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Fractions Egyptiennes - ISISS M. Casagrande (Pieve di Soligo)

Titre du sujet: 
Fractions Egyptiennes
We worked with Egyptian fractions. Egyptians used only fractions like 1/n such that n is a non-zero natural number and did not know negative fractions. These fractions will be called unitary fractions or unit fractions. We want to write proper irreducible fractions a/b, such that a and b are non-zero natural numbers, as a sum of distinct unit fractions. This sum is called Egyptian fraction. The aims of our article are to verify if we can always write a proper irreducible fraction a/b as an Egyptian fraction; to verify if there are different and eventually in finite possible expansions; to explore different ways to expand a proper fraction, comparing various methods in order to understand if there is a preferable one, depending on the results they lead to. We studied Fibonacci's method, Golomb's method and a method based on practical numbers, retracing the original proofs, introducing new results and proposing some variants to the methods. Most importantly, we observed that through Fibonacci's algorithm every proper fraction can be expanded into Egyptian fractions, and the ways to do that are in nite in number. We proposed a new original method based on a geometric approach to the problem. We studied the tree composed of the unit fractions that expand a given proper fraction, designing a function that allows to determine the terms of the tree. Thanks to the tree we can also expand natural numbers and improper fractions.

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