Egyptian Mathematics
How do we know how the Egyptians did math?
For a long time the mysterious Egyptian hieroglyphic language befuddled scholars, and guarded the secrets of the Egyptians.
Then the Rosetta Stone was discovered and a quartercentury later deciphered, and provided the means for scientists to begin to understand exactly what the Egyptians had understood and discovered in ancient times:
Napoleon's Campaign in Egypt led to the discovery of the stone. To Napoleon's credit, his army traveled with a slew of scientists (the famous mathematician Fourier among them), who studied the history, architecture, and culture as his military wreaked havoc.... http://upload.wikimedia.org/wikipedia/commons/thumb/2/29/FrancoisLouisJoseph_Watteau_001.jpg/300pxFrancoisLouisJoseph_Watteau_001.jpg  An image of the stone, and its translation.
About the Rosetta Stone:
"The decree is inscribed on the stone three times, in
hieroglyphic (suitable for a priestly decree), demotic
(the native script used for daily purposes), and Greek
(the language of the administration). The importance
of this to Egyptology is immense."
(From the British Museum website about the stone) The Greek text was deciphered in 1803, so it's astonishing that it took another 20 years to decipher the demotic, and then the hieroglyphics!  Its history and meaning: "Soldiers in Napoleon's army discovered the Rosetta Stone in 1799 while digging the foundations of an addition to a fort near the town of elRashid (Rosetta). On Napoleon's defeat, the stone became the property of the British under the terms of the Treaty of Alexandria (1801) along with other antiquities that the French had found."

Jean Francois Champollion deciphered the stone, around 1822, and his work is described in his text Grammaire Egyptienne en Encriture Hieroglyphique http://upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Leon_Cogniet__JeanFrancois_Champollion.jpg/220pxLeon_Cogniet__JeanFrancois_Champollion.jpg  The translation of the Rhind Papyrus introduced mathematicians to the delights of Egyptian arithmetic.
Egyptian multiplication
is a simple outgrowth of the Binary Card Trick. Multiplication is accomplished through binary decomposition (successive doublings). Consider, for example, 57*63. We'll double the larger of the two numbers, and then select the rows that add up to the smaller:
1  63  * 
2  126  
4  252  
8  504  * 
16  1008  * 
32  2016  * 
64  too big! 
Now add up those rows marked with an asterix (*), because 57=32+16+8+1. You'll get your answer: 3591.
 How would the Egyptians do 8*17?
This works because of this amazing fact: Every natural number is either a power of two, or can be expressed as a sum of distinct powers of two in a unique way. So 57=32+16+8+1 is the only way of writing 57 as a sum of distinct powers of two.
Egyptian division
Division is also carried out in binary, but fractions make it more interesting:
Let's look at an example: divide 35 by 8.
In a way we turn it into a multiplication problem: what times 8 equals 35? So we know the 8, and use it to "double"  but then to "halve", when 8 won't go evenly into 35:
1  8  
2  16  
4  32  * 
1/2  4  
1/4  2  * 
1/8  1  * 
So the answer is 4+1/4+1/8
The Egyptians restricted themselves to the socalled "unit fractions", which are fractions of the form 1/m: unit fraction table, which is found on the Rhind Papyrus (which dates to around 1650 BCE).
But they didn't restrict themselves to "halving", as our next example shows. Divide 6 by 7:
1  7  
1/2  3+1/2  * 
1/4  1+1/2+1/4  * 
1/7  1  
1/14  1/2  * 
1/28  1/4  * 
So the answer is 6/7 = 1/2+1/4+1/14+1/28. We see in this case that at some point we divided 7 by 7, to get to a unit, then began dividing by twos again. This is a standard trick.
Some Examples
 Here's a relatively easy story problem: Suppose Fatima had 3 loaves to share between 4 people. How would she do it?
 A little trickier problem: How would you divide 5 by 7?
Links
 Fractions  binary decimals, oh Ra!
 Division is also carried out in binary, but fractions make it more interesting: they restricted themselves to the socalled "unit fractions", which are fractions of the form 1/m
 The unit fraction table, which is found on the Rhind Papyrus (which dates to around 1650 BCE).
 How would you like to do story problems like this one?!
 Why did Egyptians do things this way? (an example division problem, using binary). The reason I find most plausible is that the "shares" upon dividing always look the same. So there shouldn't be any arguing about the distribution.
 The best Egyptian fractions
 Ancient Egypt
 Thebes, Egypt
 Discovery of ancient Egyptian manuscripts
 Participation of Gaspard Monge and JeanBaptiste Fourier in the scientific expedition attached to Napoleon's army
 Rhind papyrus (owned by A. Henry Rind, purporting (by Ahmes) to be mathematics of 1800 BC or so).
 Moscow (or Golenischev) papyrus
 Its discovery isn't too interesting.... This link gives its history, and a list of the 25 problems.
 The Egyptian Mathematical Leather Roll (EMLR)  a paper elucidating
 From the Rhind: "Knowledge of all obscure secrets" comes down to multiplication and division.
 Multiplication is accomplished through binary decomposition (successive doublings).
 Unit fractions  the key and the curse
 The Rhind papyrus contains the Table of unit fractions multiplied by 2, which is critical for doublings (this table constitutes 1/3 of the Rhind papyrus!  p. 40).
 An example problem from the Moscow Papyrus
 Unit fractions:
 What's with 2/3?
 The splitting identity 
allows us to represent all rational numbers as sums of unit fractions.
 Fibonacci also developed a method for doing so.
 A fantastic site on Egyptian Fractions (including a calculator to turn your rationals into sums of unit fractions).
 Another
 Ancient Egyptian Science: A Source Book (by Marshall Clagett)