What half-life actually means
Half-life is the time it takes for half of a given amount of a radioactive substance to decay, whatever amount you started with and whatever amount is left. Ten grams of a substance with a five-year half-life becomes five grams after five years and two and a half grams after ten, exactly as one hundred grams becomes fifty then twenty-five over the same two steps. The proportion halved is fixed; the absolute amount is not, which is exactly why this calculator asks for a starting amount and a half-life separately rather than baking one into the other.
What half-life does not mean is that every atom in a sample somehow survives for precisely one half-life and then decays on schedule. Radioactive decay is genuinely random at the level of a single atom: there is no way to predict which atom in a lump of carbon-14 will decay next, or when, only the probability that any given atom decays within a given stretch of time. Half-life is a statement about huge numbers of atoms behaving predictably in aggregate, the same way a fair coin flipped a million times lands close to half heads even though no individual flip is predictable. With a sample containing anything like a mole's worth of atoms, on the order of 1023, that statistical regularity is reliable enough that the decay curve looks perfectly smooth, which is why the formula below works as cleanly as it does.
The decay law, and its three rearrangements
Every calculation on this page comes from one equation:
N₀ is the amount you start with, T is the half-life, t is however much time has passed (in the same unit as T), and N(t) is what's left. Set t equal to T and the exponent is 1, halving N₀ once; set t equal to 2T and the exponent is 2, halving it again to a quarter. Because the same equation holds for any of its four quantities once the other three are known, it rearranges into three more useful questions, each reached by ordinary algebra, taking a logarithm to pull the unknown out of the exponent when it needs to:
Those three rearrangements are exactly this calculator's other three solve-for options: work out what you must have started with, how long decay has been running, or how long the substance's half-life itself must be, given the other three figures.
Two related figures describe the same decay in different units, and this page shows both alongside every answer. The decay constant, λ, is the fraction of what's left that decays per unit time; a substance with a five-year half-life has a bigger λ than one with a five-hundred-year half-life, because it is disappearing faster. It's tied to the half-life by λ = ln2 ÷ T, where ln2 (the natural log of 2, about 0.6931) turns up because halving is the same event as multiplying by e raised to the power −λt, and setting that equal to one half and solving for t gives back T = ln2 ÷ λ.
Mean lifetime, τ, is the average time a single atom survives before decaying, and it's simply the reciprocal of the decay constant: τ = 1 ÷ λ = T ÷ ln2. Because ln2 is a little less than 1, τ is always somewhat longer than T, by a factor of about 1.4427. Carbon-14's 5,730-year half-life carries a mean lifetime of roughly 8,267 years: most individual atoms don't decay right at the half-life mark, some go much sooner and some much later, and τ is the average of all of those individual survival times, not T.
Reading the chart and the two tables
The decay curve plots N(t) smoothly across roughly six half-lives, with a dashed line stepping down by exactly half at every half-life mark laid over the top of it, so the same decay is visible two ways at once: as a continuous curve and as the discrete halvings that give the process its name. Your own point, wherever it falls among the four fields above, is marked with a dotted vertical line.
Underneath that, a table runs the halvings out to ten steps, from one half remaining down to roughly a thousandth. It rewards sitting with for a moment, because the fall is faster than intuition usually expects: half is gone after one half-life, three-quarters after two, seven-eighths after three, and by the tenth half-life only about 0.098% is left, close enough to nothing for most practical purposes even though the equation itself never technically reaches exactly zero.
Carbon dating, worked through
Radiocarbon dating is the half-life formula run in the direction that solves for elapsed time. A living organism keeps a roughly constant ratio of carbon-14 to ordinary carbon-12 while it's alive, replenished by whatever it eats or breathes; once it dies, that intake stops and the carbon-14 already inside it decays away on its own schedule with nothing topping it back up. Measuring how much carbon-14 is left compared with what a living sample would show, then, tells you how long ago that stopping point was.
Say a bone sample tests at 12.5% of the carbon-14 a living bone would carry. 12.5% is one eighth, and each half-life halves whatever's left, so getting from a full amount down to one eighth takes exactly three halvings: one eighth is a half of a half of a half. Running this page's own solve-for-time mode with N₀ at the living-sample level, N at one eighth of it, and T at carbon-14's accepted 5,730-year half-life gives t = 5,730 × log₂(8) = 5,730 × 3 = 17,190 years. It's the same three-half-lives arithmetic either way, just carried out with the formula instead of counted on fingers.
