Count Like an Egyptian: A Hands-on Introduction to Ancient Mathematics

Image of Count Like an Egyptian: A Hands-on Introduction to Ancient Mathematics
Author(s): 
Release Date: 
April 26, 2014
Publisher/Imprint: 
Princeton University Press
Pages: 
256
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“David Reimer succeeds in keeping the mathematics in Count Like an Egyptian clever and light, raising this book into a rare category: a coffee table book that is serious and fun.”

Count Like an Egyptian is a richly illustrated and deeply fascinating coffee table book providing a mix of ancient Egyptian culture and mathematics. “Egyptian mathematics has an alien feel to it . . . but . . . to someone who’s mastered it, Egyptian mathematics is beautiful.”

Egyptian mathematics is pre-algebra and expressed geometrically through objects. The author of Count Like an Egyptian, David Reimer goes back to the source, not of the Nile but to the source of what we know of Egyptian mathematics, the original translation of the Rhind Mathematical Papyrus.

What makes ancient Egyptian mathematic different is its odd use of fractions. Reimer tells the reader, “. . . Egyptian fractions are not fractions in the modern sense. There is no 1 in the numerator because there is no numerator.” Egyptians only used unit fractions—the denominator is always used by itself and the numerator is implied and always one. For example, the sum of the fractions 1/5 + 1/5 would be written as the symbol for 1/3 with the symbol for 1/15 next to it. As the sum 1/3 plus 1/3 has no reduction, the sum has its own symbol. For us moderns, Reimer uses a bar above a number to represent the number as a unit fraction.

The analogy used by the author for the reader to understand ancient Egyptian mathematics is to compare unit fractions to slices of bread in a loaf. And as the reader may not be able appreciate how truly weird this is, the author also provides numerous examples along with algebra to compare methods. Reimer cautions, “. . . I doubt few people could argue that the . . . equations give any real insight into the mind of an ancient mathematician.” This reviewer in following along with the examples felt that Egyptian math compared to modern math was like having to doing math with one hand tied behind your back.

There is plenty of humor in the Reimer’s presentation, which is done in the manner of 1950s musicals. In a musical, dialog ends on a phrase and a song starts with that phrase as its theme. In Count Like an Egyptian instead of a song, the reader gets a math problem. And Count Like an Egyptian is not just about mathematics. The reader will learn about Egyptian geography, mythology, hieroglyphics, religion, diet and magic.

Ancient Egyptian society remained stable for 3000 years for the most part because of its geography; waterless barriers kept violent outsiders away. In Egyptian agriculture, “water was so precious that restricting flow onto your neighbor’s farm was an offence punishable by damnation during your soul’s final judgment in the Hall of Osiris.”

As the ancient Egyptian civilization evolved over its 3000 years, cities grew or lost importance and as they did so did that city’s gods and mythologies. Competing theologies led to competing views of the afterlife and its people chose the afterlife that most suited them, and the favorite gods of a city’s members varied by their profession and social class.

The blue-skinned god who created the world, Ptah was the patron god of the craftsmen who built and decorated the tombs and temples. Thoth was the god of scribes and wisdom and could change into an ibis form. The sun god Ra merged with the god Amun to become the god Amun-Ra. The eyes of the gods were beings in themselves having their own personalities and could be sent out to accomplish difficult tasks independent of the divine body.

The reader learns mainly about the scribal class, as scribes were also mathematicians and historians. Children entered school at the age of five to learn more than six hundred hieroglyphs and many hieroglyphs had multiple meanings. Scribes wrote on scrolls made of papyrus though their writing also appears on leather and broken pottery. The value of the material written on indicated its owner’s social status.

Reimer also translates fragments for us. One scribe warns in a letter to his son of the hardships of other professions such as blacksmith, craftsmen, and bricklayer. The father hoped to impress that only by learning to write could his son escape the horrors of physical labor—no scribe ever went hungry.

Reimer also offers a translation of a writing exercise, a faux letter similar in spirit to typing “a quick brown fox” but in the tone of a teacher in admonishing a less than serious student. The letter begins, “I hear. . . that you are neglecting your writing and spending your time dancing, going from tavern to tavern, always reeking of beer . . .”

Hieroglyphs were too awkward (or too slow to write) for scribes, who they created their own symbols called hieratic, which are similar to hieroglyphs but different, using simpler strokes. Egyptian writing seldom used vowels and their absence provides no clue as to how to sound out the ancient Egyptian language. Intelligent guesses are based on ancient Coptic, which evolved from ancient Egyptian.

