Over the years, different civilisations have had very different ways of looking at maths and arithmetic. One of these was (as you can imagine from the title of this post) the Ancient Egyptians. They had a very interesting way of looking at fractions, that is quite different, but still similar, to what we learn and use today. It is worth noting before we begin that there is every chance that the Egyptians didn't actually use these, and instead used another long-lost system (or just didn't use them at all).
The essential part of Egyptian fractions follows the baseline that only unit fractions (fractions with a 1 as the numerator) are allowed. e.g to write ⅔, you would have to write ⅓ + ⅓. Often however, repeat fractions would not be allowed, so you would have to write ½ + ⅙. This is a very interesting way of looking at fractions.
The question becomes: Can this be done for every fraction. Can you, at will, take any fraction, and express it as the sum of only unit fractions? Furthermore, can you always avoid repeating a fraction?
CHALLENGE 1: Try to find a way to express a few different fractions as the sum of non-repeating, unit fractions. If you're really feeling up to it, try to find an 'algorithm' that works for any fraction. The answer is below.
The answer to that, of course, has been pondered at by many people for a long time. Eventually, people came up with what's called 'The Greedy Algorithm'.
The essential steps behind the greedy algorithm state that you must always take the biggest unit fraction that is lower than the original fraction that you are trying to express, hence it's name.
e.g:
For our example earlier of ⅔, the biggest unit fraction that is less than or equal to ⅔ is ½. So ½ will be the first fraction in our chain. Taking ½ away from ⅔ leaves just ⅙, which is a unit fraction, so we can finish there. For some fractions, this chain will get very complicated and take a long time to finish.
CHALLENGE 2: Find the fraction chains for more complicated fractions like 11/12, 15/16 and so on. Post any that you have found in the comments.
This algorithm, by the way, ALWAYS WORKS, and isn't only used for Egyptian fractions. Any problem where you are required to break something down into smaller parts can use this algorithm.
If you have any questions, let me know (or just google them i'm busy ok),
Noah
The essential part of Egyptian fractions follows the baseline that only unit fractions (fractions with a 1 as the numerator) are allowed. e.g to write ⅔, you would have to write ⅓ + ⅓. Often however, repeat fractions would not be allowed, so you would have to write ½ + ⅙. This is a very interesting way of looking at fractions.
The question becomes: Can this be done for every fraction. Can you, at will, take any fraction, and express it as the sum of only unit fractions? Furthermore, can you always avoid repeating a fraction?
CHALLENGE 1: Try to find a way to express a few different fractions as the sum of non-repeating, unit fractions. If you're really feeling up to it, try to find an 'algorithm' that works for any fraction. The answer is below.
The answer to that, of course, has been pondered at by many people for a long time. Eventually, people came up with what's called 'The Greedy Algorithm'.
The essential steps behind the greedy algorithm state that you must always take the biggest unit fraction that is lower than the original fraction that you are trying to express, hence it's name.
e.g:
For our example earlier of ⅔, the biggest unit fraction that is less than or equal to ⅔ is ½. So ½ will be the first fraction in our chain. Taking ½ away from ⅔ leaves just ⅙, which is a unit fraction, so we can finish there. For some fractions, this chain will get very complicated and take a long time to finish.
CHALLENGE 2: Find the fraction chains for more complicated fractions like 11/12, 15/16 and so on. Post any that you have found in the comments.
This algorithm, by the way, ALWAYS WORKS, and isn't only used for Egyptian fractions. Any problem where you are required to break something down into smaller parts can use this algorithm.
If you have any questions, let me know (or just google them i'm busy ok),
Noah
Have fun with the challenges!!
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