in Ancient Egypt”, 1894] Herodotus wrote: Ramses II “divided the land into lots and gave a square piece of equal size, from the produce of which he exacted an annual tax. [If] any man's holding was damaged by the encroachment of the river ... [The Pharoah] ... would send inspectors [and surveyors] to measure the extent of the loss, in order that he pay in future a fair proportion of the tax at which his property had been assessed. Perhaps this was the way in which geometry was invented, and passed afterwards in to Greece.

Building great monuments like the pyramids also took some mathematical knowledge, Democritus (~410 B.C.) wrote: No one surpasses me in the construction of lines with [the help of a ruler and compass], not even the so-called rope-stretchers (surveyors) among the Egyptians.” Jimmy Dunn of touregypt.net wrote: “Some of the pyramids indicate an accurate understanding of Pi, but the mathematical knowledge of the Egyptians did not include the ability to arrive at this by calculation. It is possible that this could have been arrived at "accidentally" through a means such as counting the revolutions of a drum.”

Ancient Egyptian Mathematical Papyri

Pam Belluck wrote in the New York Times The Rhind papyrus, which dates to 1650 B.C., is one of several precocious papyri and other artifacts displaying Egyptian mathematical ingenuity. There is the Moscow Mathematical Papyrus (held at the Pushkin State Museum of Fine Arts in Moscow), the Egyptian Mathematical Leather Roll (which along with the Rhind papyrus is housed at the British Museum) and the Akhmim Wooden Tablets (at the Museum of Egyptian Antiquities in Cairo). [Source: Pam Belluck, New York Times, December 6, 2010]

The Rhind Mathematical Papyrus is one of the best known examples of Egyptian mathematics. Named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, it was apparently found during illegal excavations in or near the Ramesseum. Dated to around 1550 B.C., it contains 84 different calculations to help with various aspects of Egyptian life such as building a pyramid or determining the amount of grain needed to fatten a goose. [Source: Wikipedia +]

The Moscow Mathematical Papyrus is an ancient Egyptian mathematical papyrus, also called the Golenishchev Mathematical Papyrus. Dated to the 13th dynasty (1803- 1649 B.C.) and based on older material probably dating to the 12th dynasty, it is older but smaller than the Rhind Papyrus. The Moscow Papyrus is approximately 5½ m (18 ft) long and varies between 3.8 and 7.6 cm (1.5 and 3 in) wide.

The Moscow Mathematical Papyrus is divided into 25 problems with solutions. The problems are organized no particular order, and the solutions feature much less detail than those in the Rhind Papyrus. The Moscow papyrus is well known for its geometry problems. Problems 10 and 14 compute a surface area and the volume of a frustum respectively. The remaining problems deal with common subjects. Problems 2 and 3 are related to ship’s parts.Most of the problems are algebraic pefsu problems. A pefsu measures the strength of the beer made from a hekat of grain. +

Math in the Ancient Egyptian Mathematical Texts

Pam Belluck wrote in the New York Times The ancient Egyptian math texts include "methods of measuring a ship’s mast and rudder, calculating the volume of cylinders and truncated pyramids, dividing grain quantities into fractions and verifying how much bread to exchange for beer. They even compute a circle’s area using an early approximation of pi. (They use 256/81, about 3.16, instead of pi’s value of 3.14159....) [Source: Pam Belluck, New York Times, December 6, 2010]

And the documents were practical guides to navigating a maturing civilization and an expanding economy. “Egypt was going from a centralized, structured world to partially being decentralized,” Milo Gardner, an amateur decoder of Egyptian mathematical texts who has written extensively about them, told the New York Times. “They had an economic system that was run by absentee landowners and paid people in units of grain, and in order to make it fair had to have exact weights and measures. They were trying to figure out a way to evenly divide the hekat so they could use it as a unit of currency.”

So the Akhmim tablets, nearly 4,000 years old, contain lists of servants’ names, along with a series of computations concerning how a hekat of grain can be divided by 3, 7, 10, 11 and 13. The Egyptian Mathematical Leather Roll, also from about 1650 B.C., is generally considered a kind of practice test for students to learn how to convert fractions into sums of other fractions.