Medical isotopes: why short is the point
Not every use of half-life wants a long one. Technetium-99m is injected into patients for diagnostic imaging precisely because its half-life is short, about 6.01 hours, so it delivers a clear scan and then largely clears the body within a day or two, keeping the radiation dose to the patient low. It is reported to be involved in the large majority of nuclear medicine procedures carried out worldwide, standing in for a whole category of otherwise much longer-lived tracers that would linger in the body for no diagnostic benefit. The same short half-life that makes it clinically convenient also makes it awkward to stock: it has to be generated close to where it's used or delivered fresh from a nearby source, since a batch made even a few days in advance would have decayed away to a small fraction of its starting activity before it could be used.
Biological half-life: caffeine as the everyday example
Half-life isn't only a nuclear physics idea. Plenty of drugs and other compounds leave the body in a way well approximated by the same halving pattern, usually called a biological or elimination half-life, even though nothing is radioactive about it: the liver, the kidneys, or both simply clear a roughly constant fraction of what's left per unit time, the same shape of decline as the nuclear version.
Caffeine is the everyday example most people already have some feel for. In a healthy adult its half-life is usually put at around five hours, though the honest answer is that it varies a good deal from person to person, roughly one and a half to nine or so hours, mostly down to differences in a liver enzyme called CYP1A2. It runs markedly slower in pregnancy, particularly by the third trimester, and faster in smokers, whose livers metabolise it noticeably quicker than non-smokers'. None of that changes the shape of the curve, only how quickly it plays out, which is exactly what changing the half-life figure does to every other result on this page.
Exponential decay well beyond physics
The same halving mathematics turns up anywhere a quantity shrinks by a constant proportion per step rather than by a constant amount, from a hot drink cooling towards room temperature to a wireless signal fading with distance, which is one reason the solve-for options on this page are written in the general terms N₀, N, t and T rather than tied to any one substance.
Questions people ask
Does half-life mean exactly half the atoms decay at that instant?
Statistically, yes, for any sample large enough to matter: after one half-life, close to exactly half the atoms present at the start have decayed. For a single atom, or a tiny handful, there is no "instant" it happens at, only a probability; half-life describes the average behaviour of enormous numbers of atoms, not a countdown timer on any individual one.
Why does the remaining amount never reach exactly zero?
Each half-life only ever removes half of whatever is currently left, so mathematically there is always something remaining, however small. In practice, after around ten half-lives the leftover amount, roughly a thousandth of the original, is small enough to be treated as gone for almost any real purpose, even though the equation keeps producing a smaller, nonzero number forever.
What's the difference between half-life and mean lifetime?
Half-life is the time for half of a sample to decay; mean lifetime is the average survival time of a single atom within that sample. They describe the same decay but are not the same number: mean lifetime is always about 44% longer than half-life, since τ = T ÷ ln2 and ln2 is a little under 1.
Can I use any time unit I like?
Yes, as long as the elapsed time and the half-life are both in the same one. The unit selector on this page only changes the label shown next to each figure; it does not convert a number already typed in, so switch it before entering your figures rather than after.
Why do some isotopes have half-lives of seconds and others billions of years?
Half-life comes from how unstable a particular atomic nucleus is, which depends on its exact mix of protons and neutrons; there's no single reason it should land on any particular timescale. Some arrangements are barely stable and shed energy almost immediately, while others, such as uranium-238, sit only fractionally off a stable configuration and take billions of years on average to tip over.
Is carbon dating exact?
It's a well-tested estimate rather than an exact reading. Atmospheric carbon-14 levels have varied a little over history, measurement always carries some uncertainty, and contamination can distort a sample, so real dates are usually given with a margin of error and cross-checked with calibration curves built from tree rings and other independently dated records, rather than trusted as a single bare number the way this simplified calculator presents one.
Isotope half-lives shown here are typical published figures gathered from nuclear data references; real measurements carry their own uncertainty and are periodically refined as measurement techniques improve, so treat them as a reliable guide rather than a certified constant.