# Add fractions like an Egyptian

Egyptians and Babylonians around 3000 BC had a different way of working with fractions. The ancient Egyptian way also served a unique purpose that we might not appreciate. For example, in paying 8 farm workers with 5 loaves of Barley bread, how can the farm owner split the loaves into 8 equal shares? In our modern notation, each worker would get $\frac{5}{8}$ loaves. But for the ancient Egyptian farm owner, $\frac{5}{8}$ would be more like the problem than the answer because $\frac{5}{8}$ does not tell him how to divide the loaves of bread.

It turns out that the farm owner could give every worker half a loaf (totaling 1/2 x 8 = 4 loaves) and then give each worker one eighth of the remaining loaf. In modern notation, the thought process of the farm owner is expressed in the following way:

$\displaystyle \frac{5}{8}=\frac{1}{2}+\frac{1}{8}$

Another problem that an ancient Egyptian boss might encounter – how to divide 13 loaves of bread among 12 workers? A natural way is to let each worker receive one loaf and then receive one-twelfth of the remaining loaf. If a loaf is small, then making 12 cuts may destroy the loaf. A better way is the following.

$\displaystyle \frac{13}{12}=\frac{1}{2}+\frac{1}{3}+\frac{1}{4}$

Each worker would receive half a loaf (for a total of 6 loaves). Then each worker receives one-third of a loaf (for a total of 4 loaves). The remaining three loaves would produce one-quarter of a loaf for each worker.

Any fraction with 1 in the numerator and a positive integer in the denominator is called a unit fraction. In our notation, $\frac{1}{3}$ and $\frac{1}{10}$ are unit fractions. The ancient Egyptians expressed the unit fractions in hieroglyphic notation. For example:

The above image is found in the Wikipedia entry on Egyptian fractions.

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Egyptian Fractions

An Egyptian fraction is a sum of several unit fractions where all the unit fractions are different. Thus, $\frac{1}{3}+\frac{1}{10}$ is an Egyptian fraction. The sum $\frac{5}{8}=\frac{1}{2}+\frac{1}{8}$ is an Egyptian fraction that encodes the instruction on how to divide 5 loaves of bread among 8 workers. The sum $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}$ is an instruction on dividing 13 units into 12 equal shares.

Fractions, in the way ancient Egyptians used them, clearly were useful, especially in a barter economy, e.g. exchanging labor for foods or other goods. Thus for ancient Egyptians, sum of unit fractions as a way to express fractions had practical advantages over other forms of expression.

On the other hand, in our modern way of life, we think of fractions differently. Fractions still tell us the relative size of a given transaction or action (e.g. the cost for half a pizza is \$5). Arithmetic calculation can be performed in decimal numbers. Thus we do not have a need for the Egyptian form of fractions. However, Egyptian fractions are studied for historical purposes. Egyptian fractions are also an object of study in number theory and are also a source of interesting recreational math problems or puzzles. See here for a very comprehensive look at Egyptian fractions from a mathematical point of view. The Wikipedia entry on Egyptian fractions is also a good resource.

Egyptian fractions can also be a good way to teach fractions to children. Including Egyptian fractions into the lesson plan can provide additional insight. In preparing for this post, I worked out a few Egyptian fraction calculations to get a sense of how it works. I found that dividing loaves of bread among workers is a great way to think about fractions. The dividing of bread as a device can actually be an alternative mechanism for representing and adding fractions.

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How to Obtain Egyptian Fractions

In this section, we explain how Egyptian fractions work. For example, given a fraction $\frac{T}{B}$, we show how to derive an Egyptian fraction:

$\displaystyle \frac{T}{B}=\frac{1}{i}+\frac{1}{j}+\frac{1}{k}+ \cdots + \frac{1}{w}$

In fact, for any given fraction $\frac{T}{B}$, there can be infinitely many ways to express it as Egyptian fractions. Consider the following examples.

Example 1
Let’s say the farm owner pays 2 loaves of bread among 5 workers. How can this be done?

The first step is to give each worker 1/3 loaves for a total of 5/3 loaves. Then 1/3 loaves remain and each worker would get 1/5 of that or 1/15 loaves. The following is the answer.