The problems in these ancient texts are not difficult by modern mathematical standards. The challenge for scholars has come in deciphering what the problems are saying and checking their accuracy. Some of the numerical equivalents are written in a symbolic system called the Eye of Horus, based on a drawing representing the eye of the sky god Horus, depicted as a falcon. Sections of the falcon’s eye are used to represent fractions: one-half, one-quarter and so on, up to one sixty-fourth.

Scholars have found a few errors in the problems, and Ahmes even wrote an incorrect number in his St. Ives problem. But over all, the equations are considered remarkably accurate. “The practical answers are solved,” Mr. Gardner said. “What is unsolved about them is the actual thinking in the scribe’s head. We don’t know exactly how he thought of it.”

Math in the Rhind Papyrus

The first part of the Rhind papyrus consists of reference tables and 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 second part of the Rhind papyrus, being problems 41-59, 59B and 60, consists of geometry problems. Problems 41 – 46 show how to find the volume of both cylindrical and rectangular granaries. 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. [Source: Wikipedia]

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. +

John Burnet wrote in “Early Greek Philosophy”: “The Rhind papyrus in the British Museum gives us a glimpse of arithmetic and geometry as they were understood on the banks of the Nile. It is the work of one Aahmes, and contains rules for calculations both of an arithmetical and a geometrical character. The arithmetical problems mostly concern measures of corn and fruit, and deal particularly with such questions as the division of a number of measures among a given number of persons, the number of loaves or jars of beer that certain measures will yield, and the wages due to the workmen for a certain piece of work.

It corresponds exactly, in fact, to the description of Egyptian arithmetic Plato gives us in the Laws, where he tells us that children learnt along with their letters to solve problems in the distribution of apples and wreaths to greater or smaller numbers of people, the pairing of boxers and wrestlers, and so forth. This is clearly the origin of the art which the Greeks called logistikê, and they probably borrowed that from Egypt, where it was highly developed; but there is no trace of what the Greeks called arithmêtikê, the scientific study of numbers. [Source: John Burnet (1863-1928), “Early Greek Philosophy” London and Edinburgh: A. and C. Black, 1892, 3rd edition, 1920, Evansville University]

“The geometry of the Rhind papyrus is of a similar character, and Herodotus, who tells us that Egyptian geometry arose from the necessity of measuring the land afresh after the inundations, is clearly far nearer the mark than Aristotle, who says it grew out of the leisure enjoyed by the priestly caste. The rules given for calculating areas are only exact when these are rectangular. As fields are usually more or less rectangular, this would be sufficient for practical purposes. It is even assumed that a right-angled triangle can be equilateral. The rule for finding what is called the seqt of a pyramid is, however, on a rather higher level, as we should expect. It comes to this. Given the "length across the sole of the foot," that is, the diagonal of the base, and that of the piremus or "ridge," to find a number which represents the ratio between them. This is done by dividing half the diagonal of the base by the "ridge," and it is obvious that such a method might quite well be discovered empirically. It seems an anachronism to speak of elementary trigonometry in connection with a rule like this, and there is nothing to suggest that the Egyptians went any further.

Ancient Egyptian Geometry

The Egyptians knew still less of geometry than they did of arithmetic, though surface-measurement was most necessary to them, because of the destruction of so many field boundaries in the inundation every year. All their calculations were founded on the right angle, the content of which they correctly determined as the product of the two sides. But in the strangest way they quite overlooked the fact that every quadrilateral figure, in which the opposite sides are of the same length, could not be treated in the same manner. Now as they treated each triangle as if it were a quadrangle, in which two sides are identical, and the others are half the size, they carried this error into the calculation of triangles also. To them also an isosceles triangle equalled half the product of its short and its long side, because they would in all cases determine the quadrangle corresponding to it by the multiplication of its two sides, though it were nothing but a right angle. The error which would arise from this kind of misconception might be considerable under some circumstances. [Source: Adolph Erman, “Life in Ancient Egypt”, 1894]

The calculation of the trapezium ' suffered also from the same error; in order to find its content, they would multiply the oblique side by half the product of the two parallel sides. As we see, the fundamental mistake of these students of surface-measurement was that they ncver realised the value of the perpendicular; instead of the latter they used one of the oblique sides, and therewith from the outset they excluded themselves from the correct manner of working. It is remarkable that, with such errors, they should have rightly determined approximately the difficult question of the area of a circle; in this case they deducted a ninth of the diameter, and multiplied the remainder of the same by itself Thus if the diameter of a circle amounted to 9 rods, they would calculate its area to be 8 X 8 = 64 square rods, a result which would deviate from the correct result by but about r of a square rod.

