# Egyptian Fractions: From Mythology to Computation

While we in the United States often associate the All Seeing Eye with the Illuminati or the little eye on top of the pyramid on the dollar bill, in Egypt it is better known as the Eye of Horus – a symbol of healing, life, and resurrection. In Egyptian mythology, Horus was the son of Isis and Osiris, king of Egypt. Osiris had a brother named Set, who after years of watching his brother bring prosperity and joy to the people of Egypt became jealous of his brother and killed him to claim his throne. Once he had come of age, Horus sought to reclaim the throne that was rightfully his. In a battle between Horus and Set, Set gouged out Horus’s eye and ripped it into six pieces (the fight wasn’t too one sided – Set lost a testicle). Seeing the damage of the rightful heir to Egypt, the god Thoth collected the pieces of Horus’s eye and restored them using magic. Thus, the Eye of Horus represents health or wholeness.

The Egyptians ascribed a specific set of unit fractions to the six parts of the Eye of Horus. Each fraction is associated with one of the pieces that the eye was ripped into as well as each of the five senses (sight, smell, taste, hearing, touch) plus a sixth sense represented by the Egyptians as thought. These fractions, 1/2 for smell, 1/4 for sight, 1/8 for thought, 1/16 for hearing, 1/32 for taste, and 1/64 for touch, were often used to measure amounts of pigment, ingredients for medicines, and grain.

It is interesting to note that the Horus Eye fractions were the only unit fractions used in dividing grain and that they were written differently from other unit fractions when used in this context. The base unit of grain used by the Egyptians was the hekat which is approximately 1/8 of a modern day bushel. Rather than using standard unit fraction notation for hekat, each piece of the Horus Eye hieroglyphic became a symbol for each fraction (see table below). That way, if the symbol for a specific part of the Horus Eye was used, it was known to measure a fraction of hekat.

The only exception to this was ro. If the fraction required was less than 1/64, then a seventh unit fraction, ro, was used. Like the Horus Eye fractions, ro was not written as a typical unit fraction, but was symbolized by an open mouth hieroglyphic. One ro was somewhere between a teaspoon and a tablespoon and was used to measure 1/320 hekat. I was not able to find any writings about the origin of ro or why 1/320 might be a significant unit fraction other than 5 ro = 1/64 hekat.

From today’s perspective, this collection of numbers makes up the first six terms of a geometric series whose sum converges to one. It is interesting that the symbol for “whole” would only sum to 63/64 in Egyptian times. Some scholars think that the missing 1/64 represents the magic used to reassemble the eye to make it whole. Some consider it intentionally imperfect because mortals cannot achieve the perfection of the gods. Still others think it has value in relation to the Egyptian unit fraction ro. Whatever the reason may be, this collection of sacred unit fractions of the form 1/2n is just one of many forms of Egyptian mathematics that is derived from the fraction 1/2.

As a matter of fact, most of Egyptian mathematics is centered around halving or doubling quantities and manipulating those halved or doubled quantities to perform calculations. The only additional multiplier often seen in Egyptian mathematics is 2/3, but even this appears to be a result of manipulations of 1/2 (discussed below). The fraction 3/4 is also seen on rare occasion, but I couldn’t find any records or calculations based on this.

Archaeologists have found many Egyptian artifacts containing mathematical tables that were most likely used as reference materials by scribes during computation. Many examples of such tables can be found in the Rhind Mathematical Papyrus (RMP). The Recto of the RMP contains a table of unit fraction sums where the sum is equal to 2/n and n is an odd numbers between 3 and 101. Another table in the RMP lists the results of finding 2/3 of unit fractions with odd denominators. Problem 61 of the RMP begins with a table of five unit fractions and the results of halving those unit fractions. Elsewhere in the RMP, there is a table of doubled odd unit fractions. The lack of tables to describe operations with fractions other than 1/2 and 2/3 further supports the hypothesis that Egyptians strictly used these fractions to obtain results for other fractional quantities. Also, the lack of scratch work or steps along with the ordering of the problems suggests that these were reference tables rather than independent problems. This is particularly clear in Problem 61 of the RMP because the table is given before calculations were made in the first and second parts of the problem that clearly used the values in the table in their computations.

