The Egyptian number system 
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. For example, to illustrate the idea with an English sentence, we can
see how "I hear a barking dog" might be represented by:
"an eye", "an ear", "bark of tree" + "head with crown", "a dog".
Of course the same symbols might mean something different in a
different context, so "an eye" might mean "see" while "an ear" might signify
"sound".
The Egyptians had a bases 10 system of hieroglyphs for numerals. By this we
mean that they has separate symbols for one unit, one ten, one hundred, one
thousand, one ten thousand, one hundred thousand, and one million.
Here are the numeral hieroglyphs.
To make up the number 276, for example, fifteen symbols
were required: two "hundred" symbols, seven "ten" symbols, and six "unit"
symbols. The numbers appeared thus:
276 in hieroglyphs.
Here is another example:
4622 in hieroglyphs.
Note that the examples of 276 and 4622 in hieroglyphs are
seen on a stone carving from Karnak, dating from around 1500 BC, and now
displayed in the Louvre in Paris.
As can easily be seen, 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. 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 a "mouth", which meant
"part", above the number. Here are some examples:
Notice that when the number contained too many symbols for the "part" sign to
be placed over the whole number, as in 1/249, then the "part" symbol was just
placed over the "first part" of the number. [It was the first part for here the
number is read from right to left.]
We should point out that the hieroglyphs did not remain the same throughout
the two thousand or so years of the ancient Egyptian civilisation. This
civilisation 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 memorised. There were separate 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
Here are versions of the hieratic numerals
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 one way the Egyptians wrote 2765 in hieratic
numerals
Here is a second way of writing 2765 in hieratic
numerals with the order reversed
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.