Amongst the volumetric problems which they attempted, they would calculate, for instance, the quantity of grain which would go into a granary of a certain size. Pam Belluck wrote in the New York Times: The Rhind papyrus contains geometry problems that compute the slopes of pyramids and the volume of various-shaped granaries. And the Moscow papyrus, from about 1850 B.C., has about 25 problems, including ways to measure ships’ parts and find the surface area of a hemisphere and the area of triangles. Especially interesting are problems that calculate how efficient a laborer was by how many logs he carried or how many sandals he could make and decorate. Or the problems that involve a pefsu, a unit measuring the strength or weakness of beer or bread based on how much grain is used to make it.One problem calculates whether it’s right to exchange 100 loaves of 20-pefsu bread for 10 jugs of 4-pefsu malt-date beer. After a series of steps, the papyrus proclaims, according to one translation: “Behold! The beer quantity is found to be correct.” [Source: Pam Belluck, New York Times, December 6, 2010]

Ancient Egyptian Fractions

Mark Millmore wrote in discoveringegypt.com: “All ancient Egyptian fractions, with the exception of 2/3, are unit fractions, that is fractions with numerator 1. For example 1/2, 1/7, 1/34. Unit fractions are written additively: 1/4 1/26 means 1/4 + 1/26. and 1/4 + 1/28 = our 2/7. The hieroglyph for ‘R’ was used as the word ‘part’. In one of the ancient stories the god Seth attacked his brother the god Horus and gouged out his eye and then tore it to pieces. Fortunately for Horus the god Thoth was able to put the pieces back together and heal his eye. In honour of this story the ancient Egyptians also used the pieces of Horus’s eye to describe fractions. The right side of the eye = 1/2. The pupil = 1/4. The eyebrow = 1/8. The left side of the eye = 1/16. The curved tail = 1/32. The teardrop = 1/64. [Source: Mark Millmore, discoveringegypt.com discoveringegypt.com ^^^]

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According to the website “Egyptian Fractions” put out by the University of California, Irvine: “Nowadays, we usually write non-integer numbers either as fractions (2/7) or decimals (0.285714). The floating point representation used in computers is another representation very similar to decimals. But the ancient Egyptians (as far as we can tell from the documents now surviving) used a number system based on unit fractions: fractions with one in the numerator. This idea let them represent numbers like 1/7 easily enough; other numbers such as 2/7 were represented as sums of unit fractions (e.g. 2/7 = 1/4 +1/28). Further, the same fraction could not be used twice (so 2/7 = 1/7 + 1/7 is not allowed). We call a formula representing a sum of distinct unit fractions an Egyptian fraction. [Source: Egyptian Fractions, University of California, Irvine]

“This notation is cumbersome and difficult to compute with, so the Egyptian scribes made large tables so they could look up the answers to arithmetic problems. However there is also some interesting mathematics associated with the problem of converting modern fraction notation into the Egyptian form. A number of famous mathematicians have looked at this problem, and invented different ways of doing this conversion process. Each of these methods has advantages and disadvantages in terms of the complexity of the Egyptian fraction representations it produces and in terms of the amount of time the conversion process takes. There are still some unsolved problems about whether some of these processes finish, or whether they get into an infinite loop.

