Throughout history, different civilisations have had different ways of representing numbers. Some of these systems seem strange or complicated from our perspective. The ancient Egyptian ideas about fractions are quite surprising. They wrote,for example

(but with their numerals), these are $1/5$, $1/16$ and$1/429$ in modern notation.

The ancient Egyptians had no notation for and probably no concept of, say $2/7$ as a fraction. They didn't have any fraction that we would write with a numerator greater than 1*. To them, "2 shared between 7" (i.e. $2/7$) was a problem, and writing = $1/7 +1/7$ was just a restatement of that problem. Their answer to this problem was:

2 shared between 7 : or alternatively
there are also other answers, can you find any?

You may have questions of your own you'd like to investigate about this fraction system.
Some questions which occured to us:
A little research on this topic will show that famous mathematicians have asked and answered questions about the Egyptian fraction system for hundreds of years. I quickly found references to results in this field proved in the 1200s and in the 2000s. I also found some open questions - things mathematicians think are true, but no-one has thought of how to prove them yet.

Why not stop reading now, and find various Egyptian fraction sums for yourself. Can you answer any of your own questions, or the ones above?

... or try one of the following:

Fibonacci's greedy algorithm

At every stage, write down the biggest possible unitary fraction that is less than the fraction you're working on.
Does this work?
Does it always find the shortest possible sum?
Does the method always work - do the sums always finish? Can you prove this?
Why is it called a greedy algorithm? - what do these words mean in a mathematical context?


Formulae for $1/n$ and $2/n$

It is likely that the Egyptians developed specific methods for converting one of their fractions (ie numerator=1) into Egyptian fraction sums. Experiment with this and see what methods you can find.

The Rhind Mathematical Papyrus is an important historical source for studying Egyptian fractions - it was probably a reference sheet, or a lesson sheet and contains Egyptian fraction sums for all the fractions 2/3, 2/5, 2/7 ...2/101. Why did they only include the odd ones? Can you find sums for these? Can you find formulae or methods for these? What else can you find out about the Rhind Mathematical Papyrus?


$4/n$ and $3/n$

In the 1940s, the mathematiciansPaul Erdos and Ernst G. Straus conjectured that every fraction with numerator = 4 can be written as an Egyptian fraction sum with three terms.
If you have found an example that appears to need more than three, can you find an alternative sum?
Can you find a reason why it must work, or a counter-example - the conjecture isn't yet proved. It is proved for $3/n$.


The Eye of Horus

Often it was good enough to use only the fractions {2,4,8,16,32 and 64} to get a fraction that is close enought to any specific fraction. Pick some fractions and convert them to this form of Egyptian fraction. You might like to research how these particular fractions were written down in pictorial form.



*There is evidence that the specific fraction $2/3$ was used by the Egyptians, and $3/4$ sometimes as well. They had special symbols for these two fractions.



old version:

Did you know that the Egyptians wrote all their fractions using what we call unit fractions? A unit fraction has $1$ as its numerator (top number). Here are some examples:

$$\frac{1}{5}, \frac{1}{3}, \frac{1}{2}$$

They expressed all fractions as the sum of unit fractions, but they weren't allowed to repeat the same unit fraction in the addition. So we couldn't write:

$$\frac{3}{8} = \frac{1}{8}+\frac{1}{8}+\frac{1}{8}$$

because we've used $\frac{1}{8}$ three times.

However, this would be fine:

$$\frac{3}{8}= \frac{1}{4}+\frac{1}{8}$$

How could the Egyptians write $\frac{3}{4}$? Are there any other ways?

What is $\frac{2}{3}$ written as the sum of unit fractions? Again, investigate different ways of doing this.

Find some more fractions (say three or four) which you can write as the sum of unit fractions.