A single radioactive nucleus is completely unpredictable. It might decay in the next second, or it might sit there, unchanged, for a thousand years. There is no ageing, no wearing out, no way to tell which nucleus goes next — every one is a roll of the dice. So how can a physicist look at a lump of uranium and say, with real confidence, exactly how radioactive it will be in a hundred years' time?

The trick is that a tiny sample still contains a staggering number of nuclei — a speck of dust holds billions upon billions. When you throw that many dice at once, the average stops being a guess and becomes a near-perfect rule. And the rule is beautifully simple: in a fixed span of time, half of the undecayed nuclei will decay. Wait the same time again, and half of what's left decays. That fixed span is the half-life, and it is the heartbeat of every radioactive material.

What "half-life" actually means

We measure how radioactive a sample is by its activity (or count rate on a detector) — the number of nuclei decaying each second, measured in becquerels (Bq), where 1\ \text{Bq} is one decay per second. Because activity depends on how many undecayed nuclei are left, when the number of nuclei halves, the activity halves too. So the half-life can be stated two equivalent ways.

The words "on average" matter. You cannot say which nuclei will decay, only that, across the whole crowd, close to half of them will. With billions of nuclei that "close to" becomes so tight that the halving is, for all practical purposes, exact.

The halving pattern: \left(\tfrac{1}{2}\right)^{n}

Here is the key idea. Each half-life doesn't remove a fixed amount — it removes a fixed fraction, namely half of whatever is currently left. Start with a full sample and count the half-lives:

After n half-lives the fraction of the original nuclei still undecayed is

And because the activity is proportional to the number of undecayed nuclei, the count rate follows the exact same law:

where A_0 is the starting activity and A_n is the activity after n half-lives. Notice the steps get smaller as you go: the drop from full to a half is huge, but the drop from \tfrac{1}{32} to \tfrac{1}{64} is tiny. That is why a decay curve swoops down steeply at first and then flattens into a long, shallow tail.

Watch it halve

The blue curve below shows the count rate of a source that starts at 64\ \text{Bq}. The faint dots sit one half-life apart, so their heights march down the halving ladder: 64 \to 32 \to 16 \to 8 \to 4 \to 2 \to 1. Drag the half-life slider to make the isotope decay quickly or slowly (the shape is always the same — only the timescale stretches), and drag the half-lives elapsed slider to walk the marker along the curve and read off how much is left.

Two things to notice. First, the curve never touches the time axis — each step only halves what's there, so a little always remains. Second, the equal horizontal gaps between the dots (each one half-life) produce ever-shrinking vertical drops. Equal time, halving height: that is the signature of radioactive decay.

Worked examples

Example 1 — how much is left? A sample of a radioactive isotope has an activity of 800\ \text{Bq}. Its half-life is 5 years. What is the activity after 15 years?

First count the half-lives: 15 \div 5 = 3 half-lives. Now halve three times:

Or straight from the formula: A_3 = 800 \times \left(\tfrac{1}{2}\right)^{3} = 800 \times \tfrac{1}{8} = 100\ \text{Bq}.

Example 2 — how many half-lives? A source drops from 2400\ \text{Bq} to 300\ \text{Bq}. How many half-lives have passed?

Keep halving until you reach the target and count the arrows:

That is 3 halvings, so 3 half-lives. Check with the fraction: 300 / 2400 = \tfrac{1}{8} = \left(\tfrac{1}{2}\right)^{3} — yes, three. (If the half-life were, say, 20 minutes, that would be 60 minutes of real time.)

Example 3 — a fraction, not a count. After 6 half-lives, what fraction of the original nuclei are still undecayed?

So even after six half-lives, about 1.6\% of the sample is still there, ticking away.

Reading a half-life off a graph

Often you're handed a decay curve and asked to find the half-life from it. The method is always the same: pick a starting count rate on the curve, find half of it, and read off how much time passed in between.

Suppose a graph starts at 80\ \text{counts/s} and you read that the count rate has fallen to 40\ \text{counts/s} after 6 hours. Halving from 80 to 40 took 6 hours, so the half-life is 6 hours.

You can check it anywhere on the curve, and this is the best tip for exams: the half-life is the same no matter where you start. Going from 40 down to 20 should also take 6 hours; so should 20 to 10. If two different halvings give the same time, you have read the graph correctly.

Long life, short life — and why it matters

Half-lives span an almost unbelievable range. Some isotopes have half-lives of a fraction of a second; others outlast the age of the Universe. That single number decides how an isotope is used — and how dangerous it is.

• Very long half-lives (thousands to billions of years) — like some of the isotopes in nuclear waste. They stay weakly radioactive for an enormous time, which is exactly why waste must be stored safely for so long: you cannot simply wait for it to fade.
• Medium half-lives — like carbon-14 (about 5700 years). Steady and predictable over human and archaeological timescales, which makes them perfect natural clocks for dating old things.
• Short half-lives (hours) — like the technetium-99m used as a medical tracer. A doctor wants an isotope that lights up a scan and then decays away quickly, so the patient isn't left radioactive for long. A short half-life is a feature, not a flaw.