A radioactive nucleus has no memory and no schedule. It does not age, it does not wear out, and one that has sat there for a billion years is exactly as likely to go in the next second as one made this morning. There is no mechanism that makes half of them decay in a half-life — and yet half of them do, every time, to as many decimal places as you care to measure.
Each square is one nucleus. On every tick, each surviving one is given the same small chance of going — independently, with no reference to the others and no memory of how long it has been sitting there. Press Run. The grid empties in a ragged, random way, and the count underneath traces a smooth curve that nobody drew.
Halve the sample and it still takes the same time for half of what remains to go. That is what makes an exponential an exponential, and it is why the quantity is quoted per isotope rather than per lump. The half-lives below span twenty-four orders of magnitude — from a fraction of a millisecond to longer than the universe has existed.
| isotope | half-life | emits | where you meet it |
|---|
Living things take up carbon-14 from the air while they are alive and stop when they die. From that moment the clock runs. Measure how much is left, invert the exponential, and you have the age — no assumptions beyond a steady starting proportion. It runs out at about ten half-lives, when there is too little left to measure against the background.
Alpha is a helium nucleus — enormous, slow, and stopped by paper or skin, which makes it almost harmless outside you and extremely dangerous inside. Beta is an electron, stopped by a few millimetres of aluminium. Gamma is light with no mass at all, and only thick lead or concrete cuts it down. The danger of a source depends far more on where it is than on how loudly it ticks.
The grid in step 1 runs a real per-atom trial each tick: every surviving nucleus gets an independent chance p = 1 − e−λΔt of decaying, and the count is whatever survives. Nothing anywhere multiplies by e−λt to produce the line; the exponential is what a large number of independent coin-flips looks like. Run it with sixteen atoms and the curve is a jagged staircase that misses the theory badly. Run it with two thousand and it lies on the theory. That is the whole of why radioactive dating works on a lump of rock and would be meaningless for a single atom.
Which also settles a common confusion: a half-life does not mean a nucleus is halfway through anything. Nuclei do not age. If one survives ten half-lives it is in exactly the state it started in, with exactly the same chance of going in the next instant — the mathematics of a process with no memory, the same shape as waiting for a bus that arrives at random.