The Egyptians appear to have developed various algorithms for solving specific types of problems. For example, Problems 32 and 34 of the RMP show two sets of equations that exhibit the characteristic, if a*x=b, then (1/b)*x=(1/a). No additional work is provided around the problems, indicating that either scratch work was done elsewhere or this was a general rule known to Egyptian scribes.

In calculating halves of unit fractions described as a sum of other unit fractions, the Egyptians knew to double the denominator of each unit fraction in the sum. An example of this is found in problem 1 the Egyptian Mathematical Leather Roll (EMLR). In this problem, 1/4 is equated to 1/7+1/14+1/28. By doubling, the result is 1/8 is equal to 1/14+1/28+1/56.

A similar process is used to calculate 1/3 of a known unit fraction sum. Interestingly, Egyptians did not simply multiply their denominators through by three, likely because they were so dependent on their tables of halves and two-thirds. Instead multiple problems in the RMP show that scribes would first calculate 2/3 of each unit fraction and then calculate 1/2 of each of the resultant fractions to get to 1/3 of each of the original unit fractions in the sum.

The use of 2/3 instead of 1/3 is generally thought to be a result of manipulations of 1/2. There are many problems in the RMP that show the process of finding 2/3 of any unit fraction as a sum of other unit fractions. Problem 61B of the RMP show the Egyptians understood that 2/3 of any unit fraction, 1/n can be written as (1/2n)+(1/6n). For the case of even unit fractions, this is always appears to be simplified to 1/(n+(n/2)). Both of these calculations simply rely on manipulating halves or wholes of the original fraction to calculate the desired fraction.

Another interesting relation known the Egyptian scribes is demonstrated by ten, mostly successive lines in the EMLR. In Mathematics in the Time of the Pharaohs, Richard J. Gillings calls this the G Rule: “If one unit fraction is double another then their sum is a different unit fraction if and only if the larger denominator is divisible by 3. The quotient of the division is the unit fraction of the sum.” Many examples of this rule are written with no scratch work or intermediate steps shown such as line 11 where 1/9+1/18=1/6. But amongst those problems are problems of a slightly different form. Lines 1, 2, and 3 of the EMLR show that 1/10+1/40=1/8, 1/5+1/20=1/4, and 1/4+1/12=1/3 respectively. These seem to follow the same principal of the G Rule, but for cases when one fraction is triple or quadruple another and the larger fraction is divisible by four or five respectively. According the Gillings, this suggests that the Egyptians had a more inductive understanding of the G Rule: “If one of two unit fractions is K times the other, then their sum is found by dividing (K-1) into the larger number, providing the answer is an integer.”

From all these translations and analyses of original sources, I found it interesting that the groupings of problems on different papyri suggest mathematical rules that Egyptian scribes were familiar with but are not explicitly proven. This suggests to me that the Egyptians discovered rules of mathematics by discovering patterns through repeated computation and if those patterns were regularly helpful in further computations they were organized and made into reference tables. It was strange to me that many of these similar problems are not listed in any particular order or group all together unless they appear in a designated table. To me, this would suggest that these problems were done for practice, but so little information is known about the origins of these sources that it is difficult for even experts to deduce their original purpose.

Sources:

http://www.laits.utexas.edu/cairo/teachers/osiris.pdf

http://www.egyptianmyths.net/mythisis.htm

http://www.landofpyramids.org/eye-of-horus.htm

http://www.ancientegyptonline.co.uk

http://www.britishmuseum.org

Gillings, Richard J. Mathematics in the Time of the Pharaohs. Cambridge, Mass. MIT, 1972. Print

# Egyptian Numbers: Counting By Ten

When most people think of early mathematics, the first thing that comes to mind is probably Archimedes or Pythagoras of ancient Greece; or perhaps ancient Mesopotamia and its thousands of clay tablets, many of them containing math problems.  Few people think of ancient Egypt. This is mainly due to the fact that little is known about ancient Egyptian math in comparison to these other civilizations. However, by looking at what we do know about Egyptian math and the feats of engineering they accomplished with it, we can recognize how complex and sophisticated it was. A fragment from the Rhind Mathematical Papyrus. Image: Public domain, via Wikimedia Commons.