Egyptian numbers were not hieroglyphs but their own symbols. The Egyptian symbol for the number one was a single stroke just like we use today. Their symbol for ten was an upside down U, representing a cattle hobble. There were unique symbols for the numbers 100 and 1000, indicating the Egyptians used a base ten system. There is no direct evidence as to how the ancient Egyptians added as the referenced papyrus solutions are given without showing their work, however their arrangement on scrolls do look like adding Roman numerals.

We can learn a lot about ancient mathematics by looking at texts with the pedagogical point of view. For example, one scroll starts with fraction tables without defining what a whole number is. The assumption by a researcher would be that if you used this scroll you would already know what a whole number was.

There was no need for an ancient Egyptian to memorize multiplication tables, as it appears they never multiplied by any number greater than two. Performing mathematics before algebra, Egyptians could make mathematical statements but not prove them; for the ancient Egyptian, proof was by example only.

Ancient Egyptian measurements are unfamiliar to us today. A “palm” is four fingers, a “cubit” is six palms, a “khet” is 100 cubits, a “hekat” is approximately a modern gallon, a “khar” is 20 hekat. A “cubit strip” is a plot of land 1 cubit by 100 cubits, and Reimer offers the reader math problems for calculating the area of triangles and trapezoids in cubits. Egyptians knew that area of a right triangle was one-half times the base times height, and the Great Pyramid of Khufu is off by only 1/17th of a degree

One may presume that accurate measurements were reached by accurate computations though the Egyptians apparently had issues in architecting. The first pyramids, made from limestone blocks, collapsed because the angles were too steep. Egyptian architects had to discover the proper rise-over-run, which they called the “seked.” If you are curious, google the phrase “bent pyramid” for a link to the Wikipedia page that shows an ancient pyramid that makes a rise over run change in its middle.

Of all possible shapes for tombs, why a pyramid? The pyramid represents the ramp to heaven, and the (ideal) pyramid got its shape from a symbolic ray of light, called the benbenet—the sacred icon of Heliopolis, the Egyptian city of the sun. Pyramids were aligned north-south and east-west and were built on the western side of the Nile because the entrance into the netherworld was in the land of the setting sun. The image intended is of a dead king walking up to the stars on a ray of sunlight.

Temples were not for worship but used to cater to the needs of the gods, often in the form of small statuettes that had to be clothed, fed, washed, entertained, and guarded. Many tombs had a false door painted on the western inner wall—accessible only to the spirits of the dead. And what we today see as artistically odd design choice in the flattened depictions of people temples and pyramids was not intended to be art but was intended as a specification for their afterlife. The size of the drawn figures determined their importance, and figures were drawn “idealistically” because Egyptians expected that one day their portraits would come to life. This belief followed into the hieroglyphs, too. The hieroglyph for the “j” sound was expected to become a venomous snake in the afterlife, and had to be used carefully

Temple staff (as were all workers who were paid) were paid in shares instead of a salary, the greater your rank, the greater the number of your shares. Shares were paid in grain, bread, and beer, and we know this because all shares were calculated and recorded on scrolls by scribes.

Ancient Egyptians drank beer instead of water (alcohol kills bacteria), but the beer was watered down. Women were not treated as equals to men though there is evidence that the wives of scribes were highly regarded as they could read and also take men who had wronged them to court.

All civilizations decline and fall. By the time of Alexander the Great (~350 BCE), Egyptian culture had been in decline since the rule of Ramses (~1200 BCE). The era of Ramses is of note because it recorded the first strike in history, a reaction of workers to a corrupt vizier who had robbed the granaries leaving the granaries empty of shares for workers’ pay.

Reimer provides the reader comparisons of ancient Egyptians to other ancient cultures. The travel of the constellations in the night sky were used as markers in the calendar for time keeping across cultures; you can tell the day of the year by the stars you see at sunset. We still measure time in Babylonian hours, minutes, and seconds. The Mesopotamians, the Babylonians, and the Amorites through the epic of Gilgamesh connect the Zodiacal constellations back to ancient Mesopotamian myth. The constellation of Taurus is only half a bull because Taurus was cut in half by Gilgamesh’ sword.

The last chapter compares Babylonian math to Egyptian and modern math, explaining why modern math is better, and what “better” means. This chapter provides grab bag of mathematical odds and ends, and puzzles and tricks as well.

David Reimer succeeds in keeping the mathematics in Count Like an Egyptian clever and light, raising this book into a rare category: a coffee table book that is serious and fun.