half-life

When something decays exponentially, there is no single moment when it all disappears — it just keeps halving. The half-life is the steady time it takes for the quantity to fall to half of whatever it currently is. After one half-life, half remains; after two, a quarter; after three, an eighth; and so on, regardless of how much you started with.

If a quantity has half-life T, you can write it as A(t) = A_0 · (1/2)^(t / T), where A_0 is the starting amount and t is elapsed time. Equivalently, in natural form, A(t) = A_0 · e^(-kt) with the decay constant k = ln(2) / T. The two forms are the same model written with different bases.

Half-life is most famous in radioactive dating and in pharmacology, where it sets how long a drug stays active. A key intuition: equal time spans always multiply the amount by the same factor, so the curve never reaches zero — it only ever gets half as close. To find the time for some other fraction to remain, solve with a logarithm.