How half-life calculator works
Half-life is the time a decaying population takes to halve, and the mathematics is identical whether the population is carbon-14 nuclei, caffeine molecules in your blood or residents of a shrinking town. After one half-life half remains; after two, a quarter; after three, an eighth. The closed form is N(t) = N₀ × 2^(−t/h), where N₀ is the starting amount, h the half-life and t the elapsed time — measured in the same units for h and t, whatever they are.
The halving ladder is worth internalising because it is brutally forgiving of imprecision: 100 → 50 → 25 → 12.5 → 6.25 → 3.125 → 1.5625 → 0.78 → 0.39 → 0.20 → 0.10 percent over ten half-lives. Two disciplines share the same curve through two constants: the decay constant λ = ln 2 / h, the per-unit-time decay rate in the exponential form N = N₀e^(−λt), and the mean lifetime τ = 1/λ = h / ln 2 ≈ 1.44 × h, how long an average nucleus (or molecule) lasts before decaying. Carbon-14's half-life of 5730 years gives λ ≈ 0.000121 per year and τ ≈ 8267 years.
The calculator also runs the curve backwards: give it a target amount and it returns the time to reach it, t = h × log₂(N₀/N). That is radiocarbon dating in one line — a sample at 25% of its original carbon-14 is two half-lives old, about 11,460 years. One caution the ladder already suggests: the remaining amount approaches zero but never reaches it, so a target of zero has no finite answer; ten half-lives, at 0.1% remaining, is the working convention for "effectively gone". For the caffeine case specifically — with doses, times of day and a personal threshold — the Caffeine Half-Life Calculator packages the same math for daily use.