$\displaystyle \frac{2}{5}=\frac{1}{3}+\frac{1}{3} \times \frac{1}{5}=\frac{1}{3}+\frac{1}{15}$

In this scenario, each worker gets 1/3 of a loaf and then gets 1/5 of 1/3 of a loaf. How do we know to start with 1/3? The key is to find the largest unit fraction less than 2/5. Since 2/5 = 0.4, the largest unit fraction in this case is 1/3. $\square$

Example 2
The problem now is to divide 4 loaves of bread among 5 workers.

Note that 4/5 = 0.8. The largest unit fraction less than 0.8 is 1/2. So distribute 1/2 loaves to each worker for a total 5/2 = 2.5 loaves. Then 3/2 = 1.5 loaves remain, which are divided by 5 workers again, giving each worker 3/10 loaves. The following shows the calculation.

$\displaystyle \frac{4}{5}=\frac{1}{2}+\frac{3}{2} \times \frac{1}{5}=\frac{1}{2}+\frac{3}{10}$

The fraction 3/10 is not a unit fraction. We can use the same idea to break it up. Now 3/10 is like dividing 3 loaves among 10 workers. What is the largest unit fraction less than 3/10=0.33? It is 1/4. So give each worker 1/4 loaves for a total of 10/4 = 2.5 loaves. The remaining 1/2 loaves are divided by 10 workers.

$\displaystyle \frac{4}{5}=\frac{1}{2}+\frac{3}{10}=\frac{1}{2}+ \biggl[ \frac{1}{4}+\frac{1}{2} \times \frac{1}{10} \biggr]=\frac{1}{2}+\frac{1}{4}+\frac{1}{20}$

The answer is that in distributing 4 loaves to 5 workers, each worker gets 1/2 of a loaf, then 1/4 of a loaf. What remains would be 1/4 of a loaf. From the remainder of 1/4, each worker gets 1/5 of that, meaning that each worker gets 1/20 of a loaf. $\square$

The method described in the above two examples is called the greedy algorithm. This is because at each step, the algorithm is to (greedily) find the largest share of bread that can be taken away. Then repeat the same greedy process on the remaining portion of the bread. In this process, the series of unit fractions always decrease and eventually the process will stop. Interestingly, the greedy algorithm can be relaxed and still produces an equivalent Egyptian fraction. Thus there are more than one way to obtain Egyptian fractions. The following example shows how it is done.

Example 3
Let’s repeat Example 2 but this time do not take the largest unit fractions less than the given fraction at each stage.

In the first step take 1/3 away from 4/5 (instead of taking 1/2). In the bread analogy, each worker gets 1/3 loaves for a total of 5/3 loaves. Then 7/3 loaves remain, which are shared by 5 workers.

$\displaystyle \frac{4}{5}=\frac{1}{3}+\frac{7}{3} \times \frac{1}{5}=\frac{1}{3}+\frac{7}{15}$

To break up 7/15, the greedy algorithm would take 1/3 since 1/3 is the largest unit fraction less than 7/15 = 0.467. For illustration, we take 1/5 away from 7/15 for a total of 15/5 = 3 loaves. Then the remaining 4 loaves are shared by 15 workers.

$\displaystyle \frac{4}{5}=\frac{1}{3}+\frac{7}{15}=\frac{1}{3}+ \biggl[ \frac{1}{5}+4 \times \frac{1}{15} \biggr]=\frac{1}{3}+\frac{1}{5}+\frac{4}{15}$

Next, divide 4 loaves of bread among 15 workers. The largest unit fraction less than 4/15 = 0.267 is 1/3. But we take 1/6 away. Each of 15 workers get 1/6 loaves for a total of 15/6 = 2.5 loaves. Then 3/2 = 1.5 loaves remain, which are shared by 15 workers.

