In my previous article we discussed an overview of the Ancient Egyptian mathematics : Part I where we saw a glimpse of the only evidences we have : The Rhind and Moscow papyri. We will discuss more detailed analysis of these evidences. But in order to truly understand Egyptian Mathematics, it is important to discuss Egyptian numerals in the first hand.
The Egyptians had a writing system based on hieroglyphs from around 3000 BC. Hieroglyphs are little pictures representing words. It is easy to see how they would denote the word ‘bird’ by a little picture of a bird but clearly without further development this system of writing cannot represent many words. The way round this problem adopted by the ancient Egyptians was to use the spoken sounds of words. In this article we will not discuss about how to read or write hieroglyphs, but if you are interested you can find more detailed discussion here.
In the Arabic number system, we have ten digits (from 0-9) and we can make as big a number as we want with these. We use all ten digits to count to nine, then we combine them to make bigger numbers. So we never run out of numbers, as long as there is room to write them down. The ancient Egyptians didn’t think this. They had a simple line to mean one, like us, but instead of a new symbol for two, they used two lines. There were three lines for three, four lines for four, and so on, up to nine lines for nine. By now, there were rather a lot of lines! So they introduced a new symbol for ten. Then they carried on adding lines for units and ten symbols for ten, until they got to a hundred, which needed a new symbol. This sort of system is called unary. It is common among ancient civilizations. One advantage of unary systems is that it doesn’t matter what order you write the number. You can jumble them up and you can still work out what they mean. But in our system, 123 means something different to 321. The Egyptians used ten as a base for their number systems. We have ten fingers on our hands, so base 10 is common.
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.
Here are two examples: 3244 and 21237
Some of the examples in hieroglyphs are seen on a stone carving from Karnak, dating from around 1500 BC, and now displayed in the Louvre in Paris.
As we can see, adding numeral hieroglyphs is easy. One just adds the individual symbols, but replacing ten copies of a symbol by a single symbol of the next higher value. The most interesting concept described in the Rhind papyrus is the treatment of fractions since, of the 87 problems it contains, all but six deal with fractions. Fractions to the ancient Egyptians were limited to unit fractions (with the exception of the frequently used 2/3 and less frequently used 3/4). A unit fraction is of the form 1/n where n is an integer and these were represented in numeral hieroglyphs by placing the symbol representing an ‘open mouth’, which meant ‘part’, above the number. The hieroglyph for ‘ro’ was used as the word ‘part’. Here are some examples:
Ro is the smallest part of a hekat largest measure for grain. The only fractions of hekat used were 2, 4, 8, 16, 32 and 64. The smallest of these fractions, 64, contained 5 ro . These fractions were written in a special way, quite unlike ordinary fractions and the symbols when put together, form the Eye of Horus, one of the most well known symbols of ancient Egypt, and were used solely for the measure of grain.
Horus was the son of Isis and Osiris, who was conceived after Isis revived Osiris. Horus is also one of the names of the sun, and had his myths independently from either Ra or Osiris. The myth of Blind Horus describes the victory of Darkness over Light. A legend contained in the 112th chapter of the Book of the Dead describes Horus as wounded in the eye by Seth (or Set) in the form of a black boar. Set swallowed the eye, and was compelled to vomit it from the prison in which he was confined with a chain of steel fastened about his neck. The Eye of Horus is afterwards spoken of as a distinct deity, terrible to the enemies of Light.
According to another version of the legend, Horus’ eye is restored to him by Tehuti (or Teth, who is known to be, among other things, a God of Mathematics), who is the Egyptian Hermes. Tehuti is sometimes called also “the measurer of this earth”. He is said to have “calculated the heaven and counted the stars”, and to have “calculated the earth and counted the things which are in it”. In honor 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
Now, what is so remarkable about the use of unit fractions is the fact that they are in fact used to reduce a division between two numbers to the sums of unit fractions. An example would be dividing 2 by 43:
You will notice that no fraction is allowed to appear twice, but is not known how these particular unit fractions were derived. It is evident that there are many possible solutions, but it is not so evident why some solutions would be preferential to others. It is interesting to note that the first section of Rhind Papyrus is a table of the division of 2 by every odd integer from 3 to 101, and it may be that the Egyptian scribes realized that the result of multiplying by 2 is the same as that of dividing 2 by n. In fact, it may be that the decomposition for fractions of the form was the only necessary decomposition since Egyptians used the dyadic multiplication. This means that, instead of using times tables (as we do), they used two operations to multiply, doubling and adding.
But it should be noted that the hieroglyphs did not remain the same throughout the two thousand or so years of the ancient Egyptian civilization. This civilization is often broken down into three distinct periods:
Old Kingdom – around 2700 BC to 2200 BC
Middle Kingdom – around 2100 BC to 1700 BC
New Kingdom – around 1600 BC to 1000 BC
Numeral hieroglyphs were somewhat different in these different periods, yet retained a broadly similar style. Another number system, which the Egyptians used after the invention of writing on papyrus, was composed of hieratic numerals. These numerals allowed numbers to be written in a far more compact form yet using the system required many more symbols to be memorized. There were separate yet similar symbols for
1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 20, 30, 40, 50, 60, 70, 80, 90,
100, 200, 300, 400, 500, 600, 700, 800, 900,
1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000
With this system numbers could be formed of a few symbols. The number 9999 had just 4 hieratic symbols instead of 36 hieroglyphs. One major difference between the hieratic numerals and our own number system was the hieratic numerals did not form a positional system so the particular numerals could be written in any order. Here is an example:
Like the hieroglyphs, the hieratic symbols changed over time but they underwent more changes with six distinct periods. Initially the symbols that were used were quite close to the corresponding hieroglyph but their form diverged over time. The versions we give of the hieratic numerals date from around 1800 BC. The two systems ran in parallel for around 2000 years with the hieratic symbols being used in writing on papyrus, as for example in the Rhind papyrus and the Moscow papyrus, while the hieroglyphs continued to be used when carved on stone.
In the next article we will discuss more about Egyptian Papyri and their mathematics. Till then stay tight and let me know your views in the comment section.