Historical Orientation--Egypt
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We are now ready to begin a more detailed
historical study of the mathematics of several
of the ancient civilizations that had especially
large influence on where the mathematics you
have learned originally “came from”
“The past is a foreign country; they do things
differently there” L.P. Hartley, The GoBetween
Today, we'll start that with a bit of orientation
(in location and time) for the first
Geography of Egypt
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The “gift of the Nile”
Egyptian history
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3200 - 2700 BCE -- predynastic period
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~3300 - 3100 BCE first hieroglyphic writing
Eventful history, stable culture
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~2650 - 2134 BCE -- Old Kingdom (pyramidbuilding period)
~2134 - 2040 BCE -- First intermediate period
~2040 - 1640 BCE -- Middle Kingdom
(Moscow mathematical papyrus)
1640 - 1550 BCE -- Second intermediate
period (Hyksos) Rhind (Ahmes) mathematical
papyrus (possibly copying an older work from
Middle Kingdom)
Egyptian timeline, continued
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1550 - 1070 BCE -- New Kingdom (some wellknown pharaohs include Thutmose III, whose
name appears the first example of
hieroglyphics from above, Amenhotep III,
Akhenaten, Tutankhamen, Ramses II)
after 1070 BCE -- Third intermediate period
then Egypt ruled by Nubians, Assyrians,
Persians, Ptolemaic Greek dynasty (until
Cleopatra), Romans, Byzantines, Islamic
caliphate, Ottomans, British, …
History lost and regained
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How do we know about a lot of this?
Much of this history was lost when the
hieroglyphic system fell completely out of use
in the Roman period (it had become extremely
archaic and probably readable only by a few
trained priests long before that)
But, inscriptions could be read again after
Jean-Francois Champollion (1790 – 1832 CE)
began the decipherment, with the help of the
inscriptions on the Rosetta Stone
The Rosetta Stone
Egyptian hieroglyphics
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A very rich system with phonetic signs for
single sounds, combinations of sounds, plus a
few ideographs (signs representing ideas)
Pretty much the antithesis of the cuneiform
script from Mesopotamia that we'll study later
in terms of the variety of signs!
Some of the most recognizable symbols are
the names of kings and queens given in the
oval signs called “cartouches” (Champollion's
detective work used cartouche for Ptolemy!)
Tutankhamen's Cartouches
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Each king had a principal pair of names –
“birth name” and “throne name” (as well as
several others)
(note how hieroglyphs also function as decoration)
Other Egyptian writing
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Hieroglyphics were “formal” Egyptian written
language, used mostly for temple or tomb
inscriptions carved in stone, grave goods
(coffins, etc.) – meant to last.
The Egyptians also used a paper-like writing
medium called papyrus manufactured from
plant material grown along the Nile for
“everyday” writing – scrolls with stories,
business records, school exercises, …
Hieratic and demotic (as in middle panel of
Rosetta Stone) writing forms as well
An Egyptian mathematical papyrus
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A portion of the Rhind papyrus:
Egyptian number symbols
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The Egyptians, like us, used a base 10
representation for numbers, with hieroglyphic
symbols like this for powers of 10:
Egyptian numbers
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The Egyptians did not really have the idea of
positional notation in this system, though.
To represent a number like 4037 (base 10) in
hieroglyphics, the Egyptians would just group
the corresponding number of symbols for each
power of 10 together – four lotus flowers, 3
“hobbles,” 7 strokes (something like a simpler
version of Roman Numerals).
There were separate and more involved
number systems used in hieratic writing.
Egyptian arithmetic
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Even though the Egyptians used a base 10
representation of numbers, interestingly
enough, they essentially used base 2 to
multiply (!)
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Called multiplication by successive doubling
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Example: Say we want to multiply
47 x 26
“The Egyptian way”
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Successively double:
1 x 26 = 26
2 x 26 = 52
4 x 26 = 104
8 x 26 = 208
16 x 26 = 416
32 x 26 = 832
(stop here since 32 x 2 = 64 > the first factor,
which is 47)
The calculation concluded
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Then to get the product 47 x 26, we just need
to add together multiples to get 47 x 26:
47 = 32 + 8 + 4 + 2 + 1, so
47 x 26 = 32 x 26 + 8 x 26 + 4 x 26 + 2 x 26 +
1 x 26
= 832 + 208 + 104 + 52 + 26
= 1222
Note: this essentially uses 47 (base 10) =
101111 (base 2)!
Comments
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Important to realize that the calculations here
were just doubling and addition, not calculation
of all the intermediate products by
multiplication(!)
We used modern numerals here; the
Egyptians would have used their own symbols,
of course!
More efficient to reorder the factors as 26 x 47
– would require fewer doublings
Egyptian scribes would have been very adept
at this and other “tricks” for using this system
“Egyptian fractions”
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Probably the most distinctive feature of the
way the Egyptians dealt with numerical
calculations was the way they handled
fractions.
They had a strong preference for fractions with
unit numerator, and they tried to express every
fraction that way, for example to work with the
fraction 7/8, they would “split it up” as:
7/8 = ½ + ¼ + 1/8.
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More on this next time!