$\displaystyle \frac{4}{5}=\frac{1}{3}+\frac{1}{5}+\frac{4}{15}=\frac{1}{3}+ \frac{1}{5}+\biggl[ \frac{1}{6}+\frac{3}{2} \times \frac{1}{15} \biggr]=\frac{1}{3}+\frac{1}{5} +\frac{1}{6}+\frac{1}{10}$

The answer is that each worker gets 1/3 of a loaf, 1/5 of a loaf, then 1/6 of a loaf and finally 1/10 of a loaf. Of course, this division would be equivalent to the one in Example 2 (i.e. the same of amount of bread for each worker). $\square$

Example 3 shows that there are more than one way to express a fraction in Egyptian fractions (in fact, infinitely many ways as shown below). Example 3 also shows that if the greedy algorithm is not used (if the largest share of bread is not taken away at each step), it is possible it will take more steps to distribute the bread.

Example 4
We now demonstrate that an Egyptian fraction can be expressed in an infinite number of ways. The example we use is

$\displaystyle \frac{5}{8}=\frac{1}{2}+\frac{1}{8} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$

We demonstrate that starting with (1), we can derive infinitely many new (but equivalent) Egyptian fractions. The following relationship will be useful.

$\displaystyle 1=\frac{1}{2}+\frac{1}{3}+\frac{1}{6} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*)$

The idea is that we replace the last term in a given Egyptian fraction with three new terms using relation (*). The last term in the first Egyptian fraction (1) is $\frac{1}{8}$. So multiply (*) by $\frac{1}{8}$ to obtain

$\displaystyle \frac{1}{8}=\frac{1}{8} \biggl[\frac{1}{2}+\frac{1}{3}+\frac{1}{6} \biggr]=\frac{1}{16}+\frac{1}{24}+\frac{1}{48}$

which then replaces the $\frac{1}{8}$ in (1) to produce

$\displaystyle \frac{5}{8}=\frac{1}{2}+\frac{1}{16}+\frac{1}{24}+\frac{1}{48} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$

The last term in (2) is $\frac{1}{48}$. Multiply (*) by $\frac{1}{48}$ and plug the result back into (2) to obtain

$\displaystyle \frac{5}{8}=\frac{1}{2}+\frac{1}{16}+\frac{1}{24}+\frac{1}{96}+\frac{1}{144}+\frac{1}{288} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$

By continuing the same process on replacing the last term, the fraction $\frac{5}{8}$ can be expanded to include more and more unit fractions. By performing the same process, any Egyptian fraction has infinitely many different expressions as Egyptian fractions. $\square$

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Rhind Papyrus

Both of the references cited above (links are repeated here and here) mention the Rhind papyrus. Both have an image of a portion of the papyrus. Rhind papyrus was named after Henry Rhind, the Scottish antiquarian who purchased it in 1858 in Luxor, Egypt. It dates to around 1650 BC and is one of the best known original documents on Egyptian mathematics. A table for Egyptian fractions $\frac{2}{n}$ is found in the early part of the papyrus along with problems and puzzles. The table lists $\frac{2}{n}$ as sum of unit fractions for all odd values of $n$ from 3 to 101.

As observed earlier, Egyptian fraction representation is not unique. It is clear that the author of the papyrus preferred some forms over the others. For example, $\frac{2}{13}=\frac{1}{8}+\frac{1}{52}+\frac{1}{104}$ is found in the papyrus while the simpler form $\frac{2}{13}=\frac{1}{7}+\frac{1}{91}$ is not. The scribe seemed to favor unit fractions with even denominators, possibly to make multiplication and division easier.

Some of the $\frac{2}{n}$ fractions in the Rhind papyrus are derived from smaller calculations. For example,

$\displaystyle \frac{2}{81}=\frac{1}{54}+\frac{1}{162}$

is found in the papyrus. Since 27 x 3 = 81, $\frac{2}{81}$ is obtained by multiplying $\frac{2}{3}=\frac{1}{2}+\frac{1}{6}$ by $\frac{1}{27}$.

$\displaystyle \frac{2}{81}=\frac{1}{27} \times \frac{2}{3}=\frac{1}{27} \biggl[ \frac{1}{2}+\frac{1}{6} \biggr]=\frac{1}{54}+\frac{1}{162}$

Another example, $\frac{2}{87}=\frac{1}{58}+\frac{1}{174}$ is obtained by multiplying $\frac{2}{3}=\frac{1}{2}+\frac{1}{6}$ by $\frac{1}{29}$. Thus, whenever the denominator is a composite number, the given fraction can be obtained from a smaller Egyptian fractions.