Ancient Egyptian Math Puzzles

The Rhind Mathematical Papyrus is a 3,600-year-old Egyptian document with roughly 85 problems and puzzles. The brain teasers found in this document — the world’s first known ones — were practical: problems to calculate how efficient a laborer was by how many logs he carried, or to gauge the potency of beer. The very British-sounding St. Ives conundrum (the one where the seven wives each have seven sacks containing seven cats who each have seven kits, and you have to figure out how many are going to St. Ives) is found in the papyrus in an earlier form. [Source: Pam Belluck, New York Times, December 6, 2010]

Pam Belluck wrote in the New York Times, “The Rhind Mathematical Papyrus contains a puzzle of sevens that bears an uncanny likeness to the St. Ives riddle. It has mice and barley, not wives and sacks, but the gist is similar. Seven houses have seven cats that each eat seven mice that each eat seven grains of barley. Each barley grain would have produced seven hekat of grain. (A hekat was a unit of volume, roughly 1.3 gallons.)The goal: to determine how many things are described. The answer: 19,607. (The method: 7 + 7² + 7³ + 74 + 75.)

It all goes to show that making puzzles is “the most ancient of all instincts,” said Marcel Danesi, a puzzle expert and anthropology professor at the University of Toronto, who calls documents like the Rhind papyrus “the first puzzle books in history.” Dr. Danesi says people of all eras and cultures gravitate toward puzzles because puzzles have solutions. “Other philosophical puzzles of life do not,” he continued. “When you do get it you go, “Aha, there it is, damn it,” and it gives you some relief.”

But the Egyptian puzzles were not just recreational diversions seeking the comforting illusion of competence. They were serious about their mission. The Rhind papyrus’s scribe, known as Ahmes, introduced the problems by saying that he is presenting the “correct method of reckoning, for grasping the meaning of things and knowing everything that is, obscurities and all secrets.”

Influence of Egyptian Math on the Ancient Greeks

John Burnet wrote in “Early Greek Philosophy”: “That the Greeks learnt as much from them is highly probable, though we shall see also that, from. the very first, they generalized it so as to make it of use in measuring the distances of inaccessible objects, such as ships at sea. It was probably this generalization that suggested the idea of a science of geometry, which was really the creation of the Pythagoreans, and we can see how far the Greeks soon surpassed their teachers from a remark attributed to Democritus. It runs (fr. 299) : "I have listened to many learned men, but no one has yet surpassed me in the construction of figures out of lines accompanied by demonstration, not even the Egyptian arpedonapts, as they call them." [Source: John Burnet (1863-1928), “Early Greek Philosophy” London and Edinburgh: A. and C. Black, 1892, 3rd edition, 1920, Evansville University]

“Now the word arpedovaptês is not Egyptian but Greek. It means "cord-fastener," and it is a striking coincidence that the oldest Indian geometrical treatise is called the Sulvasutras or "rules of the cord." These things point to the use of the triangle of which the sides are as 3, 4, 5, and which has always a right angle. We know that this was used from an early date among the Chinese and the Hindus, who doubtless got it from Babylon, and we shall see that Thales probably learnt the use of it in Egypt. There is no reason for supposing that any of these peoples had troubled themselves to give a theoretical demonstration of its properties, though Democritus would certainly have been able to do so. As we shall see, however, there is no real evidence that Thales had any mathematical knowledge which went beyond the Rhind papyrus, and we must conclude that mathematics in the strict sense arose in Greece after his time. It is significant in this connection that all mathematical terms are purely Greek in their origin.”

Image Sources: Wikimedia Commons, The Louvre, The British Museum, The Egyptian Museum in Cairo

Text Sources: UCLA Encyclopedia of Egyptology, escholarship.org ; Internet Ancient History Sourcebook: Egypt sourcebooks.fordham.edu ; Tour Egypt, Minnesota State University, Mankato, ethanholman.com; Mark Millmore, discoveringegypt.com discoveringegypt.com; Metropolitan Museum of Art, National Geographic, Smithsonian magazine, New York Times, Washington Post, Los Angeles Times, Discover magazine, Times of London, Natural History magazine, Archaeology magazine, The New Yorker, BBC, Encyclopædia Britannica, Time, Newsweek, Wikipedia, Reuters, Associated Press, The Guardian, AFP, Lonely Planet Guides, “World Religions” edited by Geoffrey Parrinder (Facts on File Publications, New York); “History of Warfare” by John Keegan (Vintage Books); “History of Art” by H.W. Janson Prentice Hall, Englewood Cliffs, N.J.), Compton’s Encyclopedia and various books and other publications.

Last updated August 2024

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