Likewise, with a pie instead of a second, geometry is not content to cut off some small portion at random: it draws a regular polygon with q sides on the pie, divides it into equal slices and takes p of them.
Dividing, breaking up, cutting, amputating… the many situations that involve fractioning are closely bound up with the idea of fracturing. The term fractal itself was coined to evoke infinitely many breaks. This historical association of fractions with fracture comes through clearly in the expression broken numbers, a now obsolete term once used for fractions of a certain kind.
Wherever a part of a whole must be quantified, fractions are a fundamental tool. Yet they soon move beyond their original role, combining the elegance of integers with the ability to perform division. Once they become greater than 1, they are no longer confined to representing the division of a whole: they become a privileged vantage point from which to observe a wide variety of phenomena, chief among them proportionality, accompanied by its emblematic rule of three.
**Besides allowing us to observe and explore,
fractions provide a springboard for analysis.**
Far from being merely a world unto itself, the world of fractions is therefore also a springboard. Analysis uses fractions as a lever to reach and study other numbers, taking them beyond the algebraic framework to which their original definition confines them. Continued fractions use the Euclidean algorithm to explore the properties of numbers in depth. Decimal notation provides a standard representation of every real number as a sum of fractions whose denominators are powers of ten. Finally, the notion of a series extends this idea to sums of arbitrary fractions, yielding formulas regarded as some of the most beautiful in all of mathematics.
