Mathematics 436:History of Mathematics Assignment 2

Due 1/24/06

Here is a table of the hieroglyphics used in Egyptian mathematics to represent the powers of 10 up to one million:

Number 100 101 102 103 104 105 106

1. Express the following numbers in Egyptian hieroglyphics

a) 436 b) 2006 c)3,010,279

2. In the Egyptian book of the dead the Egyptian god Nu is recorded as saying:

I work for you, O ye spirits, we are in number

Express this number in our system.

3. The Rhind Papyrus contains tables of 2/n expressed as a sum of Egyptian unit fractions. Using this and writing 3=2+1 it is easy to write 3/n as a sum of unit fractions. For example the papyrus gives 2/11=1/6+1/66. Thus 3 divided by 11 is (2+1)/11=2/11+1/11=1/6+1/66+1/11. Similarly 4 divided by 11 would be written in the Egyptian system as 1/3+1/33 (the Egyptians were aware that the double of 1/(2n) is 1/n).

Find Egyptian fraction expressions for 5 divided by 11, 6 divided by 11 and 7 divided by 11 as a sum of Egyptian unit fractions.

4. Recall the method of Fibonacci described in class to express a fraction as a sum of Egyptian unit fractions ( to express p/q as a sum of unit fractions begin by finding an integer m so that q/p is between m and m+1, and then consider p/q-1/(m+1) as a new fraction and continuing until the numerator of the result is 1 ).

Use this to express 2/13 as a sum of Egyptian unit fractions.

The expression for 2 divided by 13 in the Rhind Papyrus was not that obtained by Fibonacci's method. The Egyptians seemed to prefer an expression in which unit fractions had even denominators. Can you find such an expression for 2/13? (Hint: In the Rhind Papyrus expression for 2/13 one term was 1/8).