half-life
How do you describe the speed of a decaying quantity in a way that anyone can grasp? Not with the abstract decay constant lambda, but with a vivid, concrete number: the time it takes for half of whatever you have to disappear. That is the half-life. After one half-life, half remains; after two, a quarter; after three, an eighth — the same fraction vanishes in each equal stretch of time.
For exponential decay y(t) = y0 e^(-lambda t), the half-life t_(1/2) is the time when y drops to y0/2. Setting e^(-lambda t) = 1/2 and taking logarithms gives lambda · t_(1/2) = ln 2, so t_(1/2) = (ln 2)/lambda ≈ 0.693/lambda. The striking feature is that this time does not depend on how much you started with: a kilogram of carbon-14 and a single gram both lose half their atoms in the same 5730 years. That constancy is the signature of exponential decay.
Half-life is the everyday currency of decay, used for radioactive isotopes, drug clearance in the body, and the discharge of a capacitor. A common error is to think two half-lives wipe the substance out — they leave a quarter, not nothing. Decay is geometric: a long enough wait makes the amount tiny, but it never reaches exactly zero in finite time.
Carbon-14 has a half-life of about 5730 years, so its decay constant is lambda = (ln 2)/5730 ≈ 1.21 × 10^(-4) per year. After 11460 years (two half-lives), a sample retains 1/4 of its original carbon-14.
Same fraction lost per equal interval: 1 → 1/2 → 1/4 → 1/8.
Half-life is meaningful only for exponential decay. For a process with a different rate law (say, decay proportional to y^2), 'the time to halve' depends on the starting amount and is not a fixed constant.