Historical Orientation--Egypt

1
Historical Orientation--Egypt

• 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 Go-Between
• Today, we'll start that with a bit of orientation
  (in location and time) for the first

2
Geography of Egypt

• The gift of the Nile

3
Egyptian history

• 3200 - 2700 BCE -- predynastic period
• 3300 - 3100 BCE first hieroglyphic writing

4
Eventful history, stable culture

• 2650 - 2134 BCE -- Old Kingdom (pyramid-building
  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)

5
Egyptian timeline, continued

• 1550 - 1070 BCE -- New Kingdom (some well-known
  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,

6
History lost and regained

• 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

7
The Rosetta Stone
8
Egyptian hieroglyphics

• 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!)

9
Tutankhamen's Cartouches

• 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)
10
Other Egyptian writing

• 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

11
An Egyptian mathematical papyrus

• A portion of the Rhind papyrus

12
Egyptian number symbols

• The Egyptians, like us, used a base 10
  representation for numbers, with hieroglyphic
  symbols like this for powers of 10

13
Egyptian numbers

• 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.

14
Egyptian arithmetic

• Even though the Egyptians used a base 10
  representation of numbers, interestingly enough,
  they essentially used base 2 to multiply (!)
• Called multiplication by successive doubling
• Example Say we want to multiply
• 47 x 26

15
The Egyptian way

• Successively double
• 1 x 26 262 x 26 524 x 26 1048 x 26
  20816 x 26 41632 x 26 832
• (stop here since 32 x 2 64 gt the first factor,
  which is 47)

16
The calculation concluded

• 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)!

17
Comments

• 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

18
Egyptian fractions

• 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.
• More on this next time!