The majority of our knowledge of ancient Egyptian math comes from two papyri: the Rhind Mathematical Papyrus, which was originally written about 1985-1975 BCE; and the Moscow Mathematical papyrus, dating from around 1850 BCE. Math problems were not written on the walls of temples or great monuments in Egypt, so all math was probably done on papyrus or other perishable media, meaning that many mathematical works have not come down to us. This in in contrast to the mathematical documents from Mesopotamia, which were primarily done on clay tablets that are not as susceptible to the elements and ravages of time.

The ancient Egyptians probably developed the first base ten numerical system in human history. It was fully in use around 2700 BCE and perhaps even earlier. They had different symbols representing the powers of ten. A straight vertical line represented one, 10 was shown by a drawing of a hobble for cattle, a picture of a coil of rope served as a symbol for 100, a lotus plant delineated 1,000, a bent finger for 10,000, a tadpole or a frog for 100,000 and the picture of a god, perhaps Heh, represented 1 million. Although the Egyptian numerals were in a decimal system, it was not a positional place value system like the decimal system we use, or the sexagesimal (base 60) system developed by the Mesopotamians. Multiples of the powers of ten were written by repeating the symbol as many times as needed, and although they had a symbol for 1 million, other large numbers would have been very tedious to write. For instance, the number 987,654 would have required 39 characters. This Egyptian system possibly had an influence on the later Greek numerical system, but the Greeks improved upon it, creating a different symbol for each number 1-9, and other symbols for 20, 30 and so on. Parts of the Eye of Horus were used to write fractions. Image: BenduKiwi, via Wikimedia Commons.

The Egyptians also had notation for fractions, although all Egyptian fractions were unit fractions (meaning they always had one in the numerator), with only two exceptions, 2/3 and 3/4. Fractions were marked by the hieroglyph for ‘R’ which is a long skinny oval, very similar in shape to the Eye of Horus. In fact, in an ancient Egyptian myth, the evil god Seth attacked his brother the god Horus and in the fight Seth gouged out the eye of Horus and tore it into pieces, fortunately the god Throth was able to put Horus’s eye back together and heal it. Thus, to honor the gods and this myth, whenever the Egyptians used fractions in relation to their measurement of volume, which is the hekat, the commonly used fractional parts were transcribed by using the corresponding parts of the Eye of Horus.

The great feats of engineering that the ancient Egyptians accomplished would have required an enormous amount of sophisticated math. The pyramids, for instance, are considered a marvel of mathematics and engineering. The base of the pyramids are almost perfect squares which the Egyptians would have achieved by using trigonometry, like the 3-4-5 trick. It was known in Egypt that a triangle with 3, 4 and 5 unit sides would always be a perfect right triangle. This Pythagorean triple, observed by the Egyptians long before Pythagoras, is sometimes called an “Egyptian triangle.” The Egyptians would have utilized this rule of geometry when laying out the base of a pyramid by tying knots in rope at 3, 4 and 5 unit intervals. Other tricks of trigonometry were known as well. The Rhind Mathematical Papyrus contains an equation for calculating the slope of a pyramid’s face, which is the same thing as finding the cotangent. It is also likely that they knew how to find a pyramid’s volume.

It is unfortunate that more is not known about the mathematics that were used by the ancient Egyptians, because they were obviously very skillful and innovative, and they must have been among the first people to develop important mathematical principals.

Sources:

http://discoveringegypt.com/egyptian-hieroglyphic-writing/egyptian-mathematics-numbers-hieroglyphs/

http://www.britishmuseum.org/explore/highlights/highlight_objects/aes/r/rhind_mathematical_papyrus.aspx

http://www.touregypt.net/featurestories/numbers.htm