exponential decay

A hot cup of coffee cools fast at first, then more and more slowly as it nears room temperature. A radioactive sample loses the same fraction of its atoms in every fixed stretch of time. These are examples of exponential decay: a quantity shrinking by a constant percentage (or factor) per unit of time. Because you always remove a slice of a shrinking total, the actual drop gets smaller and smaller.

We model decay with f(t) = a · b^t, where a is the starting amount and the factor b lies strictly between 0 and 1. If something loses 20% each year, then 80% remains, so b = 0.8 and the amount is multiplied by 0.8 every year. The graph slides downward, leveling off toward zero without ever reaching it — zero is a horizontal asymptote.

A subtle but important point: decaying by a constant percentage never gets you to exactly zero in finite time, only ever closer. This is why we measure the speed of decay with a half-life (the time to lose half) rather than a time-to-empty, which does not exist for true exponential decay.