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When Am I Ever Going to Use This?
When Am I Ever Going to Use This?
Egyptian Fraction Representations have ties to number theory, algorithms, and the history of mathematics. If you want to learn more about how Egyptians did mathematics, you could consider checking out this article, though you may need an account or to check out the article with database access from a library that has it. As you'll learn if you keep reading, the final question of this month is still an open question: Is there an algorithm for the most efficient Egyptian Fraction Representation? If you like looking into these algorithms, you might enjoy a career in pure mathematics, studying this problem!
Also... Understanding the history of mathematics, how ideas have developed over time, and how they were used when they were first developed can be interesting and worthwhile in itself. However, if you are interested in using math to better understand past societies, you might look into Mathematical Anthropology (here is somewhere you could start).
What We Learned
What We Learned
(or watch on YouTube here)
Does every fraction has an EFR?
Does every fraction has an EFR?
Yes, every fraction can be represented as a sum of unique unit fractions! One way of finding such a sum is called the greedy algorithm. The Numberphile video linked to the left does a great job explaining this algorithm and using general statements to do so (thus proving that it will always work).
The gist of the algorithm is this: If the fraction we are working with is a/b, then we can find the largest unit fraction, 1/n, by taking b divided by a and rounding up (in other words, n, is the ceiling of b/a).
For example, with 4/5, I would do 5/4=1.25, and round up to 2. So, the largest unit fraction less than 4/5, is 1/2.
Then, I would subtract that largest fraction, and keep repeating this process until I end up at a unit fraction.
Continuing my example of 4/5, I would do the following:
4/5 - 1/2 = 3/10
10/3 = 3.3ish, round up to 4
3/10 - 1/4 = 1/20
I have reached a unit fraction so I am done!
I have: 4/5 = 1/2 + 1/4 + 1/20
Is there more than one EFR for a given fraction?
Is there more than one EFR for a given fraction?
Yes! Consider the fraction 5/6. This fraction could be written as 1/2 + 1/3, or 1/3 + 1/4 + 1/6 + 1/12, and many more.
In fact, there are infinitely many Egyptian Fraction Representations for a number because we can always rewrite one of our unit fractions as sums of other unit fractions.
One way we might do this is by applying the greedy algorithm to our fraction. Let's use the Greedy Algorithm to rewrite 1/3 as a sum of other unit fractions: The largest unit fraction that is less than 1/3 is 1/4. Now, 1/3 - 1/4 = 1/12. So, 1/3 = 1/4 +1/12, and thus 5/6 = 1/2 + 1/4 + 1/12
And here we have another equivalent fraction, not already shown in our image. And in fact, when we subtract the largest unit fraction less than our current unit fraction, we will always get a unit fraction. So, any unit fraction 1/n can be rewritten as the sum of 1/(n+1) + 1/(n(n+1)), as shown below.
Is there an algorithm that gives the most efficient EFR? (least number of unit fractions)
Is there an algorithm that gives the most efficient EFR? (least number of unit fractions)
While the Greedy Algorithm, described earlier, always gives an Egyptian Fraction Representation, it's not always the most efficient one.
For example, the Egyptian Fraction Representation for 5/31 using the Greedy Algorithm would give:
1/7 + 1/55 + 1/3979 + 1/23744683 + 1/1127619917796295
However another EFR for 5/31 is:
1/7 + 1/62 + 1/434
Currently, there is no known algorithm for producing the Egyptian Fraction Representation with the fewest number of terms, but there are many other algorithms you can use for creating Egyptian Fraction Representations, which you can learn about in this Wolfram Alpha article.
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