According to the greedy algorithm, the fraction $\frac{2}{2m+1}$ can be expressed as:

$\displaystyle \frac{2}{2m+1}=\frac{1}{m+1}+\frac{1}{(m+1) \ (2m+1)}$

Once the formula is known, the entire $\frac{2}{n}$ in the Rhind papyrus can be obtained easily. However, the creator of the papyrus chose not to use the above short and seemingly convenient form of Egyptian fractions. We do not know fully why. One reason may be that using the above form would require large denominators in the second unit fraction. All the denominators in the $\frac{2}{n}$ table are no larger than 999. Thus $\frac{2}{83}$ is

$\displaystyle \frac{2}{83}=\frac{1}{60}+\frac{1}{332}+\frac{1}{415}+\frac{1}{498}$

rather than

$\displaystyle \frac{2}{83}=\frac{1}{42}+\frac{1}{3486}$

Deriving at the first form of $\frac{2}{83}$ would require more work. After making some calculations on Egyptian fractions on my own, I cannot help but have admiration for the person who made the calculations that appeared in the Rhind papyrus. I have help in the form of a hand held calculator. They would have to go through a lot of trials and errors to get at the results.

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A Number Theory Problem

Egyptian fractions are a source of interesting problems in the mathematical area of number theory. One example is the Erdos-Straus conjectures, which states that for any positive integer $n \ge 2$, the fraction $\frac{4}{n}$ can be expressed as a sum of unit fractions as follows:

$\displaystyle \frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$

Another way to state the problem is that the above equation always has solutions in positive integers $x$, $y$ and $z$.

Based on the above discussion, the fraction $\frac{4}{n}$ can always be expressed as a sum of unit fractions (just use the greedy algorithm). But the result may have just two terms or more than 3 terms. The requirement is that it is a sum of three unit fractions (not necessarily all different). Example 3 shows that $\frac{4}{5}=\frac{1}{2}+\frac{1}{4}+\frac{1}{20}$. The question is, can something similar be done for $\frac{4}{n}$ for all other values of $n$? Computer calculations had shown that $\frac{4}{n}$ can be expressed a sum of three unit fractions for all $n \le 10^{17}$. At this point, there is no answer for values of $n$ above $10^{17}$.

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Concluding Remarks

Egyptian fraction is a fascinating topic in many respects. The concept has a practical significance for Egyptians and Babylonians of 3000 BC, e.g. to address the division of wages. Though the practical reason is no longer relevant for us, learning about Egyptian fractions does add to our understanding of fractions. Thus Egyptian fraction is a great educational tool for teaching fractions. There is also the historical connection. Learning about Egyptian fractions makes the history real and vivid. Knowing how to express $\frac{2}{5}$ or $\frac{2}{7}$ as sum of unit fractions, for example, can in a sense transport us back to the time the scribe wrote on the Rhind papyrus. We would be doing the same math problems they were doing in ancient Egypt. We are far apart from the Egyptians of 3000 BC in time and space and also in culture and languages. Yet their concept of fractions, though different in usage, are accessible to us.

The title of this post could also be “Add fractions like an ancient Egyptian” or “how ancient Egyptians add fractions with a purpose.” One thing is clear. The concept opf fractions is purpose driven math for ancient Egyptians. It should be for us too.

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Exercises

The $\frac{2}{n}$ table in the Rhind papyrus is a good source for exercises to practice Egyptian fractions. The table contains fractions for $\frac{2}{n}$ for values of odd integers $n$ from 3 to 101. See if the Egyptian fractions you come up with agree with the ones in the papyrus (see the table in the first two links below).

Another good source of practice problems is through the Erdos-Straus conjecture discussed above. The goal is to express $\frac{4}{n}$ as a sum of three unit fractions. Note that $\frac{4}{n}$ can be expressed as a sum of unit fractions using the greedy algorithm. But the results may have more than three terms. So the challenge is to look for three terms.

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$\copyright$ 2017 – Dan Ma