Yet another possible explanation of Egyptian fractions: motivated by fairness

Ancient Egyptians represented fractions as sums of inverses of natural numbers, and they made sure that all these natural numbers are different. The representation as a sum of inverses makes some sense: it is known to lead to an optimal solution to the problem of dividing bread between workers, a problem often described in the Egyptian papyri. However, this does not explain why the corresponding natural numbers should be all different: some representations with the same natural number repeated several times lead to the same smallest number of cuts as the representations that the ancient Egyptians actually used. In this paper, we provide yet another possible explanation of Egyptian fractions – based on fairness; this idea explains also why all the natural numbers should be different. 1 Mystery of Egyptian Fractions: Reminder How we usually represent fractions. We are accustomed to representing fractions as ratios p q : • whether in the explicit form like 2 3 , • or in the implicit form like 0.25 which means 25 100 . The fact that we are actively using this representation seems to indicate that this representation is convenient for computations. So, not surprisingly, this is the way fractions have been represented in many cultures.