Between Zero and One

Proper fractions populate the number line between 0 and 1 — a region that turns out to be astonishingly crowded. No matter how close two proper fractions sit, you can always squeeze another one between them. Take 1/3 and 1/2: their average is 5/12, which slides between them. Repeat the averaging trick forever; the supply never runs out.

This denseness is why probability, statistics, and machine-learning scores all live in the 0-to-1 interval. A probability of 0.73 is a proper fraction. A batting average of 0.285 is a proper fraction. The unit interval became mathematics' favorite playground because it holds every scale of part-of-a-whole you will ever need.