Mathematics
Exponential Decay
Quick fact
In exponential decay, the time it takes for a quantity to halve is constant, no matter when you start measuring—this is the 'half-life', a concept used to date ancient artifacts.
Why this is interesting
You have a cup of hot coffee on a cold morning. It cools quickly at first, then slower. Why doesn't it just cool at a constant rate?
Read the full explanation
Understanding Exponential Decay
Imagine a savings account that loses half its money every year. Start with $100. After one year: $50. After two: $25. After three: $12.50. The amount drops by half each time, but the dollars lost per year shrinks. That's exponential decay. The key idea is that the change depends on the current amount—so it's 'self-slowing'. In math, we write N(t) = N0 e^(-λt), where N0 is the starting amount, λ is the decay constant, and t is time. The curve starts steep and then flattens, asymptotically approaching zero but never quite reaching it in finite time.
A deeper explanation
Exponential decay arises when a process has a constant probability of happening per unit time. For example, in radioactive decay, each atom has the same chance of decaying each second, independent of age. That results in a fixed fraction decaying per unit time, which mathematically leads to the exponential function. The rate of change dN/dt = -λN. This differential equation shows the rate is proportional to the current amount. Why does it matter? It models countless natural phenomena: cooling (Newton's law), drug elimination from the body (first-order kinetics), absorption of light through a medium (Beer-Lambert law), and even the fading of a memory trace. Understanding exponential decay lets you predict how long a process will take to reach a certain level (using half-life) and grasp why some things diminish rapidly then seem to linger.