1. Historical Orientation--Egypt
We are now ready to begin a more detailed study of the mathematics of one of the ancient civilizations that had an 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)

2. Geography of Egypt
The “gift of the Nile”

3. Egyptian history 3200 - 2700 BCE -- predynastic period
~ BCE first hieroglyphic writing

4. Eventful history, but stable culture
~ BCE -- Old Kingdom (pyramid- building period)
~ BCE -- First intermediate period
~ BCE -- Middle Kingdom (Moscow mathematical papyrus)
BCE -- Second intermediate period (Hyksos) Rhind (Ahmes) mathematical papyrus (possibly copying an older work from Middle Kingdom)

5. Egyptian timeline, continued
BCE -- New Kingdom (well-known pharaohs include Thutmose III in 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 (the period described by Herodotus), 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 specially trained priests long before that)
But, inscriptions could be read again after Jean-Francois Champollion (1790 – 1832 CE) began decipherment, with the help of the inscriptions on the Rosetta Stone

7. The Rosetta Stone

8. Egyptian hieroglyphs
A very rich system with phonetic signs for single sounds, combinations of sounds, plus a few ideographs (signs representing whole words or ideas)
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 the cartouche for Ptolemy!)

9. “King Tut's” Cartouches
Each king had two principal names – “birth name” and “throne name:” Tut-ankh-amen (heqa iunu shema), Neb-kheperu-re:
(note how hieroglyphs can also be decoration)

10. Other Egyptian writing
Hieroglyphics were the “formal” Egyptian writing system, 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 plants grown along the Nile for “everyday” writing – scrolls with stories, business records, school exercises, and simplified
Hieratic and demotic (as in middle panel of Rosetta Stone) writing forms

11. An Egyptian mathematical papyrus
A portion of the Rhind papyrus (from Second Intermediate Period, ~1650 BCE)

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 group the corresponding number of symbols for each power of 10 together – four lotus flowers, 3 “hobbles,” 7 strokes (very like the Aztec system, but base-10).
There were separate and more involved number systems used in hieratic writing.

14. Egyptian arithmetic
The Egyptians used their base 10 representation of numbers to do addition in pretty much the “obvious” way: “carrying” into the next digit as necessary.
But they had a distinctive way of doing multiplication called multiplication by successive doubling
Example: Say an Egyptian wanted to multiply
47 x 26

15. “The Egyptian way” Successively double, starting from 26:
* 1 x 26 = 26 Asterisks mark needed ones: * 2 x 26 = = * 4 x 26 = 104 * 8 x 26 = x 26 = 416 * 32 x 26 = 832
(Next one 64 x 26 not needed since 64 > 47.)
Then 47 x 26 = = 1222

16. An interesting observation
Successively double, starting from 26
* 1 x 26 = Asterisks mark needed ones * 2 x 26 = = * 4 x 26 = 104 * 8 x 26 = x 26 = 416 * 32 x 26 = 832
Note that this calculation is essentially using the base-2 (binary) expansion of the factor 47:
(47)-base 10 = (101111)-base 2(!)

17. More Comments
What the Egyptians were doing: just doubling and addition, not calculation of intermediate products, so relatively simple – didn't require lots of memorization of times tables: doubling a number is just adding it to itself(!)
We used modern numerals here; in Egyptian symbols, Joseph has an example on page 88.
Would be 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 division
The Egyptians did division by a process very closely related to their multiplication
Where we might say “compute x/y” they would say “reckon with y to obtain x”
That is, in modern terms, a = x/y <=> ay = x
So, given x and y, the Egyptians would try to identify the multiple a of y that would yield x.
Many simple examples were set up so the value of a would turn out be a whole number.

19. A division example Say we want to compute 2184/56 ``the Egyptian way''
We “reckon with 56 to yield 2184” like this:
* 1 x 56 = = ,
* 2 x 56 = so 2184/56 =
* 4 x 56 = = 39,
8 x 56 = which gives the answer.
16 x 56 = Note: finding this takes some
* 32 x 56 = “fiddling” (trial and error)

20. Some Comments
When dividing x/y, in case the quotient is a whole number, the operations involved are just doubling and addition – same as for multiplication(!)
The difference was that the Egyptians would find a combination of the numbers obtained by doubling adding up to the dividend x, not adding up to the other factor as in a multiplication.
The Egyptians also had techniques to work with fractions x/y when the quotient was not a whole number.

21. “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 like 1/2, 1/3, etc. (the only other fraction they allowed was apparently 2/3); to work with the fraction 7/8, they would “split it up” as:
7/8 = ½ + ¼ + 1/8.