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 > 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!