Egyptian Mathematics

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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.

EgyptianArt.jpg (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:

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:

64too 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:


So the answer is 4+1/4+1/8

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:


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?