Le frazioni Egizie nella didattica

Cerasaro, Silvia ; Tomassi, Laura

Matematica, cultura e società, Série 1, Tome 8 (2023) no. 2, pp. 157-173

Résumé

Every rational number can be written as the sum of unit fractions, ie as an 'Egyptian fraction', by a process called `disintegration'. Some ideas of historical origin are illustrated, applied in didactic activities related to learning the notion of fraction and rational number in lower secondary school or in the first two years of upper secondary school. The proposed topic also allows us to approach and illustrate an open problem in current mathematics. Historical texts provide indication of various applications. Problems of dividing goods among anumber of people are found in the Rhind papyrus with the constraint of making the dividing easier to do in practice; in the case in which the division concerned land, disaggregating had to correspond to making the division not onlys impler but also more equitable and with the least number of cuts of the plots. The solution corresponded to the sum of a certain number of fractional units. A demonstration of the existence of disintegrations and a reasoned illustration on calculation methods can be found in Fibonacci's Liber Abaci. Several algorithms based on ascending and descending continued fractions are illustrated, emphasizing their historical presence in Hellenistic and medieval times. With this study it is possible to gradually understand the concept of approximation to a given rational number through its convergents, considering the graphic representation on a Cartesian plane of a rational number using Ford circles, or representing a rational number a/b as a vector (b, a) on a grid plane, taking intoconsideration the results of Farey and Pick. Graphically seeing how to “get” to a given rational number in smallsteps, obtaining it as the sum of unit fractions, allows us to understand the concept of convergence at a perceptual level, in continuity with what has been affirmed by neuroscientific studies.ì