Egyptian Mathematics

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1 Egyptian Mathematics Sources Rudman, Peter S. (007). How Mathematics Happened: The First 50,000 Years. Amherst, NY: Prometheus Books. Benson, Donald C. (003) A Smoother Pebble: Mathematical Explorations. New York, NY: Oxford University Press. Joseph, George G. (000). The Crest of the Peacock: The Non-European Roots of Mathematics. Princeton, NJ: Princeton University Press. Notes Egyptians used a hieroglyphic numeration system based on groups of 10, with symbols standing for 1, 10, 100, 100, etc. There was no 0, no positional notation, and no place value. They added and subtracted by combining symbols together and doing 1-for-10 exchanges when necessary. We will take up multiplication and division below. The emphasis in Egyptian mathematics was decidedly practical. They knew some rules, such as (the converse of) the Pythagorean Theorem, that allowed them to make right angles. They were also able to solve some specific problems in geometry, such as finding the volume of a truncated square pyramid, etc. However, they did make some advances. They developed the concept of a square unit for area measure. They also used the method of false position to solve certain linear equations: Problem: A quantity and its quarter added become 15. What is the quantity? Solution: If the quantity were 4, and I added 4 and a quarter of 4 (i.e. 1) I would get 5. To get the real answer of 15 from 5, I would multiply by 3. Thus I multiply 4 by 3 to get the quantity (1). (In general, the method is as follows: if L(x) is a linear combination of x s and L(x) = A. let G be an initial guess. If L(G) = A, we re done. Otherwise, A G is the answer. This method was in use in Europe until about a century LG ( ) ago. d They approximated the area of a circle of diameter d as ( d ), which yields an 9 16 approximate value of π = ( ) Multiplication and Division were accomplished by doubling and halving:

2 Examine the following algorithm for division. The problem is dividing 19 by 16. Here, the numbers in each column are repeatedly doubled until doubling again would result in a number larger than 19. Then, numbers in the left column are examined to find a combination adding to 19. The corresponding numbers in the right column are added to obtain the answer Try the algorithm on 186 divided by 3. It s started below: Why are you guaranteed of always being able to get numbers in the left column that add up to the dividend? Now examine the following algorithm for multiplication. The problem is multiplying 0 by 4. Again, we build two columns, starting with 1 on the left side and the larger of the two numbers, 4, on the right. We double both numbers until the left side gets as close as possible to the other number (0) without going over (just like The Price is Right!)

3 Then, starting at the bottom of the left column, find a combination of numbers that add up to 0, and mark those rows Add the corresponding right column numbers: = 480 to get the answer. How do we know we can always find numbers in the left column that will add to the desired factor? Egyptian Fractions: Ancient Egyptians used fractions in their measurements and calculations, but used only unit fractions fractions of the form 1 for positive integer n. They represented all n fractions as the sums of unit fractions. For example, = In order to aid computation, they established tables given fractions written as sums of unit fractions. Below, where is a table from the Rhind Papyrus that gives the denominators of unit factions that add to, for n odd between 5 and 101. n

4 Thus, for example, = Every fraction has a representation (indeed, more than one) as the sum of unit fractions. Leonardo of Pisa showed this to be true. One way of finding such a representation is using the greedy algorithm that subtracts the largest unit fraction possible from the given fraction, then repeatedly subtracts the largest unit fraction possible from each successive difference: = +, then 3 = So, 5 = Of course, = + + = as well. The Egyptians did not always work with the representation given by the greedy algorithm. An Example of Computations with Fractions: /7 * ¼ + 1/8 4 ½ + 1/14 * 5 ½ +1/7 +1/14 Note that all fractions with a denominator of 7 can be determined from this table. To give you some idea of how Egyptians would work with fractions, I will present a nottoo-messy problem below: 5 :

5 I begin by writing ¾ in terms of unit fractions as + ½ + ¼. Then I do my usual doubling algorithm, with the added twist of fractions: * * * These two unit fractions are easy to double; a 1/3 would take using the above table. For reasons that are not clear, the only non-unit fraction the Egyptians were comfortable with was /3. Now for computing 1 1 will become +, and 3 6 we combine things to get: Which, using tables, may have been converted to: See The Crest of the Peacock, by Joseph, for more and messier examples.

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