Radioactive decay is the cleanest example of a memoryless process in nature. An atom that has survived a million years is exactly as likely to decay in the next second as one created a moment ago. There is no ageing, no accumulated fatigue, nothing that makes decay more imminent.
N = N₀ × e^(−λt), where λ = ln(2) ÷ half-life
For carbon-14, with a half-life of 5,730 years, the decay calculator gives a decay constant of 0.000121 per year. After 1,000 years, 88.6 per cent of the original material remains.
The average atom lives longer than the half-life
Half the atoms are gone by 5,730 years, so it is tempting to call that the typical lifetime. It is not the average. The mean lifetime is one divided by the decay constant, which is the half-life divided by the natural log of two.
mean lifetime = half-life ÷ ln(2) = 5,730 ÷ 0.6931 ≈ 8,267 years
The average is always about 44 per cent longer than the half-life, for every isotope, because the distribution has a long tail. Most atoms go early, but the ones that survive keep not ageing, and a small number last an extremely long time. Those stragglers pull the mean well past the median, which is what the half-life actually is.
Ten half-lives is the practical end
Two half-lives leaves 25 per cent, not nothing, which is the other common slip: halving twice is a quarter rather than zero. Ten half-lives leaves 0.098 per cent, or about one part in a thousand. For carbon-14 that is 57,300 years, and it is roughly where radiocarbon dating runs out, not because the physics changes but because the signal falls below what can be distinguished from contamination and background.
The same ten half-life rule is used for radioactive waste handling and for drug clearance in pharmacology, where the elimination half-life does the same work. In both cases it is a convention about when a remainder stops mattering rather than a point where the exponential stops.
The same curve, different subjects
Exponential decay is not a nuclear phenomenon. It is what happens whenever the rate of loss is proportional to the amount present, which covers drug elimination, capacitor discharge, atmospheric pressure with altitude and the cooling of a hot object toward room temperature.
For the chemistry version, where the question is how a reactant concentration falls over time rather than how a nucleus decays, the integrated rate law calculator handles zero, first and second order reactions, and only the first order case produces a constant half-life. That is worth knowing: a constant half-life is a property of first-order processes specifically, not a general feature of things that decline.
For the counting-rate side of the same isotope, there is radioactive activity, which converts between becquerels, curies and the number of atoms present.