Archaeologists dating an ancient bone, doctors calculating how long a radioactive tracer stays active in a patient's body, and engineers estimating how long nuclear waste remains hazardous all rely on the same simple mathematical idea: half-life, the time it takes for exactly half of a decaying substance to disappear.
How to Use the Half-Life Calculator
Enter the initial amount of the substance, its half-life, and the elapsed time — all three in whatever consistent units you're working with (this calculator labels the time fields "years" for the built-in carbon-14 example, but the math works identically in seconds, days, or any other unit, as long as half-life and elapsed time use the same one). The calculator returns the remaining amount, what percentage of the original is left, and how many half-lives have elapsed.
What This Calculator Doesn't Cover
This calculator models a single decay process with one fixed half-life, which describes the overwhelming majority of practical radioactive decay and other exponential decay scenarios (drug elimination from the body, capacitor discharge, and cooling all follow the same mathematical shape). It doesn't model decay chains, where one radioactive substance decays into another radioactive substance with its own separate half-life — those situations require tracking multiple linked exponential decays rather than the single equation used here.
The Half-Life Formula
N(t) = N₀ × (1/2)^(t / half-life)
The remaining amount N(t) equals the initial amount N₀ times one-half raised to the power of elapsed time divided by half-life. The calculator's default example uses carbon-14, which has a half-life of 5,730 years, starting with 100 grams and checking in after 2,000 years: the exponent works out to 2,000 ÷ 5,730, or about 0.349 half-lives elapsed, and 0.5^0.349 is about 0.785 — meaning roughly 78.5 grams, or 78.5%, of the original carbon-14 remains.
Medical and Industrial Half-Lives Are Much Shorter
Not every half-life spans thousands of years. Technetium-99m, a radioactive tracer used in medical imaging, has a half-life of only about 6 hours — short enough that it's mostly gone from a patient's body within a day or two, minimizing radiation exposure while still lasting long enough to complete a scan. Iodine-131, used in some thyroid treatments, has an 8-day half-life. Plugging these much shorter half-lives into the same formula, with elapsed time measured in hours or days instead of years, shows exactly how a medical team plans dosing and scheduling around a tracer's decay.
Why It's Called "Half"-Life, Not "All"-Life
A decaying substance never mathematically reaches zero — after one half-life, half remains; after two half-lives, a quarter remains; after three, an eighth; and so on, forever approaching zero without ever quite arriving, at least in the idealized math (in reality, once you're down to a handful of atoms, decay becomes a matter of discrete random events rather than a smooth curve). This is why radioactive materials are described by half-life rather than a fixed "lifespan" — there's no single moment when a decaying substance is definitively "gone."
Carbon Dating: Half-Life in Action
Carbon-14 dating is the most famous practical use of this formula. Living organisms constantly exchange carbon with the atmosphere, keeping a steady ratio of radioactive carbon-14 to stable carbon-12. Once an organism dies, that exchange stops, and the carbon-14 already present starts decaying on its known 5,730-year half-life while the carbon-12 stays constant. Measuring how much carbon-14 remains relative to carbon-12, and comparing it to what a living organism would have, tells archaeologists how long ago death occurred — this calculator's remaining-amount output is exactly the number that measurement is compared against.
