# 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. (source)

Then the Rosetta Stone was discovered and a quarter-century 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.... • 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 el-Rashid (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 • 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. $\left.57*63= (32+16+8+1)*63= 32*63+16*63+8*63+1*63= 2016+ 1008+504+63\right.$

# 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 *

The Egyptians restricted themselves to the so-called "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?