Old-fashioned fractions

When discussing fractions, we often begin by evoking the idea of fair sharing. Yet, even when sticking to their common definition, this approach does not do justice to the richness and variety of uses one can make of them, ranging from the definition of our musical notes to the handling of gears. For its part, Thales' theorem makes geometry equally fond of fractions. The so-called Egyptian fractions, that is to say those with numerator 1, seem to have been the first to have been considered. Their name alone indicates that their age is counted in millennia, and we may explain this by the fact that in a sense a third, a quarter, a fifth, and more generally an nth are the prime fractions, from which all others are derived by simple multiplication by an integer.