Egyptian fractions
The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.
Problem
Egyptian Fractions printable sheet
The ancient Egyptians didn't write fractions with a numerator greater than 1 - they wouldn't, for example, write $\frac{2}{7}$, $\frac{5}{9}$, $\frac{123}{467}$.....
Instead they wrote fractions like these as a sum of different unit fractions.
There are several NRICH problems based on Egyptian fractions. You can start by exploring unit fractions at Keep it Simple
In this problem we are going to start by considering how the Egyptians might have written fractions with a numerator of 2 (i.e. of the form $\frac{2}{n}$).
For example
$\frac{2}{3} = \frac{1}{3} + \frac{1}{3}$ (but since these are the same, this wasn't allowed.)
or
$\frac{2}{3} = \frac{1}{3} + \frac{1}{4} + \frac{1}{12}$
or
$\frac{2}{3} = \frac{1}{3} + \frac{1}{5} + \frac{1}{20} + \frac{1}{12}$
or
$\frac{2}{3} = \frac{1}{3} + \frac{1}{6} + \frac{1}{30} + \frac{1}{20} + \frac{1}{12}$
or
$\frac{2}{3} = \frac{1}{4} + \frac{1}{12} + \frac{1}{7} + \frac{1}{42} + \frac{1}{31} + \frac{1}{930} + \frac{1}{21} + \frac{1}{420} + \frac{1}{13} + \frac{1}{156}$
and so on, and so on!!
You might want to check that these are correct.
(If you can't see how these have been generated, take a look at Keep it Simple )
BUT wouldn't it be simpler to write it as the sum of just two different unit fractions?
For $\frac{2}{3}$ that's quite easy.........$\frac{2}{3} = \frac{1}{2} + \frac{1}{6}$
But is it always so easy?
Try some other fractions with a numerator of 2.
Can they also be written as the sum of just two different unit fractions?
Can all fractions with a numerator of 2 (i.e. of the form $\frac{2}{n}$) be written as the sum of just two different unit fractions?
You might want to explore fractions of the form $\frac{3}{n}$, $\frac{4}{n}$, $\frac{5}{n}$...... and think about how the Egyptians would have represented these, using sums with the least number of unit fractions.
You might like to take a look at a follow up problem, The Greedy Algorithm
NOTES AND BACKGROUND
The ancient Egyptians lived thousands of years ago, how do we know what they thought about numbers? 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. You can find references to results in this field that were proved in the 1200s and in the 2000s, and you can also find some open questions - things mathematicians think are true, but have not been proved yet.
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.
For example, they wrote $\frac{1}{5}$, $\frac{1}{16}$ and $\frac{1}{429}$ as
(but using their numerals)
They didn't write fractions with a numerator greater than 1 - they wouldn't, for example,write $\frac{2}{7}$, $ \frac{5}{9}$, $ \frac{123}{167}$.... although there is evidence that the specific fraction $\frac{2}{3}$ was used by the Egyptians, and $\frac{3}{4}$ sometimes as well. They had special symbols for these two fractions.
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 $\frac{2}{3}$, $ \frac{2}{5}$, $ \frac{2}{7}... \frac{2}{101}$.
Why did they only include the odd ones?
$\frac{4}{n}$ and $\frac{3}{n}$
In the 1940s, the mathematicians Paul 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 $\frac{3}{n}$.
Getting Started
Before attempting this problem, it might be a good idea to take a look at Keep it Simple
Student Solutions
This was a tough one, well done to lots of you who sent in lots of different examples, but only Rosie gave a reason and general rule for her findings:
Great, can anyone use this to find $\frac{3}{n},\frac{4}{n},\frac{5}{n}$ and so on?
Graham and Nolan from Denver in Colorado, USA, sent in this slightly different result which shows how any fraction with 2 in the numerator can be written as a the sum of three unit fractions.
$\dfrac{2}{n}=\dfrac{1}{n}+\dfrac{1}{n+1}+\dfrac{1}{n(n+1)}$.
You could test this result by trying out some values or prove it with algebra, but can you also explain how it relates to Rosie's work from Keep it simple above?
Teachers' Resources
Why do this problem?
Possible approach
This problem follows on from Keep it Simple
You could choose to set the scene briefly by asking students what they know about mathematics throughout history, establishing the idea that some historical maths is distinctly odd to our modern view point.
Explain that the ancient Egyptians didn't write fractions with a numerator greater than 1 but expressed every fraction as the sum of different unit fractions.
Work through the example of $\frac{2}{3}$ as the problem suggests, asking students to lengthen each successive row by substituting each unit fraction by a different pair, using methods the students met in Keep It Simple
Establish that we can keep lengthening the expression for any $\frac{2}{n}$ fraction but what would have been of real value to the Egyptians would have been a method for expressing these fractions in the shorteset possible way, i.e. using just two different unit fractions.
Confirm that this is possible for$\frac{2}{3}$ and then set the challenge to choose their own $\frac{2}{n}$ fraction and express it as the sum of just two unit fractions. Any that can't be done can be written up on the board for the rest of the class to attempt.
Stop the class, and ask them to step back from number crunching and share any discoveries. Listen for any generalisations and record them for discussion.
Students could follow this up by exploring fractions of the form$\frac{3}{n}$, $\frac{4}{n}$ etc and be challenged to express them in as short a way as possible.
Key questions
What do we already know that could help?
Possible support
Keep It Simple provides a good introduction to this activity.
Possible extension
Some students might wish to undertake research about the Rhind Mathematical Papyrus.
Students could be directed to The Greedy Algorithm