History and Numbers
Introduction
The civilization of Ancient Egypt spans some 3000 years of history. In ancient times the country was first unified under a king named Narmer - sometimes called Menes - in ca 3000 BCE. The ruling families of ancient Egypt were roughly divided into dynasties. There are all together some 30 dynasties that span the history of this ancient civilization. The fourth dynasty was the ruling family responsible for the construction of the famous pyramids at Giza. The Great Pyramid of Khufu is the topic of many scholarly (and not so scholarly) books. The architects of the time must have had knowledge of mathematics - and more specifically of geometry - in order to construct not only their pyramids, but also the temples they constructed. For more information about the pyramids see: The Pyramids
We have no mathematical papyri from the time of the pyramids of Giza. There are however some temple reliefs predating that time period by several decades showing rituals associated with planning the temples that accompanied the pyramids. The reliefs show the King marking off predetermined lengths with rope to create the correct dimensions for the temples.
Our present day knowledge of much of the mathematics of the Ancient Egyptians comes from the 12th dynasty. This family ruled Egypt some 1000 years after the pyramids at Giza were constructed. There are two major sources: the Moscow Papyrus - which dates to ca. 1800 BCE - and the Rhind or Ahmes papyrus - which dates to ca 1900 BCE, but is likely a copy from an original dating to ca 1800 BCE. These papyri are discussed in more detail in The Mathematical Papyri
In addition to these two papyri there are several other smaller fragments as well as tomb inscriptions and notes written on pottery shards (so called ostraca) that give us a glimpse into the mind of the Egyptian scientists.
Experts have cautioned against drawing too wide ranging conclusions about what the Egyptians may or may not have known. We only have a handful of ancient sources. There is no way to know if we have a reasonable cross-section of Egyptian knowledge of mathematics, or if we only have a glimpse.
The Egyptian Number System and Mathematical Notation
The Ancient Egyptians used a base 10 number system. The number one was depicted by a simple stroke, the number 2 was represented by two stokes, etc.
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The numbers 10, 100, 1000, 10,000 and 1,000,000 had their own hieroglyphs. Number 10 is a hobble for cattle, number 100 is represented by a coiled rope, the number 1000 is represented by a lotus flower, the number 10,000 is represented by a finger, the number 100,000 is represented by a frog and a million was represented by a god with his hands raised in adoration. [1]
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These numbers occur in for instance scenes depicting cattle counts and in reliefs with offering scenes.
The scene above was copied by the German Egyptologist Lepsius. The scene depicts a cattle count. In the middle register we see 835 horned cattle on the left, right behind them are some 220 animals (cows?) and on the right 2235 goats. In the bottom register we see 760 donkeys on the left and 974 goats on the right. These scenes would have depicted the wealth of the tomb owner and the depiction on the tomb wall also meant the tomb owner would have these animals with him in the afterlife.
In this stela of the King's Son Wepemnefret we see the deceased sitting before a table full of offerings. The items on the table are meant to depict bread. Notice the numbers that occur everywhere in the text. This is a depiction of all the items Wepemnefret wanted access to in the afterlife. Next to his knee for instance you see what looks like an inverted Y and loop of some sort with the lotus flowers below them. The left symbol represents linen and the symbol next to it represents alabaster vessels. The numerals below them mean that he has offerings of 1000 pieces of linen and 1000 alabaster vessels. On the right below the table we see offerings of 1000 pieces of bread, 1000 jars of beer, 1000 antelopes and 1000 oxen. The rest of the text refers to even more offerings.
Multiplication and Division
Multiplication
The Egyptians used a technique called "repeated doubling" to compute their products. The technique is most easily explained through an example. Suppose we wish to compute 11 X 15. We first find several easy multiples of 15:
1 X 15 = 15 √
2 X 15 = 30 √
4 X 15 = 60
8 X 15 = 120 √
Note that 11 = 1 + 2 + 8. We compute the product by just adding up the values checked, and in this case the answer is 165.
What if we want to multiply by a fraction? In this case repeated halfing may be the answer. Suppose we wish to compute 3/4 X 30? We would proceed by noting that 3/4 = 1/2 + 1/4. (For more about fractions, see below.)
