Solve for any missing value in the half-life decay equation: initial amount, remaining amount, half-life, or elapsed time.
📖 Exponential Decay Formula (N(t) = N₀ × 0.5^(t / half-life))Fill in the fields on the left with your information, then press Calculate to see your result instantly. No sign-up required, and no data is sent anywhere — everything is calculated right in your browser.
The half-life calculator solves the "exponential decay" equation used in nuclear physics, chemistry, and archaeology, which is N(t) = N₀ × 0.5^(t/half-life), where N₀ is the initial quantity, N(t) is the quantity remaining after time t, and the half-life is the time needed for half of the current quantity to always decay. This calculator lets you calculate any one of the four variables making up the equation if the other three are known: simply leave the field you want to calculate empty. To calculate the "remaining quantity," the equation is applied directly; to calculate the "half-life" or "elapsed time," the equation is rearranged using the natural logarithm to isolate the required variable. This principle is used in dating fossils, calculating how long a given drug stays in the body, and determining the safety of radioactive materials after a certain period of time. Because the fraction remaining after each half-life is always exactly one-half, regardless of how much material was present to begin with, the decay curve looks identical in shape no matter the starting quantity.
Half-life describes a specific and slightly counterintuitive pattern of decline: rather than losing a fixed amount of material in each equal time interval, a decaying substance loses a fixed fraction — always exactly half — of whatever amount currently remains, no matter how much or how little that is at the time.
This has a striking mathematical consequence: exponential decay technically never reaches exactly zero. After one half-life, half the original amount remains; after two half-lives, a quarter remains; after ten half-lives, less than a tenth of a percent remains — vanishingly small, but never mathematically zero under the idealized model, even though in physical reality a finite number of atoms means the substance does eventually run out entirely once the count reaches a genuinely single remaining atom or molecule.
The half-life equation contains four variables — initial quantity, remaining quantity, elapsed time, and the half-life value itself — and knowing any three allows solving for the fourth. Calculating the remaining quantity from a known initial amount, elapsed time, and half-life is a direct substitution into the formula. Solving for elapsed time or for the half-life value itself instead requires isolating that variable using logarithms, since it appears as an exponent in the original equation — logarithms being precisely the tool designed to undo exponentiation, exactly as covered in the exponent and logarithm calculator elsewhere on this site.
Radiocarbon dating is probably the most widely recognized application: carbon-14, a radioactive isotope absorbed by living organisms throughout their life, begins decaying at a known, fixed half-life (about 5,730 years) the moment an organism dies and stops absorbing new carbon. Measuring how much carbon-14 remains in an ancient organic sample, compared to the amount expected in a living organism, allows scientists to estimate how long ago that organism died — the basis of radiocarbon dating used throughout archaeology and paleontology.
Pharmacology relies on the same mathematical model to describe how a drug's concentration in the bloodstream declines over time, informing dosing schedules and how long a medication remains detectable or effective in the body. Nuclear safety planning uses half-life calculations to estimate how long radioactive waste or contamination will remain hazardous, since a material's half-life directly determines how many years or centuries must pass before its radioactivity decays to a level considered safe.
It's the time required for exactly half of a decaying quantity to disappear, and this fixed duration stays the same no matter how much of the substance remains at any starting point.
Leave empty whichever of the four values (initial amount, remaining amount, half-life, or elapsed time) you want the calculator to solve for, and fill in the other three.
Yes, both must be expressed in the same time unit (both in years, both in days, etc.) for the calculation to be accurate.