Math 130
Fall 03
Unit Fractions
A unit fraction is a fraction whose numerator is 1. Sometimes unit fractions are also called Egyptian Fractions because in ancient Egyptian mathematics, most fractions were written as a sum of unit fractions.
1a. Which is bigger, or
?
b. Which is bigger, or
?
c. State a rule for
deciding which fraction is bigger, or
. Justify your rule.
One of the
most famous ancient mathematical documents is the Ahmes Papyrus (also known as
the Rhind Papyrus Ahmes is the name of the scribe who wrote the
text; Rhind is the name of the British man who bought it). The Ahmes Papyrus starts with a table of
“doubles,” i.e. values that are twice a unit fraction. These doubles are all written as the sums of
2, 3, or 4 different unit fractions.
Finding doubles was important in the way the ancient Egyptians did
multiplication and division. No one is
quite sure why the unit fractions all had to be different.
Here is an
example of how we might go about finding some possible representations for the
double of :
1. Start with a
multiple of 5 that has a lot of factors in this case, we’ll try 30, which we will
represent by 30 tiles.
2. Find the unit fractions easily represented with 30 tiles:
15 tiles represent 10 tiles represent
6
tiles represent
5 tiles represent
3 tiles represent
2 tiles represent
1 tile represents
3. The “double” of is represented by 12 tiles. Now we play around to find other combinations
of different unit fractions that sum to 12 tiles.
One
combination is (or 10+2 tiles). This is the combination that appears on the
Ahmes papyrus. Another combination is
.
2a. Find one more
representation for the double of ,
using the above data, with (you will need four different terms).
b. Try using a bigger denominator and see if you can find more representations.
3. Using a similar
method, find some representations for the double of .
4. Now try to double
of .
5. Compare your answers to the values on the Ahmes papyrus.
- Now go to http://www.mathcats.com/explore/oldegyptianfractions.html and play with the puzzles there. You can use the fraction pie tool to estimate. Some of the puzzles are quite difficult.
Copyright 2005, Debra K. Borkovitz. You may copy or edit this material for non-profit, educational use only.
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