1 X 30 = 30
1/2 X 30 = 15 √
1/4 X 30 = 7 1/2 √
We may then compute 3/4 X 30 by taking the checked values and adding them: 3/4 X 30 = 22 1/2
Division
A similar method can be used for division. Suppose we wish to divide 120 by 5. The we can work backwards and look at multiples of 5 until we reach 120.
1 X 5 = 5
2 X 5 = 10
4 X 5 = 20
8 X 5 = 40 √
16 X 5 = 80 √
32 X 5 = 160
Here 80 and 40 add up to 120, and hence 8+16=24 must be the answer we are looking for.
Fractions
Notation and Unit Fractions
The Egyptians almost exclusively used fractions of the form 1/n. One notable exception is the fraction 2/3 which is frequently found in the mathematical texts. Very rarely a special glyph was used to denote 3/4. The fraction 1/2 was represented by a glyph that may have depicted a piece of linen folded in two. The fraction 2/3 was represented by the glyph for a mouth with 2 (different sized) strokes. The rest of the fractions were always represented by a mouth super-imposed over a number.[1]
When working with fractions the Egyptians expressed all their fractions as a sum of unit fractions. For example:
3/4 = 1/2 + 1/4
3/8 = 1/4 + 1/8
2/5 = 1/3 + 1/15
The Egyptian mathematicians did not memorize all these fractions. The Rhind mathematical papyrus for instance contains some tables showing what these decompositions were. One table gives expressions for the fractions 2/n where n is any odd number between 3 and 101.
The Horus Eye Problems
It had long been thought that the Horus eye fractions represented a geometric series: 1/2, 1/4, 1/8, 1/16, 1/32, 1/64. These powers of 1/2 appear in some problems where scribes are computing rations. The most common example is someone computing different rations of grain. In more recent times some of the thinking about the origins of the glyphs has changed: "In older literature about Egyptian mathematics these signs are often interpreted as hieratic versions of the hieroglyphic parts of the eye of yhe Egyptian god Horus. However, texts from the early third millennium as well as depictions in tombs of the Old Kingdom, which show the same signs, prove that the Eye of Horus was not connected to the origins of the hieratic signs." [2]
But no matter where the glyphs came from, the fact remains that the Egyptians computed these fractions.
Problem 80 from the Rhind mathematical papyrus is a wonderful example to look at. It contains quite a few of the mathematical topics we have discussed. According to Clagett [1] the problem translates to:
As for vessels (debeh) used in measuring grain by the functionaries of the granary, done into henu :
1 hekat makes 10 henu
1/2 makes 5 henu
1/4 makes 2 1/2 etc.
This is not unlike a table in our cookbook which says:
1 cup makes 8 oz; 1/2 a cup makes 4 oz; 1/4 makes 2; 1/8 makes 1 etc.
Square Roots
The word for square root is written with a sign that represents either a corner or more likely a right angle. The name was kenbet in Egyptian. The underlying idea may well be that a right angle with equal arms is the root (in a sense) of the square area. I.e a right angle with equal arms measuring 3 units would give a total area of 9, so that the side measuring 3 represents the square root of the area which measures 9. (Paraphrased from Gunn and Peet via Clagett.)[1]
Several ancient sources mention square roots [1]. The Moscow Mathematical Papyrus uses the fact that the square root of 16 is 4 twice, and the fact that the square root of 100 is 10 once. Berlin Papyrus 6619 which dates to roughly the same time period as the Moscow Papyrus uses the fact that the square root of 100 is 10. The Berlin Papyrus has two much more sophisticated roots as well:
The square root of 1 + 1/2 + 1/16 (= 25/16) is 1 + 1/4 (= 5/4)
The square root of 6 + 1/4 is 2 1/2
It is not known how these square roots were computed. The results are used in the problems, but no justification is given. It is possible (likely?) that the Egyptians had tables of square roots they could consult when solving problems. Sadly no such table has ever been found however.
Measurements
The Ancient Egyptians took measurements in several different ways. Some measuring sticks have actually been found in tombs. An interesting example is for instance the measuring rod from the tomb of Maya - Tutankhamen's treasurer - which was found in Saqqara. The rod has the divisions into smaller units on the side.
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| Maya's Measuring rod | Detail showing smaller units |
Large distances were measured in cubits and the measuring device was a knotted rope. Such a rope and its use is shown in the tomb of Menna in Thebes.
Explorations
