Add fractions like an Egyptian

add-fractions-like-egyptians

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:

egyptian-hieroglyphic-unit-fractions

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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Links

The links indicated above are repeated here. The discussion in this post is a simple and basic mathematical introduction. The first link below is a particularly good resource, providing more details and insight.

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\copyright 2017 – Dan Ma

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What is purpose driven math?

The following is an adorable math cartoon that I found recently.

ye-olde-math-shoppe

\text{Roz Chast:  http://rozchast.com}

The cartoon depicts a shop for math related products and services. If there ever was such a shop, some of the advertised services are obsolete. Square root extraction was offered as a bonus for purchase of $50 or more. With computing devices so readily available (hand-held calculators, smart phones, etc), there is surely no need for square root extraction. There surely is no need for arithmetic services for anyone who knows how to punch buttons in a pocket calculator (or tab the screen of a device). I really got a chuckle with “fresh prime numbers daily!”

It turns out a daily supply of fresh prime numbers is not just something only mathematicians care about. Internet commerce depends on a daily fresh supply of prime numbers. In fact, a fresh daily supply of prime numbers that are several hundred or even thousands digits long.

Whenever you enter a credit card number on the Internet, it would be very risky for you if the credit card number is sent over the Internet without masking (encryption). Then other people can read it and make use of it without your authorization. It would be like shouting secrets in a crowded room. Before it is sent, the credit card number is encrypted. Once it gets to the destination (the online merchant), the credit card number is decrypted. One of the most commonly used encryption/decryption schemes is the RSA algorithm, which is built on prime numbers.

Every time a piece of sensitive information, e.g. your credit card number, is sent over the internet, the RSA algorithm encrypts the information using a public key, which consists of a large number N that is a product of two large prime numbers p and q (note that the two prime numbers p and q are not known to the public, and that only N is). Once the encrypted information arrives at the merchant, it is decrypted using a private key, which consists of the two individual prime numbers p and q. The security of the RSA algorithm rests on the difficulty of uncovering p and q if only the product N is known. To get a better sense of how RSA works, see this article from Slate. See here for another introduction to RSA algorithm.

For RSA algorithm to work properly and securely, a fresh daily supply of prime numbers is necessary. What is purpose driven math? Due to its wide spread use in online commerce, the study of prime numbers would be called purpose driven math.

It is interesting to point out that prime numbers were not always regarded as purpose driven math. Euclid proved that there are infinite number of prime numbers (circa 300 BC). Prime numbers had a special hold on humanity ever since, even though they were studied more in a “math for math’s sake” kind of way. Many people through the ages found prime numbers interesting and important, but never as a way to enable commerce. It is not always easy to tell ahead of time whether a given mathematical construct is purpose driven math. Thus what is considered esoteric math today may become purpose driven math tomorrow.

There are actually many other mathematical theories and notions that were not meant to be purpose driven at the beginning but were found to be useful later. Some people disdain (or at least do not value) mathematical theories that have no practical purpose. I have a friend who always ask “what is the use of this?” whenever I mention a math concept in a conversation. There are mathematicians who disdain mathematics that have a practical purpose. The notion of prime numbers is a bridge between these two extreme camps. As an abstract mathematical theory, prime numbers are beautiful and interesting. On the other hand, prime numbers make it possible to transmit sensitive information. In this blog, I will discuss bridge concepts such as prime numbers and other purpose driven math.

This blog is a small attempt to dispel the notion that math is useless or meaningless. I hope that learning purpose driven math will be an antidote to math phobia or even math hatred. I have no illusion that the blog will be a cure all for math phobia. I hope that some readers will come away with a sense that there is more to math than the math taught at school. There is purpose driven math and there is school math. For many, school math is fear and anxiety inducing. Many students do not like math but have to endure it because schools require it. This blog is to show that there is another face of math.

I hope that those who hate school math will hate math less (or tolerate more of it) after knowing more about purpose driven math through this blog or other resources. School math is required in order to get through school. But school math also gives the necessary background to understand and use purpose driven math. For example, to learn how money grows in a bank account, we need a tool that measures a growth phenomenon that doubles over a certain period of time (e.g. at 10% annual interest rate, the money doubles approximately every 7 years). This is called exponential growth and the tool for measuring this type of growth phenomenon is called the exponential function, which is taught in high school.

Whenever a teacher mentions anyone of these terms or phrases – solving for x, the slope of a line, quadratic equations, solving an inequality, fractions and sines and cosines and many more – some students are immediate turned off. I realize that sometimes the way math is taught at school is not very effective. Purpose driven math is another side of math that is often omitted when school math is taught. Math is not only useful, it is one of the ways through which we make sense of the world. Thus mathematics is on par with other worthy endeavors such as science, literature, philosophy, art and music.

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\copyright 2017 – Dan Ma