Professor Stewart's Hoard of Mathematical Treasures (15 page)

BOOK: Professor Stewart's Hoard of Mathematical Treasures

5.58Mb size Format: txt, pdf, ePub

ads

Doing arithmetic with unit fractions is weird, but possible. Our method is very different: we ‘put both fractions over a common denominator’ (page 310) like this:

We can see that the result is roughly 1

, which isn’t obvious from the Egyptian fractions.

Nevertheless, the Egyptians did amazing things with their symbolism. Our most important source for their work is the Rhind mathematical papyrus, now in the British Museum. Alexander Rhind bought the papyrus in 1858 in Luxor; it seems
to have been unearthed by unauthorised excavations near the Ramesseum.

Part of the Rhind mathematical papyrus.

The papyrus dates to around 1650 BC, in the Second Intermediate Period. The scribe Ahmose copied it from an earlier text from the time of the 12th dynasty pharaoh Amenemhat III, two centuries earlier, but the original text is lost. It measures 33 cm by 5 m, and even now scholars do not understand everything on it. However, one remarkable section, about one-third of one side, deals with unit-fraction representations of numbers of the form 2/n, where n is odd and runs from 3 to 101.

Ahmose’s results here can be summed up in a table. To simplify the printing and improve legibility, an entry like

means that

The table is impressive, but also raises a number of questions. How did whoever first found these representations discover
them? Why did the scribes prefer these particular representations?

Expressing 2/n, for n odd, as a sum of at most four unit fractions.

In 1967, at the request of Richard Gillings, C. L. Hamblin programmed an early electronic computer belonging to Sydney University to list all possible ways to represent the fractions 2/n in Ahmose’s table as sums of unit fractions. The results led Gillings to argue that:

• the Egyptians preferred small numbers;

• they preferred sums with two unit fractions to those with three, and sums with three unit fractions to those with four;

• usually they liked the first number to be as small as possible, but not when that made the last number too big;

• they preferred even numbers, even when this led to bigger numbers or more of them.

For example, the computer found that

but both numbers are odd and 703 is large. The scribes preferred

with two even numbers and nothing very big. Gillings gives an extensive discussion in his Mathematics in the Time of the Pharaohs. This book is a bit long in the tooth, and the historical study of Egyptian mathematics has moved on, but it still has a lot of interesting things to say.

The Greedy Algorithm

Egyptian fractions are obsolete for practical arithmetic, but still very much alive as mathematics, and you can learn a lot about modern fractions by thinking about Egyptian ones. For a start, it’s not obvious that every fraction less than one has an ‘Egyptian representation’ - as a sum of distinct unit fractions - but it’s true. Leonardo of Pisa, the famous ‘Fibonacci’ (Cabinet, page 98), proved this in 1202, showing that what is nowadays called the ‘greedy algorithm’ always does the job. An algorithm is a specific method of calculation that always produces an answer, like a computer program.

The greedy algorithm begins by finding the largest unit fraction that is less than or equal to the fraction you want to represent - that’s what makes it greedy. Subtract this fraction
from the original fraction. Now repeat, looking for the largest unit fraction that is different from the one you got the first time, but less than what’s left. Keep going.

Amazingly, this method eventually reaches a unit fraction and stops.

Let’s try out the greedy algorithm on the fraction